A monotone iterative technique combined to finite element method for solving reaction-diffusion problems pertaining to non-integer derivative

Abstract

This paper focuses on some mathematical and numerical aspects of reaction-diffusion problems pertaining to non-integer time derivatives using the well-known method of lower and upper solutions combined with the monotone iterative technique. First, we study the existence and uniqueness of weak solutions of the proposed models, then we prove some comparison results. Besides, linear finite element spaces on triangles are used to discretize the problem in space, whereas the generalized backward-Euler method is adopted to approximate the time non-integer derivative. Furthermore, the idea of this method is to construct two sequences of solutions of a linear initial value problem which are easier to compute and converge to the solution of the nonlinear problem. We show numerically through two examples that this convergence requires only few iterations. Some well-known examples with exact solutions and numerical results based on the finite element method in 2D are provided to validate the theoretical results. As a result, we confirm that the proposed method is efficient and easy to use to overcome the convergence and stability difficulties.

1 Introduction

Partial differential equations (PDEs) with non-integer order derivatives have gained much more attention in the last years. Such derivatives have been applied recently to many problems in several fields, such as physics [1,2,3,4,5,6], engineering mechanics [7, 8], biology and ecology [9,10,11,12,13], epidemiology [14,15,16,17,18,19], stochastic process [20, 21] image processing [22,23,24], signal processing [25], medical imaging [26] chaos theory [27,28,29] and others.

In the present investigation, we consider the following initial boundary value problem with non-integer order time derivative

$$\begin{aligned} {\left\{ \begin{array}{ll} \dfrac{\partial ^{\beta }u}{\partial t^{\beta }}(t,x)+{\mathcal {P}} u(t,x)=f(t,x, u(t,x))&{} \text{ in } Q, \\ u(t,x)=g(t,x) &{} \text{ on } \Gamma , \\ u(0,x)=u_0(x) &{} \text{ in } \Omega ,\\ \end{array}\right. } \end{aligned}$$

(1.1)

where $\Omega \subset {\mathbb {R}}^{N}$ be a bounded domain with Lipschitz boundary $\partial \Omega , Q=$ $(0, T)\times \Omega$ and $\Gamma = (0, T)\times \partial \Omega , T>0.$ ${\mathcal {P}}$ denotes the second-order partial differential operator in the divergence form

$$\begin{aligned} {\mathcal {P}} u(t,x)=-\sum _{i, j=1}^{N} \frac{\partial }{\partial x_{i}}\left( a_{i j}(t,x) \frac{\partial u}{\partial x_{j}}\right) +b(t,x) u, \end{aligned}$$

with coefficients $a_{i j}, c \in L^{\infty }(Q)$ and

$$\begin{aligned} \sum _{i, j}^{N} a_{i j}(t,x) \xi _{i} \xi _{j} \ge \alpha |\xi |^{2},\quad \text{ a.e. } \text{ in } Q, \alpha >0, \end{aligned}$$

(1.2)

The main idea is to study the situation when the involved nonlinear function f admits a splitting of a difference of two monotone functions or equivalently, of a sum of two functions $f_1$ and $f_2$, i.e.

$$\begin{aligned} f(t,x,u)=f_1(t,x,u)+f_2(t,x,u), \end{aligned}$$

which one of them is increasing and the other one is decreasing. The functions $f_1, f_2: Q \times {\mathbb {R}} \rightarrow {\mathbb {R}}$ are of Caratheodory type, that is, $f_1$ and $f_2$ are measurable in $(t,x) \in Q$ for each $u \in R$ and continuous in u for a.e. $(t,x) \in$ Q.

Generally, in applied mathematics it is difficult (if not impossible) to obtain exact solutions for most nonlinear partial differential equations (PDEs) with non-integer derivative. It is, therefore, an urgent necessity to look for efficient numerical methods to solve these types of PDEs. Indeed, several numerical methods have been proposed in the literature to solve such equations, the three best known are finite difference method (FDM), finite element method (FEM) and finite volume method (FVM). In [30], the authors have suggested a one and two dimensions finite difference method to numerically approximate the solutions of reaction-diffusion equations with Caputo time-fractional derivatives. Non-standard finite difference method and Spectral collocation method were used in [31] to obtain numerical solutions of fractional-order advection-dispersion problem. Reference [32] used the Legendre spectral finite difference method to solve non-linear reaction-diffusion equation and non-linear Burger’s-Huxleye quation with non-integer derivative in the ABC sense. In [33], the authors applied the finite difference method for solving parabolic equation with time Caputo derivative and fractional Laplacian. Error estimates for a semi-discrete finite element method for fractional order parabolic equations was obtained in [34]. The adaptive finite element method for fractional differential equations using hierarchical matrices was studied in [35]. In [36], the study used the finite element method to investigate the solutions of a space-time fractional Fokker-Planck equation. A Petrov–Galerkin finite element method was used by Ref. [37] to solve the convection-diffusion equations with non-integer derivatives. The authors of [38] presented the time discontinuous finite element method for numerical solution for the diffusion-wave equation with fractional derivative. In the Ref. [39], it was developed a finite Element Method for the numerical resolution of Space-Time Fractional Fokker-Planck Equation. Finally, [2] applied a finite element scheme with Newton’s method for solving the time-fractional nonlinear diffusion equation using fractional Crank-Nicolson scheme. Some other outstanding studies PDEs with non-integer derivatives adopting finite element method have been made in [40,41,42,43,44,45,46,47, 47] and related references.

The best computational algorithms for a numerical solution are the numerically stable and convergent schemes, for problem (1.1) this is guaranteed easily if the second member f does not depend on the unknown function u. For nonlinear systems, most of works cited above and in related references in the literature use explicit approximation for the nonlinear term f in (1.1). Such formulation gives a linear finite difference approximation at each time step and is, therefore, more suitable for digital processing. Unfortunately, this approach can also lead to incorrect or misleading information about the solution, see for example [48, 49]. The problem of stability can be solved by using the Newton method for the implicit scheme, several works in this direction were carried out, see for example [50]. Furthermore, Newton’s method has two drawbacks, the first is for solving some real multi-physical problems is the need for evaluating the entries of the Jacobian matrix. The second is that the Newton method is locally convergent.

We propose in this paper an iterative scheme to prove the existence and uniqueness of weak solutions of problem (1.1) as well as for the computation of numerical solutions. Furthermore, the iterative processes given in this article give two monotone sequences and are not more complicated than the processes used for the corresponding linear parabolic system when the spatial dimension is greater than one. This type of monotone iteration process was used firstly for elliptical equations in [51, 52], and for parabolic equations in [53]. Similar methods have been used in [54,55,56,57,58] for systems of parabolic and elliptical equations, but are mainly limited to problems with classical derivatives and by the finite difference method for numerical solutions. In this paper, we develop a monotone iterative method with upper and lower solutions for a parabolic problems with non-integer derivative, moreover we present a numerical scheme that combines this method to the finite element method. We prove firstly the existence and uniqueness of the weak solutions for the Nonlinear reaction-diffusion equation with non integer derivative. We show that the monotone sequences, which are solutions of the linear equation converge to the minimal and maximal solutions of the nonlinear equation, these comparison results are used to establish the last result. In addition, to compute the numerical solution, the linear $P_1$ finite element spaces on triangles are used for the space discretization, where the generalized backward-Euler method is used to approximate the time non-integer derivative.

The layout of this paper is as follow: Sect. 2 provides the definitions and preliminary results to be used in the article. In Sect. 3, main results are stated and proved: comparison results are obtained and the existence of a weak solution for the nonlinear equations (1.1) are proved. Some techniques to construct the lower and upper solutions are given in Sect. 4, where some application examples of the results are presented in Sect. 5. The linear finite element method on triangle is used in Sect. 6 to compute the numerical solution. In Sect. 7 some numerical simulations are discussed to confirm theoretical results. Finally, conclusion and future works are given in Sect. 8.

2 Definitions and preliminaries

In [59], the authors gave a new definition of no-integer derivative which is a natural extension to the usual first derivative as follows:

Definition 2.1

([59]) Given a function $f :[0, \infty ) \longrightarrow {\mathbb {R}}$ . Then for all $t>0, \quad \beta \in (0,1],$ let

$$\begin{aligned} {\mathcal {D}}^{\beta }_t(f)(t)=\lim _{\varepsilon \rightarrow 0} \frac{f\left( t+\varepsilon t^{1-\beta }\right) -f(t)}{\varepsilon } , \end{aligned}$$

${\mathcal {D}}^{\beta }$ is called the Conformable derivative of f of order $\beta .$ If f is $\beta$-differentiable in some $(0, b), b>0,$ and $\lim _{t \rightarrow 0^{+}} {\mathcal {D}}^{\beta }(f)(t)$ exists, then let

$$\begin{aligned} {\mathcal {D}}^{\beta }(f)(0)=\lim _{t \rightarrow 0^{+}} {\mathcal {D}}^{\beta }(f)(t) . \end{aligned}$$

Now we shall present some properties of this new derivative, more properties of this type of derivation are given in [59,60,61,62,63].

