Introduction

Fractional differential equations (FDEs) have wide variety of applications in diverse fields such as scienece, engineering, blood flow phenomenon, virology, image processing, control theory and so on. For a overview of these applications we refer readers to [11, 12, 14, 16, 19].

Linear multi-term fractional differential equations involve multiple fractional differential operators of degree one. Multi-term fractional differential equations (MTFDEs) appear in many real world problems like Bogley-Torvik equations [5, 20], Basset equations [15]. Due to wide applications to real world problems, it is an important research area to develop efficient tools to solve MTFDEs. There exists numerous research articles devoted to study numerical methods to solve FDEs and MTFDEs. Various numerical methods, series solution methods, decomposition methods have been introduced by researchers to solve MTFDEs [5, 8, 9].

A. Saadatmandi and M. Dehghan [18] introduced a new operational matrix for solving fractional differential equations. New higher order numerical methods for fractional differential equations are developed in [21]. Further a new Jacobi operational matrix application for solving FDEs has been developed in [7]. In [17], a multiterm fractional differential equation is reduced to algebraic equations using Bernstein polynomials and new efficient numerical methods has been developed to solve them along with the convergence analysis of the new method. V. Daftardar-Gejji and S. Bhalekar used method of seperation of variables to solve multi term fractional diffusion-wave equation [1].

Present work is devoted to develop a new numerical algorithm to solve MTFDEs and to check accuracy and convergence of the new algorithm.

The paper is organized as follows. In sect. 2, important preliminaries like definitions of fractional derivatives and integrals, some important relevant theorems are reviewed. In Sect. 3, general form of linear multi-term fractional differential equation with non-linear right hand side function is introduced along with its discretization using fractional integration in order to develop numerical algorithms to solve them. In the section 4, NIM is reviewed which is further used in section 5 where new NPCM-MT to solve MTFDEs is presented using NIM. In the section 6 detailed convergence analysis of the new algorithm is given showing that the new method is convergent. In sect. 7, some illustrative examples are solved to check the accuracy of new method practically. In last section 8, some important observations are made on the basis of illustrative examples and convergence analysis of newly proposed method.

Preliminaries

In this section, we recall some standard definitions of fractional derivatives and integrals.

Definition 2.1

[4] Riemann-Liouville(R-L) fractional integral operator of order \(\alpha > 0\) is denoted as \(I_{a}^{\alpha },\) and for \(f \in L^{1}[a,b]\), defined as

$$\begin{aligned} I_{a}^{\alpha }f(t)=\frac{1}{\Gamma (\alpha )}\displaystyle \int _{a}^{t}(t-\tau )^{\alpha -1}f(\tau )\,d\tau \end{aligned}$$

where \(a \le t \le b\).

Theorem 2.1

[4] For \(f \in L^{1}[a,b]\) and \(\alpha > 0\), the function \(I_{a}^{\alpha }f(t)\) is in \(L^{1}[a,b].\)

Theorem 2.2

[4] Let \(\alpha , \beta \ge 0\) and \(f \in L^{1}[a,b].\) Then

$$\begin{aligned} I_{a}^{\alpha }I_{a}^{\beta }f=I_{a}^{\alpha +\beta }f, \end{aligned}$$
(1)

almost everywhere on [a, b]. Further if f is continuous in [a, b] or \(\alpha + \beta \ge 1,\) then the equation (1) holds for all of [a, b], and hence

$$\begin{aligned} I_{a}^{\alpha }I_{a}^{\beta }f=I_{a}^{\beta }I_{a}^{\alpha }f. \end{aligned}$$

Definition 2.2

[4] The Riemann-Liouville (R-L) fractional differential operator of order \(\alpha >0 \), \(D_{a}^{\alpha }\) is defined as \(D_{a}^{\alpha }f := D^{m}\; I_{a}^{m-\alpha }\;f\) where \(m = \lceil {\alpha }\rceil \in {\mathbb {N}}\), \(\lceil . \rceil \) denotes the ceiling function.

Theorem 2.3

[4] Let \(\alpha \ge 0.\) Then for every \(f \in L^{1}[a,b],\,\,D_{a}^{\alpha }I_{a}^{\alpha }f=f\) almost everywhere.