Definition 2.2

(Conformable Integral [59]) Let $a \ge 0$ and $t \ge a$ . Also, let f be a function defined on (a, t] and $\beta \in {\mathbb {R}} .$ Then, the $\beta -$Conformable integral of f is defined by,

$$\begin{aligned} {\mathcal {I}}_{a}^{\beta }(f)(t):=\int _{a}^{t} f(x) d _{\beta }x=\int _{a}^{t} x^{\beta -1} f(x) d x , \end{aligned}$$

if the Riemann improper integral exists. When $a=0$ we write ${\mathcal {I}}_{0}^{\beta }(f)(t)={\mathcal {I}}^{\beta }(f)(t)$

$f : [a,b]\subset {\mathbb {R}} \rightarrow {\mathbb {R}}$ is $\beta$-integrable on [a, b] if and only if $t^{\beta -1} f$ is integrable on [a, b].

Lemma 2.3

( [60]) Assume that $f :[a, \infty ) \rightarrow {\mathbb {R}}$ is continuous and $0< \beta \le 1.$ Then, for all $t>a$ we have

$$\begin{aligned} {\mathcal {D}}^{\beta } {\mathcal {I}}_a ^{\beta } \,f(t) = f(t). \end{aligned}$$

Lemma 2.4

( [60]) Let $f :(a,\infty ) \rightarrow {\mathbb {R}}$ be differentiable and $0<\beta \le 1 .$ Then, for all $t>a$ we have

$$\begin{aligned} {\mathcal {I}}_a^{\beta } {\mathcal {D}}^{\beta }(f)(t)=f(t)-f(a) . \end{aligned}$$

In the rest of this section, we assume $a, b \in {\mathbb {R}}, 0<a< b$.

Definition 2.5

([61]) Let $p\in [1,+\infty ]$ and let $f : [a,b]\subset {\mathbb {R}} \rightarrow \overline{{\mathbb {R}}}$ be a measurable function. Say that f belongs to $L_{\beta }^{p}([a,b])$ provided that either

$$\begin{aligned} \int _{a}^{b}|f(t)|^{p} d_{\beta } t<+\infty \quad \text { if } 1\le p<+\infty , \end{aligned}$$

or there exists a constant $C \in {\mathbb {R}}$ such that

$$\begin{aligned} |f| \le C \quad \text{ a.e. } \text{ on } [a,b] \text{ if } p=+\infty . \end{aligned}$$

Theorem 2.6

([61]) Let $p \in [1,+\infty ]$. Then the set $L_{\beta }^{p}\left( [a, b]\right)$ is a Banach space endowed with the norm defined for $f \in L_{\beta }^{p}\left( [a, b]\right)$ as

$\Vert f\Vert _{L_{\beta }^{p}\left( [a, b]\right) }=\left\{ \begin{array}{l}{\left( \int _{a}^{b}|f(t)|^{p} d_{\beta } t\right) ^{1 / p}} \quad \quad \quad \text { if }1\le p<+\infty ,\\ { \text{ inf } \left\{ C \in {\mathbb {R}} :|f| \le C \text { a.e. on }[a, b]\right\} } \quad \text { if }p=\infty . \end{array}\right.$

Moreover, $L_{\beta }^{2}\left( [a, b]\right)$ is a Hilbert space with the inner product given for every $(f, g) \in L_{\beta }^{2}\left( [a, b]\right) \times L_{\beta }^{2}\left( [a, b]\right)$ by

$$\begin{aligned} \langle f, g\rangle _{L_{\beta }^{2}\left( [a, b]\right) }=\int _{[a, b]} f(t) g(t) d_{\beta } t. \end{aligned}$$

The integration by parts theorem and the Conformable version of granwal’s inequality and other properties of this derivative can be found in [59, 60, 64]. Before the statement of the properties, we denote

$$\begin{aligned} C_{0}\left( [a, b]\right) =\{f :[a, b] \rightarrow {\mathbb {R}}, f \text{ is } \text{ continuous } \text{ on } [a, b] \text { with compact support in} [a, b] \}. \end{aligned}$$

For $p\in [1,+\infty [$ we set the space

$$\begin{aligned} L_{\beta }^{p}\left( [a, b], E\right) =\left\{ u :[a, b] \rightarrow E : \int _{[a, b]}||f(t)||_E^{p} d_{\beta } t<+\infty \right\} . \end{aligned}$$

Let $H^{1}(\Omega )$ denote the usual Sobolev space of square integrable functions and let $\left( H^{1}(\Omega )\right) ^{'}$ denote its dual space. Then by identifying $L^{2}(\Omega )$ with its dual space, $H^{1}(\Omega ) \subset L^{2}(\Omega ) \subset \left( H^{1}(\Omega )\right) ^{'}$ forms an evolution triple with all the embedding being continuous, dense and compact.

We set $V=L_{\beta }^{2}\left( 0, T ; H^{1}(\Omega )\right) ,$ denote its dual space by $V^{'}=L_{\beta }^{2}(0, T ;(H^{1}(\Omega ))^{'}),$ and define a function space W by

$$\begin{aligned} W=\left\{ w \in V | {\mathcal {D}}_t^{\beta }u \in V^{'}\right\} , \end{aligned}$$

where the conformable derivative ${\mathcal {D}}_t^{\beta }$ is understood in the sense of distributions,

Lemma 2.7

( [64]) Let V, H,  and $V^{\prime }$ be Hilbert spaces such that $V \subset H \equiv H^{\prime } \subset V^{\prime },$ where $V^{\prime }$ is the dual of V and the injections are continuous. Suppose that ${\mathbf {u}} \in L^{2}_{\beta }([0, T] ; V)$ and ${\mathcal {D}}^{\beta }_t{\mathbf {u}} \in L^{2}_{\beta }\left( [0, T] ; V^{\prime }\right) .$ Then u is equal a.e. to a continuous function from [0, T] into H,  and the following equality holds in the distribution sense on (0, T)

$$\begin{aligned} {\mathcal {D}}^{\beta }_t(|{\mathbf {u}}|_{H}^{2})=2\langle {\mathcal {D}}^{\beta }_t {\mathbf {u}}, {\mathbf {u}}\rangle _{V^{\prime }, V}. \end{aligned}$$

(2.1)

Moreover, the embedding $W \subset C\left( [0, T] ; H\right)$ is continuous.

Remark 2.8

As a consequence of the previous identifications, the scalar product in H of $f\in H$ and $v\in V$ is the same as the scalar product of f and v in the duality between $V^{'}$ and V:

$$\begin{aligned} \langle f,v \rangle _{V^{\prime }, V} = (f,v)_{H},\quad \forall f\in H , \forall v\in V. \end{aligned}$$

(2.2)

3 Main results

Let $H_{0}^{1}(\Omega )$ be the subspace of $H^{1}(\Omega )$ whose elements have generalized homogeneous boundary values, and denote by $H^{-1}(\Omega )$ its dual space. One can consult [65, 66] for general properties of Sobolev spaces. Then obviously $H_{0}^{1}(\Omega ) \subset L^{2}(\Omega ) \subset H^{-1}(\Omega )$ forms also an evolution triple, and all statements made above remain true also in this situation when setting $V_{0}=L_{\beta }^{2}\left( 0, T ; H_{0}^{1}(\Omega )\right)$ $V_{0}^{'}=L_{\beta }^{2}\left( 0, T ; H^{-1}(\Omega )\right)$ and define

$$\begin{aligned} W_{0}=\left\{ w \in V_{0} | \quad {\mathcal {D}}_t^{\beta }w \in V_{0}^{'}\right\} . \end{aligned}$$

where ${\mathcal {D}}^{\beta }_t$ is the conformable derivative, is understood in the sense of vector-valued distributions and characterized by

$$\begin{aligned} \int _{0}^{T} {\mathcal {D}}^{\beta }_t u(t) \phi (t) d_{\beta } t=-\int _{0}^{T} u(t) {\mathcal {D}}^{\beta }_t\phi (t) d_{\beta } t, \quad \text{ for } \text{ all } \phi \in C_{0}^{\infty }(0, T). \end{aligned}$$

The space W endowed with the norm

$$\begin{aligned} \Vert u\Vert _{W_0}=\Vert u\Vert _{V_0}+\Vert {\mathcal {D}}_t^{\beta }u\Vert _{V_0^{\prime }}, \end{aligned}$$

is a Banach space which is separable and reflexive [64]. Furthermore, by Lemma 2.7 the embedding $W \subset C([0,T]; L^2(\Omega ))$ is continuous. Finally, because $H^{1}(\Omega ) \subset L^{2}(\Omega )$ is compactly embedded, by [64, Lemma 3.2] we have a compact embedding of $W \subset L_{\beta }^{2}\left( 0, T ; L^{2}(\Omega )\right)$

We denote the duality pairing between the elements of $V_{0}^{'}$ and $V_{0}$ by $\langle \cdot , \cdot \rangle ,$ and define the bi-linear form ${\mathcal {A}}$ associated with the operator ${\mathcal {P}}$ by

$$\begin{aligned} \langle {\mathcal {P}}u, \varphi \rangle = {\mathcal {A}}(u, \varphi ) \equiv \sum _{i, j=1}^{N} \int _{[0,T]}\int _{\Omega } [a_{ij}(t,x)\frac{\partial u}{\partial x_{i}} \frac{\partial \varphi }{\partial x_{j}} +c(t,x)u\phi ]\mathrm {d} x \mathrm {d}_{\beta } t, \quad \varphi \in V_{0}. \end{aligned}$$

We set $L^2_{\beta }=L_{\beta }^{2}[0,T;L^2(\Omega )]$ the cone $L_{\beta }^{2+}$ of all nonnegative elements of $L_{\beta }^{2}.$ This induces a corresponding partial ordering also in the subspace W of $L^{2}(Q),$ and if $v, w \in W$ with $v \le w$ then $[v, w] :=\{u \in W | v \le u \le w\}$ denotes the order interval formed by v and w. Let $(\cdot , \cdot )$ denote the inner product in $L_{\beta }^{2},$ define by

$$\begin{aligned} (u,v)_{L_{\beta }^{2}\times L_{\beta }^{2}}:=\int _{0}^{T}\int _{\Omega }u(t,x)v(t,x)dxd_{\beta }t=\int _{\Omega }\int _{0}^{T}u(t,x)v(t,x)d_{\beta }tdx\quad \forall u,v\in L_{\beta }^{2}, \end{aligned}$$

and denote by $F_1$ and $F_2$ the Nemytskij operators related with the functions $f_1$ and $f_2$, we have $F_1 u(t,x)=f_1(t,x, u(t,x))$ and $F_2 u(t,x)=f_2(t,x, u(t,x))$ for all $u\in V_0$.

Definition 3.1

A function $u,v \in W_{0}$ are said to be coupled weak solutions of (1.1) if

• 1. $F_1u+F_2v,F_1v+F_2u\in L^{2}_{\beta }$,

• 2. $u(0,x)=v(0,x)=u_0(x)$ in $\Omega$,

• 3. $\langle {\mathcal {D}}_t ^{\beta }(u) , \varphi \rangle + {\mathcal {A}}(u, \varphi )=(F_1 u+F_2 v, \varphi )$ and $\langle {\mathcal {D}}_t ^{\beta }(v) , \varphi \rangle + {\mathcal {A}}(v, \varphi )=(F_1 v+F_2 u, \varphi )$ for all $\varphi \in V_{0}$.

Definition 3.2

A functions $v ,w \in W_0$ are said to be a weak lower and upper solutions respectively of (1.1) if

• 1. $F_1 v +F_2w,F_1 w +F_2 v \in L^{2}_{\beta }$,

• 2. $v (t,x) \le 0\le w (t,x)$ on $\Gamma$ and $v (0,x) \le u_0(x)\le w (0,x)$ in $\Omega ,$

• 3. $\langle {\mathcal {D}}_t ^{\beta }(v) , \varphi \rangle + {\mathcal {A}}(v, \varphi )\leqslant (F_1 v+F_2 w, \varphi )$ and $\langle {\mathcal {D}}_t ^{\beta }(w) , \varphi \rangle + {\mathcal {A}}(w, \varphi )\geqslant (F_1 w+F_2 v, \varphi )$ for all $\varphi \in V_{0}\cap L_{\beta }^{2+}$.

Remark 3.3

In the definitions above, we have chosen $g\equiv 0$.

3.1 Comparison results

We can now prove the following comparison result.

Theorem 3.4

Assume that $v , w \in W_0$ are coupled lower and upper solutions of (1.1) and $f_1, f_2$ satisfy the following inequalities

$$\begin{aligned} {\left\{ \begin{array}{ll} f_1\left( t,x, u_{1}\right) -f_1\left( t,x, u_{2}\right) &{} \le c_{1}(t,x)\left( u_{1}-u_{2}\right) ,\\ f_2\left( t,x, u_{1}\right) -f_2\left( t,x, u_{2}\right) &{} \ge -c_{2}(t,x)\left( u_{1}-u_{2}\right) , \end{array}\right. } \end{aligned}$$

(3.1)

whenever $u_{1} \ge u_{2}$ for some $c_{1}, c_{2} \in L_{+}^{\infty }(Q).$ Then we have $v \le w$.

Remark 3.5

Clearly, the condition (3.1) is satisfied with $c_1=c_2=0$ when $f_1$ is monotone nonincreasing in u and $f_2$ is is monotone non-decreasing in u.

Proof

By the definition of coupled lower and upper solutions of (1.1) we get

$$\begin{aligned} v - w \le 0 \text{ on } \Gamma , v (0,x)- w (0,x) \le 0 \text{ in } \Omega , \end{aligned}$$

and

$$\begin{aligned} \langle {\mathcal {D}}_t^{\beta }( v - w ), \varphi \rangle +{\mathcal {A}}( v - w , \varphi ) \le (F_1 v -F_1 w +F_2 w -F_2 v , \varphi ), \end{aligned}$$

for all $\varphi \in V_{0} \cap L^{2+}_{\beta } .$ Choose $\varphi =( v - w )^{+}=\max \{ v - w , 0\} \in V_{0} \cap L_{\beta }^{2+},$ so that $( v - w )^{+}(0,x)=0$ and using (2.1) and the ellipticity condition (1.2), we get, for any $\xi \in [0, T],$ the following inequality

$$\begin{aligned} \begin{array}{l} \displaystyle \frac{1}{2}\left\| ( v - w )^{+}(\xi ,\cdot )\right\| _{L^{2}(\Omega )}^{2}+\alpha \left\| \nabla ( v - w )^{+}\right\| _{L^{2}_{\beta }\left( Q_{\xi }\right) }^{2} \\ \quad \le \left\| c_{1}+c_{2}-b\right\| _{L^{\infty }(Q)} \displaystyle \int _{0}^{\xi }\int _{\Omega }\left( ( v - w )^{+}\right) ^{2} d x d_{\beta } t, \end{array} \end{aligned}$$

where $Q_{\xi }=\Omega \times (0, \xi ) \subset Q .$ Now set $y(\xi )=\left\| ( v - w )^{+}(\xi ,\cdot )\right\| _{L^{2}(\Omega )}^{2}$ to obtain

$$\begin{aligned} y(\xi ) \le 2\left\| c_{1}+c_{2}-b\right\| _{L^{\infty }(\Omega )} \int _{0}^{\xi } y(t) d_{\beta } t, \quad \xi \in [0, T], \end{aligned}$$

since $v - w \in C\left( [0, T]; L^{2}(\Omega )\right) , y(\xi ) \ge 0$ and $y(0)=0,$ By applying the Conformable Gronwall inequality, we get $y(\xi )= 0$ for any $\xi \in [0, T]$ which means $( v - w )^{+}(t,x)=0$ a.e. in Q,  that is, $v \le w ,$ proving the claim. $\square$

The following corollary is also needed in our discussion.

Corollary 3.6

For any $z \in W$ satisfying $z(t,x) \le 0$ on $\Gamma , z(0,x) \le 0$ in $\Omega$ and

$$\begin{aligned} \langle {\mathcal {D}}_t^{\beta }z, \varphi \rangle +{\mathcal {A}}(z, \varphi ) \le 0 \text{ for } \text{ all } \varphi \in V_{0} \cap L_{\beta }^{2+}, \end{aligned}$$

we have $z \le 0$.

Proof

Let $\varphi (t,x)=z^{+}(t,x)=\sup \{z(t,x), 0\}$ then $\varphi \in V_0 \cap L_{+}^{2}(\Omega ) .$ Hence we have $z^+(0,x)=0 \text { in }\Omega$ and

$$\begin{aligned} \left( {\mathcal {D}}_t^{\beta }(z), z^{+}\right) +{\mathcal {A}}(z,z^{+}) \le 0 , \end{aligned}$$

we have

$$\begin{aligned} {\mathcal {A}}(z, z^{+})\ge \alpha \Vert z\Vert ^2 \ge 0, \end{aligned}$$

we get for any $\xi \in (0, T]$

$$\begin{aligned} \begin{aligned} \langle {\mathcal {D}}_t^{\beta }(z), z^{+}\rangle&={\int _{\Omega }} \int _{0}^{\xi }{\mathcal {D}}_t^{\beta }(z)z^+ d_{\beta }t \text { } dx\le 0, \end{aligned} \end{aligned}$$

then

$$\begin{aligned} \begin{aligned} 0\le \frac{1}{2}\left\| z^{+}(\xi ,\cdot )\right\| _{L^{2}(\Omega )}^{2}-\frac{1}{2}\left\| z^{+}(0,\cdot )\right\| _{L^{2}(\Omega )}^{2}=\frac{1}{2}\left\| z^{+}(\xi ,\cdot )\right\| _{L^{2}(\Omega )}^{2} \le 0, \end{aligned} \end{aligned}$$

which implies $z^+(t,x)=0$ a.e. $x\in Q$, for all $t\in [0,T]$. i.e. $z(t,.)\le 0$ a.e. in $\Omega$, for all $t\in [0,T]$. $\square$

3.2 Existence and uniqueness

We consider the linear IBVP

$$\begin{aligned} \begin{aligned}&{\mathcal {D}}_t^{\beta }u(t,x)+{\mathcal {P}}u(t,x)=g(t,x) \text { in } Q,\\&u(t,x)=0 \text { on } \Gamma ,\\&u(0,x)=\phi (x) \text { in } \Omega , \end{aligned} \end{aligned}$$

(3.2)

where $\phi \in L^{2}(\Omega )$ and $g \in L_{\beta }^{2} \subset V_{0}^{*}.$

Theorem 3.7

There exists a unique weak solution $u \in W_{0}$ to the IBVP (3.2), and for some constant $C>0$

$$\begin{aligned} \displaystyle \Vert u\Vert _{W_0} \le C\left( \Vert \phi \Vert _{L^{2}(\Omega )}+\Vert g\Vert _{L^{2}_{\beta }}\right) . \end{aligned}$$

Proof

For a similar proof using Fedo-Galerkin technique see [64]. $\square$

We shall prove the following result which includes several interesting special cases.

Theorem 3.8

Assume that

$\left( A_{1}\right)$ $v _{0}, w _{0}$ are coupled weak lower and upper solutions of (1.1) such that $v _{0} \le w _{0}$ a.e. in Q

$\left( A_{2}\right)$ $f_1, f_2: Q \times R \rightarrow R$ are Caratheodory functions such that $f_1(., ., u)$ is nondecreasing in u and g(., ., u) is nonincreasing in u for $(t, x) \in Q$ a.e.

$\left( A_{3}\right)$ for any $\mu _1, \mu _2 \in \left[ v _{0}, w _{0}\right] , h \in L_{\beta }^{2}$ where $h(t,x)=f_1(t,x, \mu _1)+$ $f_2(t,x, \mu _2)$

Then there exist monotone sequences $\left\{ v _{j}(t,x)\right\} _j,\left\{ w _{j}(t,x)\right\} _j \subset W_{0}$ such that

$$\begin{aligned} v _{j} \rightarrow v ^*, w _{j} \rightarrow w ^* \text { in } W_{0}, \end{aligned}$$

and $v ^*, w ^*$ are the coupled weak minimal and maximal solutions of (1.1).

Proof

For simplicity we set $g=u_0\equiv 0$, without loss of generality we consider the following IBVPs for each $j=1,2, \ldots$

$$\begin{aligned} {\left\{ \begin{array}{ll} {\mathcal {D}}_t^{\beta }v_j+{\mathcal {P}} v_{j}=f_1\left( t,x, v_{j-1}\right) +f_2\left( t,x, w_{j-1}\right) \quad \text{ in } Q ,\\ v_{j}(t,x)=0 \text{ on } \Gamma \text{ and } \quad v_{j}(0,x)=0 \text{ in } \Omega , \end{array}\right. } \end{aligned}$$

(3.3)

and

$$\begin{aligned} {\left\{ \begin{array}{ll} {\mathcal {D}}_t^{\beta }w_j+{\mathcal {P}}w_{j}=f_1\left( t,x, w_{j-1}\right) +f_2\left( t,x, v_{j-1}\right) \quad \text{ in } Q ,\\ w_{j}(t,x)=0 \text{ on } \Gamma \text{ and } w_{j}(0,x)=0 \text{ in } \Omega , \end{array}\right. } \end{aligned}$$

(3.4)

whose variational forms associated with (3.3) and (3.4) are given by

$$\begin{aligned} \langle {\mathcal {D}}_t^{\beta } v_j, \varphi \rangle +{\mathcal {A}}\left( v_{j}, \varphi \right) =\left( F_1 v_{j-1}+F_2 w_{j-1}, \varphi \right) , \end{aligned}$$

(3.5)

and

$$\begin{aligned} \langle {\mathcal {D}}_t^{\beta } w_j, \varphi \rangle +{\mathcal {A}}\left[ w_{j}, \varphi \right] =\left( F_1 w_{j-1}+F_2 v_{j-1}, \varphi \right) , \end{aligned}$$

(3.6)

for all $\varphi \in V_{0}$.

By $\left( A_{3}\right) , h \in L^{2}_\beta$ where $h(t,x)=f_1\left( t,x, v_{0}\right) +f_2\left( t,x, w_{0}\right) .$ Hence by Theorem 3.7 there is a unique weak solution $v_{1} \in W_{0}$ of (3.3) for $j=1$. In the same way, we get the existence of a unique weak solution $w_{1} \in W_{0}$ of (3.4). We shall first show that

$$\begin{aligned} v_{0} \le v_{1} \le w_{1} \le w_{0}. \end{aligned}$$

(3.7)

To prove $v_{0} \le v_{1},$ we find that $v_{1}$ satisfies the relation

$$\begin{aligned} \langle {\mathcal {D}}_t^{\beta } v_{1},\varphi \rangle +{\mathcal {A}}\left( v_{1}, \varphi \right) =\left( F_1 v_{0}+F_2 w_{0}, \varphi \right) , \end{aligned}$$

(3.8)

and note that $v_{0}$ satisfies

$$\begin{aligned} \langle {\mathcal {D}}_t^{\beta } v_{0}, \varphi \rangle +{\mathcal {A}}(v_{0}, \varphi ) \le \left( F_1 v_{0}+F_2 w_{0}, \varphi \right) , \end{aligned}$$

(3.9)

for each $\varphi \in V_{0} \cap L_{\beta }^{2+}$ and $v_{0}(t,x) \le 0$ on $\Gamma , v_{0}(0,x) \le 0$ in $\Omega .$ Let $z=v_{0}-v_{1} .$ Then z satisfies $z(t,x) \le 0$ on $\Gamma , z(0,x) \le 0$ in $\Omega$ and

$$\begin{aligned} \langle {\mathcal {D}}_t^{\beta } z, \varphi \rangle +{\mathcal {A}}(z, \varphi ) \le 0 \text{ for } \text{ all } \varphi \in V_{0} \cap L_{\beta }^{2+}, \end{aligned}$$

Hence by Corollary 3.6, we get $z \le 0,$ that is $v_{0} \le v_{1} .$ A similar reasoning proves that $w_{1} \le w_{0}$. To show that $v_{1} \le w_{1}$, we see from (3.6) that $w_{1}$ satisfies

$$\begin{aligned} \langle {\mathcal {D}}_t^{\beta } w_{1}, v\rangle +{\mathcal {A}}\left( w_{1}, \varphi \right) =\left( F_1 w_{0}+F_2 v_{0}, v\right) \ge \left( F_1 v_{0}+F_2 w_{0}, v\right) , \end{aligned}$$

for each $v \in V_{0} \cap L_{\beta }^{2+},$ using the monotone character of $F_1$ and $F_2$. By Corollary 3.6 we readily obtain, in view of (3.8) and (3.9), $v_{1} \le w_{1}$, proving (3.7)

Next by induction we shall prove that if for some $j>1$

$$\begin{aligned} v_{0} \le v_{j-1} \le v_{j} \le w_{j} \le w_{j-1} \le w_{0}, \end{aligned}$$

(3.10)

then

$$\begin{aligned} v_{j} \le v_{j+1} \le w_{j+1} \le w_{j} , \end{aligned}$$

(3.11)

According to $\left( A_{3}\right)$ and the relation (3.10), Theorem 3.7 guarantees the existence of unique weak solutions $v_{j}, w_{j}, v_{j+1}, w_{j+1} \in W_{0}$ of (3.3) and (3.4). To show that $v_{j} \le v_{j+1}$, consider

$$\begin{aligned} \langle {\mathcal {D}}^{\beta }_t v_{j+1}, \varphi \rangle +{\mathcal {A}}\left( v_{j+1}, \varphi \right) =\left( F_1 v_{j}+F_2 w_{j}, \varphi \right) , \end{aligned}$$

(3.12)

and

$$\begin{aligned} \langle {\mathcal {D}}_t^{\beta } v_{j}, \varphi \rangle +{\mathcal {A}}\left( v_{j}, \varphi \right) =\left( F_1 v_{j-1}+F_2 w_{j-1}, \varphi \right) \le \left( F_1 v_{j}+F_2 w_{j}, \varphi \right) , \end{aligned}$$

(3.13)

for each $v \in V_{0} \cap L_{\beta }^{2+} .$ In the above inequality (3.13), we have used the monotone nature of $F_1, F_2$ and (3.10). Corollary 3.6 then yields that $v_{j} \le v_{j+1}$. Similarly, we can show that $w_{j+1} \le w_{j} .$ To prove $v_{j+1} \le w_{j+1}$, we see, using the monotone character of $F_1, F_2$ and (3.10), that

$$\begin{aligned} \langle {\mathcal {D}}^{\beta }_t w_{j+1}, \varphi \rangle +{\mathcal {A}}(w_{j+1}, \varphi )=\left( F_1 w_{j}+F_2 v_{j}, \varphi \right) \ge \left( F_1 v_{j}+F_2 w_{j}, \varphi \right) , \end{aligned}$$

(3.14)

for each $\varphi \in V_{0} \cap L_{\beta }^{2+} .$ The relation (3.14), together with (3.12), prove by Corollary 3.6 that $v_{j+1} \le w_{j+1},$ showing that (3.11) is valid. Hence by induction, we arrive at

$$\begin{aligned} v_{0} \le v_{1} \le v_{2} \le \ldots \le v_{j} \le w_{j} \le \ldots \le w_{2} \le w_{1} \le w_{0}\quad a.e. \text { in } Q, \end{aligned}$$

(3.15)

for all j and for a.e. in Q. By the monotonicity of iterates $\left\{ v_{j}\right\} ,\left\{ w_{j}\right\}$ and (3.15), there exist pointwise limits

$$\begin{aligned} v ^*(t,x)=\lim _{j \rightarrow \infty } v_{j}(t,x), w ^*( t,x)=\lim _{j \rightarrow \infty } w_{j}(t,x), \end{aligned}$$

for a.e. in Q Furthermore, since $v_{j}, w_{j} \in \left[ v_{0}, w_{0}\right] ,$ it follows by Lebesgue’s dominated convergence theorem that

$$\begin{aligned} v_{j} \rightarrow v ^* \text{ and } w_{j} \rightarrow w ^* \text{ in } L^{2}_{\beta }. \end{aligned}$$

(3.16)

By $\left( A_{3}\right) ,$ we have for any $\mu _1, \mu _2 \in \left[ v_{0}, w_{0}\right] , F_1 \mu _1+F_2 \mu _2$ is continuous and bounded as a mapping from $\left[ v_{0}, w_{0}\right] \subset L_{\beta }^{2} \rightarrow L_{\beta }^{2}$ and therefore in view of (3.16), it follows that

$$\begin{aligned} F_1 v_{j} \rightarrow F_1 v ^*, F_2 w_{j} \rightarrow F_2 w ^*, F_1 w_{j} \rightarrow F_1 w ^*, F_2 v_{j} \rightarrow F_2 v ^* \text{ in } L^{2}_{\beta } \text{ as } j \rightarrow \infty . \end{aligned}$$

(3.17)

Since $v_{j}, w_{j}$ are solutions of the linear IBVPs (3.3) and (3.4) with homogeneous initial boundary values, the estimate of Theorem 3.7 gives

$$\begin{aligned} \left\| v_{j}\right\| _{W} \le C\left\| F_1 v_{j-1}+F_2 w_{j-1}\right\| _{L^{2}_{\beta }}, \end{aligned}$$

(3.18)

and

$$\begin{aligned} \left\| w_{j}\right\| _{W} \le C\left\| F w_{j-1}+F_2 v_{j-1}\right\| _{L^{2}_{\beta }}. \end{aligned}$$

(3.19)

Due to (3.17), the foregoing estimates imply the strong convergence of the sequences $\left\{ v_{j}\right\} ,\left\{ w_{j}\right\}$ in W whose limits must be $v ^*$ and $w ^*$ respectively, because of the compact embedding $W \subset L^{2}_{\beta }$ [64, Lemma 3.2.] . Since $v_{j}, w_{j}$ for $j \ge$ 1 belong to the subset $M=[u \in W: u(0,x)=0]$ which is closed in W,  it follows that their limits $v ^*, w ^*\in M,$ which are the limits satisfying also homogeneous initial and boundary conditions. The convergence result (3.17) together with

$$\begin{aligned} v_{j} \rightarrow v ^*, w_{j} \rightarrow w ^* \text{ in } W_0 \quad \text{ as } j \rightarrow \infty , \end{aligned}$$

permit us to pass to the limit in the corresponding variational forms (3.5) and (3.6) as $j \rightarrow \infty$. This yields

$$\begin{aligned} \langle {\mathcal {D}}^{\beta }_t v ^*, \varphi \rangle +{\mathcal {A}}( v ^*, \varphi )=(F_1 v ^*+F_2 w ^*, \varphi ), \end{aligned}$$

and

$$\begin{aligned} \langle {\mathcal {D}}^{\beta }_t w ^*, \varphi \rangle +{\mathcal {A}}( w ^*, \varphi )=(F_1 w ^*+F_2 v ^*, \varphi ), \end{aligned}$$

for all $\varphi \in V_{0},$ showing $( v ^*, w ^*)$ are coupled weak solutions of (1.1).

Let $u \in \left[ v_{0}, w_{0}\right]$ be any weak solution of (1.1). Then we show that $v_{1} \le u \le w_{1} .$ Since $v_{1}$ satisfies (3.9), we get, using monotone nature of $F_1$ and $F_2$

$$\begin{aligned} \langle {\mathcal {D}}^{\beta }_t v_{1}, v\rangle +{\mathcal {A}}(v_{1}, \varphi )=\left( F_1 v_{0}+F_2 w_{0}, \varphi \right) \le (F_1 u+F_2 u, \varphi ), \end{aligned}$$

for each $v \in V_{0} \cap L_{\beta }^{2+} .$ Since u is a weak solution of (1.1) we have

$$\begin{aligned} \langle {\mathcal {D}}^{\beta }_t u, v\rangle +{\mathcal {A}}(u, \varphi )=(F_1 u+F_2 u, \varphi ), \end{aligned}$$

for each $\varphi \in V_{0} \cap L_{\beta }^{2+} .$ Corollary 3.6 therefore yields $v_{1} \le u .$ A similar argument shows that $u \le w_{1}$. Next we suppose that for some $j>1$

$$\begin{aligned} v_{0} \le v_{j} \le u \le w_{j} \le w_{0}. \end{aligned}$$

Then utilizing the monotone character of $F_1$ and $F_2,$ we obtain

$$\begin{aligned} \langle {\mathcal {D}}^{\beta }_t v_{j+1}, \varphi \rangle +{\mathcal {A}}(v_{j+1}, \varphi )=\left( F_1 v_{j}+F_2 w_{j}, \varphi \right) \le (F_1 u+F_2 u, \varphi ), \end{aligned}$$

for each $\varphi \in V_{0} \cap L_{\beta }^{2+} .$ It then follows from Corollary 3.6 that $v_{j+1} \le u$. In the same way, we can prove that $u \le w_{j+1}$. Thus we arrive at

$$\begin{aligned} v_{0} \le v_{j} \le v_{j+1} \le u \le w_{j+1} \le w_{j} \le w_{0}. \end{aligned}$$

Hence by induction $v_{j} \le u \le w_{j}$ for all j and consequently, taking the limit as $j \rightarrow \infty ,$ we get $v ^* \le u \le w ^*,$ proving that $v ^*, w ^*$ are coupled weak minimal and maximal solutions of (1.1). The proof is now complete. $\square$

Corollary 3.9

In addition to the assumptions of Theorem 3.8, if we suppose that $f_1, f_2$ satisfy the conditions (3.1), then $u= v ^*= w ^*$ is the unique weak solution of (1.1).

Proof

Since $v^* \le w^* ,$ it is sufficient to show that $w ^*\le v ^* .$ But this follows right away from Corollary 3.6 if we set $z=w^* -v^*.$ Thus the claim of the Corollary is true. $\square$

Several particular problems can be studied by the same steps and the conclusion of Theorem 3.8 is satisfied, for example if we consider the following problem

$$\begin{aligned} \left\{ \begin{array}{l}\displaystyle {\mathcal {D}}_t^{\beta }u+{\mathcal {P}} u=h_1\left( t,x ,u\right) +h_2\left( t,x, u\right) \quad \text{ in } Q ,\\ u(t,x)=0 \text{ on } \Gamma ,\\ u(0,x)=u_0(x) \text{ in } \Omega , \end{array}\right. \end{aligned}$$

(3.20)

where the both $h_1$ and $h_2$ are not monotone. Then we find the following result

Corollary 3.10

Suppose that $f_1(t,x, u)=h_1(t,x, u)+\eta _{1}(t, x) u$ is nondecreasing in u and $f_2(t,x, u)=h_2(t,x, u)-\eta _{2}(t,x) u$ is nonincreasing in u,  where $\eta _{1}, \eta _{2} \in L_{+}^{\infty }(Q) .$ Then the same conclusion of Theorem 3.8 and Corollary 3.9 remains the same for problem (3.20).

Proof

We consider the problem (3.21) equivalent to (3.20)

$$\begin{aligned} \left\{ \begin{array}{l} {\mathcal {D}}^{\beta }_t u+{\mathcal {P}} u+\eta _{1} u-\eta _{2} u=\tilde{f_1}(t,x, u)+\tilde{f_2}(t,x, u) \quad \text{ in } Q,\\ u=0 \quad \text{ on } \Gamma ,\\ u(0,x)=0 \text{ on } \Omega . \end{array}\right. \end{aligned}$$

(3.21)

Let $v_{0}, w_{0}$ are coupled weak lower and upper solutions of (3.20), then are also for the problem (3.21). Let $\tilde{{\mathcal {P}}}u={\mathcal {P}} u+\eta _{1} u-\eta _{2} u$, all the conditions on ${\mathcal {P}}$ in Theorem 3.8 are satisfied for $\tilde{{\mathcal {P}}}$, so by application of the Theorem 3.8 and Corollary 3.1 we find the result of the Corollary 3.10. $\square$

3.3 Nonuniqueness of solutions

We have shown in the Corollary 3.9 that the solution of problem (1.1) is unique if $f_1$ and $f_2$ satisfy the condition (3.1), this result is only valid inside sector $<v, w>$, so it does not give any information outside of sector $<v, w>$. In the following example, we show that if $f_1$ and $f_2$ do not satisfy (3.1) then the problem (1.1) can have several solutions. Let us consider the problem

$$\begin{aligned} \begin{array}{ll}\displaystyle {\mathcal {D}}^{\beta }_{t}u-u_{x x}=u+\left( \frac{3}{2}\sin ^{2\beta /3}(x)\right) u^{(3-2\beta ) / 3}&{} (t>0,0<x<\pi ) ,\\ u(t, 0)= u(t, \pi )=0&{} (t>0) ,\\ u(0, x)=0 &{} (0<x<\pi ). \end{array} \end{aligned}$$

(3.22)

Clearly the functions

$$\begin{aligned} f_1(t,x, u)=u+\left( \frac{3}{2} \sin ^{2 \beta / 3} x / 2\right) u^{(3-2\beta ) / 3} \text { and }\, f_2(t,x,u)=0, \end{aligned}$$

satisfy $(A_2)$ and $(A_3)$. To apply Theorem 3.8 consider the functions v and w defined by

$$\begin{aligned} v=0, \quad w=K t^{3 / 2} \sin x, \end{aligned}$$

where $K\ge 1$ is a constant. We have

$$\begin{aligned} v(t,0)=v(t,\pi )= w(t,0)=w(t,\pi )=0, \end{aligned}$$

it remains to verify that v and w satisfy the differential inequality given by definition of upper and lower solutions, we have

$$\begin{aligned} {\mathcal {D}}_t^{\beta }w-w_{x x}=K\frac{3}{2} t^{(3-2\beta )/ 2}\sin x +Kt^{3 / 2} \sin x=w+K\frac{3}{2} t^{(3-2\beta )/ 2}\sin x , \end{aligned}$$

then the following inequality must be satisfied

$$\begin{aligned} w+K\frac{3}{2} t^{(3-2\beta )/ 2}\sin x \ge w+\left( \frac{3}{2}\sin ^{2\beta /3}(x)\right) w^{(3-2\beta ) / 3}. \end{aligned}$$

This implies

$$\begin{aligned} K\ge K^{(3-2\beta ) / 3}, \end{aligned}$$

which is clearly satisfied by any $K \ge 1$. The existence of solutions of (3.22) is guaranteed by the Theorem 3.8 but not the uniqueness. However $u_1 = 0$ and $u_2 =t^{3 / 2} \sin x$ are two different solutions to problem (3.22) in the sector $<v, w>$ and

$$\begin{aligned} 0 \le u_1(t, x),u_2(t, x) \le K t^{3 / 2} \sin x, \end{aligned}$$

Additional solutions are given in the following form

$$\begin{aligned} u_{\tau }(t, x)=\left\{ \begin{array}{ll} 0 &{} \text{ when } 0 \le t \le \tau ,\\ \left( t-\tau \right) ^{3 / 2} \sin x &{} \text{ when } t \ge \tau . \end{array}\right. \end{aligned}$$

4 Construction of lower and upper solutions

In the Conformable problem considered, to obtain the result of existence, we have to suppose that the problem has lower and upper solutions, therefore the results depend on the existence of the last ones. In many real problems the boundary and initial data are nonnegative and $f (.,. , 0)=f_1 (.,. , 0)+f_2 (.,. , 0)\ge 0$. In this situation, it is only necessary to find a suitable non-negative upper solution.

In this section, let’s use the same techniques in [56] to construct the lower and upper solutions, the construction usually depends on the nonlinear term f(t, x, u), we usually manage to solve the problem linear.

To illustrate some methods for the construction of upper and lower solutions we consider a class of nonlinear functions which are sufficient to treat the reaction-diffusion models in (1.1). For this class of functions we require the following conditions:

• $f(t,x,0) \ge 0$ and there exist two functions $K_{1}, K_{2}$ with $K_{2} \ge 0$ such that

  $$\begin{aligned} f(t,x,u) \le K_{1}(t,x)u+K_2(t,x) \quad \text{ when } u \ge 0,(t,x) \in \Omega . \end{aligned}$$

Then, the construction of upper solution w and lower solutions v it is often practical to use the solution of the Conformable linear parabolic problems in the form

$$\begin{aligned} \left\{ \begin{array}{l}\displaystyle {\mathcal {D}}^{\beta }_t w-{\mathcal {P}}w =K_1(t, x)w+K_2(t,x) \text{ in } Q_{T}=(0,T)\times \Omega \\ w(t,x)=h(t, x) \text{ on } \Gamma _{T}=[0,T]\times \partial \Omega \\ w(0, x)=u_{0}(x) \text{ in } \Omega \end{array},\right. \end{aligned}$$

(4.1)

and

$$\begin{aligned} \left\{ \begin{array}{l}\displaystyle {\mathcal {D}}^{\beta }_t v-{\mathcal {P}}v =f(t, x,0) \text{ in } Q_{T}=(0,T)\times \Omega \\ v(t,x)=h(t, x) \text{ on } \Gamma _{T}=[0,T]\times \partial \Omega \\ v(0, x)=u_{0}(x) \text{ in } \Omega \end{array},\right. \end{aligned}$$

(4.2)

where $u_0$ and h are nonnegative functions. The choice of $K_1$ and $K_2$ depends on the reaction function f under consideration. Note that the unique weak solutions of (4.1) and (4.2) exists and is nonnegative in W.

Let us note that the construction of an efficient numerical scheme of problem (4.1) and (4.2) is not difficult because it is a linear problem, therefore it is enough to use in our case the implicit Conformable Euler scheme in time and the finite element method in space.

An other simple condition on f is the existence of constant functions ${\mathbf {p}}$ and ${\mathbf {q}}$ such that

$$\begin{aligned} {\mathbf {p}} \le 0 \le {\mathbf {q}} \quad \text{ and } \quad f(.,.{\mathbf {p}}) \ge 0 \ge f(.,.,{\mathbf {q}}). \end{aligned}$$

(4.3)

It is easy to verify that the pair $w={\mathbf {q}}$ and $v={\mathbf {p}}$ are ordered upper and lower solutions.

5 Applications

In this section, we give some example of application of the existence and construction results of problem 1.1.

5.1 Nonlinear Michaelis-Menten oxygen uptake kinetics model

We consider the time Nonlinear Michaelis-Menten oxygen uptake kinetics model without advection term

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle {\mathcal {D}}^{\beta }_t C(t,x)-\frac{\partial ^{2} C}{\partial x^{2}}=\frac{-\sigma C}{C+K_{m}} \text{ in } Q_{T}=(0,T)\times (0,1)\\ C(t,0)=a,C(t,1)=b \text { in } (0,T),\\ C(0,x)=C_0(x) \text { in } (0,1), \end{array}\right. }. \end{aligned}$$

(5.1)

where $\sigma , K_m$ are positive constants. This model is a special case of (1.1) with

$$\begin{aligned} f_1\equiv 0 \text { and } f_2(t,x,C)=\frac{-\sigma C}{ C+ K_m}, \end{aligned}$$

$\partial f_2/\partial C<0$ then $f_2$ is monotone decreasing function, any constant $M\ge 0$ satisfying

$$\begin{aligned} M\ge \max \{a,b,u_0(x)/\quad x\in (0,1)\}, \end{aligned}$$

(5.2)

is an upper solution of (5.1), whose existence and uniqueness of solution u is obtained from the Theorem 3.8 and Corollary 3.9 with $0 \le u \le K .$ By application of the technique given in section 4 we can obtain a non-constant upper solution w, since

$$\begin{aligned} f(0)=0 \text { and } f(u) \le 0 \text { for all }u \ge 0, \end{aligned}$$

hence the solution w(t, x) of problem (4.1) corresponding to $K_1=K_2=0$ is an upper solution.

Another effect can make the reaction function in (5.1) in the form when the effect of inhibition in the enzyme-substrate reaction scheme is taken into consideration the reaction function becomes

$$\begin{aligned} f(t,x,C)=\frac{-\delta C}{K_m + C + \delta C^2 }, \end{aligned}$$

(5.3)

where $\delta$ is a positive constant. In this case the same function w as in the previous model is a nonnegative upper solution.

5.2 The time Conformable Fisher’s model

We consider the time Conformable Fisher’s model

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle {\mathcal {D}}^{\beta }_t u-D\Delta u=u(u - \xi )(1- u) \quad \text{ in } Q ,\\ u(t,x)=g(t,x) \text { in } \Gamma ,\\ u(0,x)=u_0(x) \text { in } \Omega . \end{array}\right. \end{aligned}$$

(5.4)

where $\sigma , \theta$ are positive constants with $0<\theta <1 .$ According to the real signification of the density function u, it must be between 0 and 1, then $0 \le u_{0}(x) \le 1$ and $0 \le h(t,x) \le 1 .$ Under this requirement, it is easy to verify that 0 and 1 are respectively lower and upper solutions of (5.4). Hence by Theorem 3.8, and Corollary 3.9 time Conformable Fisher’s model has a unique weak solution u such that $0 \le u \le 1$ .

5.3 Conformable Models in reactor dynamics and heat conduction

The model described the neutron flux with an internal source with time Conformable derivative is given by

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle {\mathcal {D}}^{\beta }_t u-D\Delta u=u(\alpha - \delta u) + \phi (x) \quad \text{ in } Q, \\ u(t,x)=g(t,x) \text { in } \Gamma ,\\ u(0,x)=u_0(x) \text { in } \Omega , \end{array}\right. \end{aligned}$$

(5.5)

where $\alpha , \delta$ are positive constants and $\phi \ge 0$. Since

$$\begin{aligned} f(0)=\phi (x)\ge 0 \text { and } f(u) \le f(\alpha /2\delta ) \text { for all }u \ge 0, \end{aligned}$$

the solutions v(t, x) and w(t, x) of (4.1) and (4.2) respectively, with $K_1=0, K_2=\alpha ^2/(4\delta )+\phi (x)$ (or $K_1=\alpha , K_2=\phi (x)$) are lower and upper solutions. Since by Theorem 3.8 and Corollary 3.9 that the weak solution of (5.5) exists and $v \le u \le {\bar{w}}$.

Another Conformable heat-conduction problem is given by

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle {\mathcal {D}}^{\beta }_t u-D\Delta u=K(\delta ^4 - u^4) \text{ in } Q_{T}=(0,T)\times \Omega \\ u(t,x)=g(t,x) \text { in } \Gamma ,\\ u(0,x)=u_0(x) \text { in } \Omega , \end{array}\right. \end{aligned}$$

(5.6)

where $\delta , K$ are positive constants. Since $f(0)=K \delta ^{4}$ and $f(\xi ) \le 0$ for $\xi \ge \delta$. We chose $w=\xi \ge \max \{g,u_0,\delta \}$, then Theorem 3.8 and Corollary 3.9 implies that the reaction model (5.6) has a unique weak solution u where $0 \le u(t, x) \le \xi .$

The nonisothennal chemical reaction model is in the form

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle {\mathcal {D}}^{\beta }_t u-D\Delta u=\mu \left( \xi -u\right) \exp (\theta u /(1+u))\\ u(t,x)=0\text { in } \Gamma ,\\ u(0,x)=u_0(x) \text { in } \Omega , \end{array}\right. \end{aligned}$$

(5.7)

where $\mu , \xi ,$ and $\theta$ are positive constants. We have for any $K \ge \xi$, $f(0)=\mu \xi$ and $f(K) \le 0$, Then by Theorem 3.8 and Corollary 3.9 the model (5.7) has a unique weak solution u where $0 \le u(t, x) \le \max \{u_{0},\xi \} .$

6 Discrete Galerkin finite element method

To solve (3.2) by the finite element method, we construct a finite dimensional subspace $V_{h}$ of $H_{0}^{1}(\Omega )$ using continuous piecewise basis functions, i.e.

$$\begin{aligned} V_{j}=\left\{ v_{j} \in C^{0}({\bar{\Omega }}):\left. v_{j}\right| _{K} \in P_{k}, \forall K \in T_{h}\right\} , \end{aligned}$$

where $T_{j}$ is a regular family of triangulations of $\Omega$, the elementary size of the elements K in $T_{j}$ as defined by:

$$\begin{aligned} h=\max _{K \in T_{j}} {\text {diam}}(K). \end{aligned}$$

Then we can define the semi-discrete finite element method for the weak problem of (1.1): Find $u_{j}:[0, T] \rightarrow V_{j}$ such that

$$\begin{aligned} \begin{aligned} \left( {\mathcal {D}}_{t}^\beta u_j, \varphi \right) +{\mathcal {A}}\left( u_{j}, \varphi \right)&=(f(u_j), \varphi ) \quad \forall \varphi \in V_{j} \\ u_{j}(\cdot , 0)&=\Pi _{j} u_{0}, \end{aligned} \end{aligned}$$

(6.1)

where $\Pi _{j}$ is the elliptic projection operator [ref 45...] into $V_{j},$ i.e., for any $u \in$ $H^{1}(\Omega ), \Pi _{j} u \in V_{j}$ satisfies the equation

$$\begin{aligned} {\mathcal {A}}\left( \left( u-\Pi _{j} u\right) , \varphi \right) =0 \quad \forall \varphi \in V_{j}. \end{aligned}$$

In semi-discrete approximation, the solution $u\in V_j$ is in the form

$$\begin{aligned} u(x, t)=\sum _{k=1}^{N} u_{k}(t) \phi _{k}(x), \end{aligned}$$

(6.2)

where $\phi _{k}$ are the basis functions of $V_{j} .$

Now the semi-discrete finite element scheme of the linear Picard problems (3.3) and (3.4) is given for $m=1,2,\ldots$ by:

Find $v^m,w^m\in V_j$ such that

$$\begin{aligned} \left\{ \begin{array}{l} \left( {\mathcal {D}}_{t}^\beta v^{m}, \varphi \right) +{\mathcal {A}}\left( v^{m}, \varphi \right) =(f_1(v^{m-1})+f_2(w^{m-1}), \varphi ) \quad \forall \varphi \in V_{j} ,\\ v^{m}(0,\cdot ) =\Pi _{j} v_{0},\\ \left( {\mathcal {D}}_{t}^\beta w^{m}, \varphi \right) +{\mathcal {A}}\left( w^{m}, \varphi \right) =(f_1(w^{m-1})+f_2(v^{m-1}), \varphi ) \quad \forall \varphi \in V_{j}, \\ w^{m}( 0,\cdot ) =\Pi _{j} w_{0}.\\ \end{array}\right. \end{aligned}$$

(6.3)

In the above approximation, the solution $v^m$ and $w^m$ are in the form

$$\begin{aligned} v^m(x, t)=\sum _{k=1}^{N} u_{k}^m (t) \phi _{k}(x), \quad w^m(x, t)=\sum _{k=1}^{N} w_{k}^m (t) \phi _{k}(x), \end{aligned}$$

(6.4)

where $\phi _{k}$ are the basis functions of $V_{j} .$ First, we partition the time interval [0, T] into N equal sub-intervals $\left[ t_{k}, t_{k+1}\right]$ of length $\Delta t=\frac{T}{N} .$ The time-discrete finite element method, such as the Conformable implicit Euler scheme [67] for (3.2) can be defined as: Find $(v,w)\in V_j\times V_j$ such that

$$\begin{aligned} \left\{ \begin{array}{l} \left( \displaystyle \frac{((n+1)^{\beta }-n^{\beta })(\Delta t)^{\beta }}{\beta }(v^m_{n+1}-v^m_{n}), \varphi \right) +{\mathcal {A}}\left( v^{m}_{n}, \varphi \right) =(f_1(v_{n+1}^{m-1})+f_2(w_{n+1}^{m-1}), \varphi ), \quad \forall \varphi \in V_{j} \\ \left( \displaystyle \frac{((n+1)^{\beta }-n^{\beta })(\Delta t)^{\beta }}{\beta }(w^m_{n+1}-w^m_{n}), \varphi \right) +{\mathcal {A}}\left( w^{m}_{n}, \varphi \right) =(f_1(w_{n+1}^{m-1})+f_2(v_{n+1}^{m-1}),\varphi ), \quad \forall \varphi \in V_{j} \\ \\ v^{m}(\cdot , 0) =\Pi _{j} v_{0},\quad w^{m}(\cdot , 0) =\Pi _{j} w_{0}. \end{array}\right. \end{aligned}$$

(6.5)

We find the flowing system for unknown vectors $U=\left( u_{1}, \cdots , u_{N}\right) ^{T}:$

$$\begin{aligned} {\left\{ \begin{array}{ll} L_n(\beta )k^{\beta }M V^{(n,m)}+AV^{(n,m)}= L_n(\beta )k^{\beta } MV^{(n-1,m)}+F_1^{n}(V^{n,m-1})+F_2^{n}(W^{n,m-1}),\\ L_n(\beta )k^{\beta } MW^{(n,m)}+AW^{(n,m)}=L_n(\beta )k^{\beta } MW^{(n-1,m)}+F_1^{n}(W^{n,m-1})+F_2^{n}(V^{n,m-1}),\\ V^{(0,m)}=V_0=\left( \Pi _1v_{0},\Pi _2 v_{0},\ldots ,\Pi _N v_0\right) ,\\ W^{(0,m)}=W_0=\left( \Pi _1w_{0},\Pi _2 w_{0},\ldots ,\Pi _N w_0\right) , \end{array}\right. } \end{aligned}$$

(6.6)

where

$$\begin{aligned} \begin{aligned} M=\left( M_{k i}\right) =\left( \phi _{k}, \phi _{i}\right) =\int _{K}\varphi _k(x)\varphi _i(x)dx,\\ A=\left( A_{k i}\right) ={\mathcal {A}}\left( \phi _{k}, \phi _{i}\right) =\int _{K} a(x)\nabla \varphi _k(x)\nabla \varphi _i(x)dx,\\ F_s^n(U^{n,m})=\int _{K}f_s^{n}(U^{(n,m)})\phi _{i}(x) dx, \quad s=1,2, \end{aligned} \end{aligned}$$

and

$$\begin{aligned} L_n(\beta )=\frac{n^{\beta }-(n-1)^{\beta }}{\beta }. \end{aligned}$$

Note that if $\beta =1$ we have $L_{n}(\beta )=1$ then we find the classical Euler finite element scheme.

7 Examples and simulations results

7.1 Example with a known analytic solution on domain disk with three holes

We consider the flowing first test problem

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle {\mathcal {D}}^{\beta }_t u-D\Delta u=\pi ^2 u \quad \text{ in } Q_T=(0,T)\times \Omega ,\\ u(t,x)=\exp (-\pi ^2 t^{\beta }/\beta )\sin (\pi x)\sin (\pi y)\quad (t,x,y)\in (0,T) \times \Gamma ,\\ u(0,x,y)=\sin (\pi x) \sin (\pi y) \quad (x,y)\in \Omega . \end{array}\right. \end{aligned}$$

(7.1)

Here, we consider in the first case the domain $\Omega$ in the flowing form (see Fig. 1a):

$$\begin{aligned} \begin{array}{l} \Omega = D\cap {\overline{D}}_1\cap {\overline{D}}_2\cap {\overline{D}}_3 \text { where }\\ D=\{(x,y)\in {\mathbb {R}}^2\quad /\quad x^2+y^2<1\}\\ {\overline{D}}_i=\{(x,y)\in {\mathbb {R}}^2 \quad /\quad (x-a_i)^2+(y-b_i)^2>r_i^2\},\quad i=1,2,3,\\ \end{array} \end{aligned}$$

with Lipschitz boundary given by

$$\begin{aligned} \begin{array}{l} \Gamma =C\cup C_1\cup C_2\cup C_3,\\ \text {where }\\ C=\{(x,y)\in {\mathbb {R}}^2\quad /\quad x^2+y^2=1\}\\ C_i=\{(x,y)\in {\mathbb {R}}^2 \quad /\quad (x-a_i)^2+(y-b_i)^2=r_i^2\},\quad i=1,2,3,\\ \end{array} . \end{aligned}$$

The exact solution of (7.1) is $u(t,x,y)=\exp (-\pi ^2 t^{\beta }/\beta )\sin (\pi x)\sin (\pi y).$

Table 1 $L_2^{\beta }(L_2)$-norm and $L_2^{\beta }(H_1)$-norm for $\Delta t=0.1$

Table 2 Table of $L_2^{\beta }(L_2)$-norm and $L_2^{\beta }(H_1)$-norm for $\Delta t=0.001$

Fig. 1

Two typs of Domain $\Omega$

Is not difficult to show that $v\equiv -1$ is a lower solution and $w\equiv 1$ is an upper solution of (7.1). By using the fully discrete finite element scheme given in (6.6) we solve the model (7.1) for two values of the non-integer order, 0.5 and 0.9. We chose firstly in our simulation $\Delta t=0.01$ and $h=0.08$. We show in Fig. 4 first 18 steps of $V_n$ and $W_n$ for $\beta =0.5$ and in Fig. 2 first 13 steps of $V_n$ and $W_n$ for $\beta =0.9$ at time step $t=100\Delta t$. Observe that the upper and lower sequence converges more quickly to the numerical solution of the model and the two sequences coincide for $k\ge 18$ for $\beta =0.5$ and $k\ge 13$ for $\beta =0.9$. The comparison between the approximate and exact solution is presented in Fig. 3 for $\beta =0.5$ and in Fig. 5 for $\beta =0.9$.

Fig. 2

Convergence of upper ($W_n$) and lower ($V_n$) sequences of Example 1 for $h=0.08$, $\Delta t=0.01$, $t_1=100\Delta t$ and $\beta =0.5$

Fig. 3

Comparison of numerical and exact solution for test Example 1 for $\Delta t=0.01$, $\beta =0.5$ at time $t_1=100\Delta t$

Fig. 4

Convergence of upper ($W_n$) and lower ($V_n$) sequences of Example 1 for $h=0.08$, $\Delta t=0.01$, $t_1=100\Delta t$ and $\beta =0.9$

Fig. 5

Comparison of numerical and exact solution for test Example 1 for $\Delta t=0.01$, $\beta =0.5$ at time $t_1=100\Delta t$

When we chose now $\Delta t=0.001$ and $h=0.04$. We compute the numerical solution for $\beta =0.5,0.9$ at time step $t=100\Delta t$. We show in Figs. 6 and 8 that the upper and lower sequence converges more quickly to the numerical solution of the model and the two sequences coincide for $k\ge 4$. The comparison between the approximate and exact solution is presented in Fig. 7 for $\beta =0.5$ and in Fig. 9 for $\beta =0.9$.

Fig. 6

Convergence of upper ($W_n$) and lower ($V_n$) sequences of Example 1 for $h=0.04$, $\Delta t=0.001$, $t_1=100\Delta t$ and $\beta =0.5$

Fig. 7

Comparison of numerical and exact solution for test Example 1 for $h=0.04$, $\Delta t=0.001$, $\beta =0.5$ at time $t_1=100\Delta t$

Fig. 8

Convergence of upper ($W_n$) and lower ($V_n$) sequences of Example 1 for $h=0.08$, $\Delta t=0.01$, $t_1=100\Delta t$ and $\beta =0.9$

Fig. 9

Comparison of numerical and exact solution for test Example 1 for $h=0.04$, $\Delta t=0.01$, $\beta =0.9$ at time $t_1=100\Delta t$

The corresponding $L^2_{ \beta }(0,T,L^2(\Omega ))$-norm errors and $L^2_{\beta }(0,T,H^1_0(\Omega ))$-norm errors for test Example 1 are listed in Tables 1 and 2 and presented in Fig. 10 for various values of $\Delta t,h$ and the non-integer order $\beta$.

Fig. 10

Convergence of $L_2$ and $H_1$ errors with increasing number of nodes in the coarse discretization for Example 1

7.2 Example on circle domain with unknown solution

We consider the generalized time Conformable Fisher-Kolmogorov equation

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle \frac{\partial ^{\beta } u}{\partial t^{\beta }}-\frac{\partial ^{2} u}{\partial x^{2}}-\frac{\partial ^{2} u}{\partial y^{2}}=\frac{\alpha }{k} u\left( 1-u^{q}\right) \text{ in } Q_{T}=(0,T)\times \Omega ,\\ u(t,x,y)=h(t,x) \text { in } (0,T]\times \Gamma ,\\ u(0,x,y)=u_0(x,y) \text { in } \Omega , \end{array}\right. \end{aligned}$$

(7.2)

where $h(t,x)=0$, $u_0(x,y)=\sin (\pi x)\sin (\pi y)$ and $k,\alpha$ are positive constants and $q \in {\mathbb {N}}$. This model is a special case of problem (1.1) with $\displaystyle f_1(t,x,y)=\frac{\sigma }{k}u$ and $\displaystyle f_2(t,x,y)=-\frac{\sigma }{k}u^{q+1}$, the domain $\Omega$ is taken as a disk with radius equal to 4 since is in the form (see Fig. 1b)

$$\begin{aligned} \Omega&= \{(x,y)\in {\mathbf {R}}^2\quad /\quad x ^2+y^2<16\}\\&\quad \text {with boundary is circle with radius 4, } \partial \Omega & =\{(x,y)\in {\mathbf {R}}^2\quad /\quad x ^2+y^2=16\}. \end{aligned}$$

We chose in our simulation $\alpha /k=1$ and $q=2$. Is not difficult to show that $v\equiv -1$ is a lower solution and $w\equiv 1$ is an upper solution of (7.2). By using the fully discrete finite element scheme given in (6.6) we solve the model (7.2) for two values of $\beta$, 0.5 and 0.95. We show in Figs. 11 and 12 first 9 steps of $V_n$ and $W_n$ for $\beta =0.95$ at time step $t=100\Delta t$ where $\Delta t=0.01$. Observe that the upper and lower sequence converges more quickly to the numerical solution the model and the two sequences coincide for $m\ge 9$, see Fig. 13. the same conclusion shown in Figs. 14,15 and 13 for the solution $\beta =0.5$ (Fig. 16).

Fig. 11

Convergence of lower sequence ($V_n$) for test Example 2 with $\Delta t=0.01$, $\beta =0.95$ and at time $t=100\Delta t$

Fig. 12

Convergence of upper sequence ($W_n$) for test Example 2 with $\Delta t=0.01$, $\beta =0.95$ and at time $t=1$

Fig. 13

Comparison between upper solutions $W_1,W_4,W_9$ and lower solutions $V_1,V_4,V_9$ for test Example 2 with $\Delta t=0.01$, $\beta =0.95$ and at time $t=100\Delta t$

Fig. 14

Convergence of lower sequence ($V_n$) for test Example 2 with $\Delta t=0.01$, $\beta =0.5$ and at time $t=100\Delta t$

Fig. 15

Convergence of upper sequence ($W_n$) for test Example 2 with $\Delta t=0.01$, $\beta =0.5$ and at time $t=100\Delta t$

Fig. 16

Comparison between upper solutions $W_1,W_4,W_9$ and lower solutions $V_1,V_4,V_9$ for test Example 2 with $\Delta t=0.01$, $\beta =0.5$ and at time $t=100\Delta t$

8 Conclusion and future works

We have shown in this current paper that the monotone iterative technique combined with the upper and lower solutions method are an efficient tools to study theoretically and numerically nonlinear PDEs with non-integer order time derivative. The numerical algorithm for the computation of the solution is obtained by using finite element method for space and the generalization to non-integer order of the implicit Euler scheme for time discetization and we presented real examples and obtained important results.

As perspective, the numerical method used can be improved to increase the speed of convergence by combining this method with Newton’s method. The stability and the convergence, thus the error estimation of the scheme proposed in Sect. 4 are considered in our future works. This method can also be applied to other problems such as hyperbolic problems, systems with several equations, problems which contain nonlinear boundary conditions or problems that contain other types of non-integer derivations and this is the subject of our next articles.