Definition 2.3

[4] Let \(\alpha \ge 0\) and \(m=\lceil \alpha \rceil .\) Then, Caputo fractional differential operator of order \(\alpha \) is denoted as \(^{c}D_{a}^{\alpha }\) and is defined as

$$\begin{aligned} ^{c}D_{a}^{\alpha }f:= I_{a}^{m-\alpha }D^{m}f \end{aligned}$$

whenever \(D^{m}f \in L^{1}[a,b].\)

Theorem 2.4

[4] Suppose \(\alpha \ge 0\) and f is a continuous function such that \(D_{a}^{\alpha }[f-T_{m-1}[f;a]]\) exists, where \(m=\lceil \alpha \rceil \) and \(T_{m-1}[f;a]\) denotes the Taylors polynomial of degree \(m-1\) for the function f, centered at a, i.e.

$$\begin{aligned} T_{m-1}[f;a](t)=\displaystyle \sum ^{m-1}_{k=0}\frac{(t-a)^k}{k!}f^{(k)}(a). \end{aligned}$$

Then

$$\begin{aligned} ^{c}D_{a}^{\alpha }f = D_{a}^{\alpha }[f-T_{m-1}[f;a]]. \end{aligned}$$

Further we observe that \(^{c}D_{a}^{\alpha }\) is the left inverse of R-L integral, namely \(^{c}D_{a}^{\alpha }I_{a}^{\alpha }f(t)=f(t)\) (cf. Theorem 3.7 of [4]) but it is not the right inverse (cf. Theorem 3.8 of [4]), since

$$\begin{aligned} I_{a}^{\alpha }\,[^{c}D_{a}^{\alpha }f(t)]=f(t)-T_{m-1}[f;a](t). \end{aligned}$$
(2)

Observing from Lemma 2.3 of [13] and from Theorem 3.8 of [4], for any \(\beta > \alpha ,\) it holds:

$$\begin{aligned} I_{a}^{\beta }\,[^{c}D_{a}^{\alpha }f(t)]=I_{a}^{\beta }\,D_{a}^{\alpha }[f(t)-T_{m-1}[f;a](t)]=I_{a}^{\beta -\alpha }[f(t)-T_{m-1}[f;a](t)].\end{aligned}$$
(3)

This Eq. (3), will be useful, in a particular way, in the next section 3.

Linear Multi-Term Fractional Differential Equations

The general form of linear MTFDEs is described as follows:

$$\begin{aligned} \lambda _{Q}\,^{c}D_{a}^{\alpha _{Q}}y(t)+\lambda _{Q-1}\,^{c}D_{a}^{\alpha _{Q-1}}y(t)+\ldots +\lambda _{2}\,^{c}D_{a}^{\alpha _{2}}y(t)+\lambda _{1}\,^{c}D_{a}^{\alpha _{1}}y(t)=g(t,y(t)),\end{aligned}$$
(4)

where \(\lambda _{1}, \lambda _{2},\ldots , \lambda _{Q-1},\lambda _{Q}\) are real numbers, reffered as coefficients of the MTFDE (4). Order of derivatives \(\alpha _{1},\alpha _{2},\ldots ,\alpha _{Q-1},\alpha _{Q}\) are sorted in an ascending order for convenience, that is, \(0<\alpha _{1}<\alpha _{2}<\ldots<\alpha _{Q-1}<\alpha _{Q}.\) Note that \(\lambda _{Q}\ne 0.\) Here we note that Eq. (4) is a linear MTFDE with respect to fractional derivatives and right hand side function g(t, y(t)) is a non-linear function of time parameter t and y(t). We denote \(M_{Q}\) be the number of the initial conditions needed to solve MTFDE and number of initial conditions needed is given by \(M_{Q}:=\max \{m_{i}\}\) and \(m_{i}=\lceil {\alpha _{i}}\rceil ,\,\,\,i=1,2,\ldots ,Q.\) Initial conditions are given as follows:

$$\begin{aligned} y(a)=y_{0},\,\frac{d}{dt}y(a)=y^{(1)}_{0},\ldots ,\frac{d^{M_Q-1}}{dt^{M_Q-1}}y(a)=y^{(M_{Q}-1)}_{0}. \end{aligned}$$
(5)

Applying \(I_{a}^{\alpha _{Q}}\) on both sides of Eq. (4) and dividing by \(\lambda _{Q}\) on both sides, we get reformulation of Eq. (4) as follows:

$$\begin{aligned}&I_{a}^{\alpha _{Q}}\,[^{c}D_{a}^{\alpha _{Q}}y(t)]+\left( \frac{\lambda _{Q-1}}{\lambda _{Q}}\right) I_{a}^{\alpha _{Q}}\,[^{c}D_{a}^{\alpha _{Q-1}}y(t)]+\ldots +\left( \frac{\lambda _{2}}{\lambda _{Q}}\right) \\&I_{a}^{\alpha _{Q}}\,[^{c}D_{a}^{\alpha _{2}}y(t)]+\left( \frac{\lambda _{1}}{\lambda _{Q}}\right) I_{a}^{\alpha _{Q}}\,[^{c}D_{a}^{\alpha _{1}}y(t)]=\frac{1}{\lambda _{Q}}I_{a}^{\alpha _{Q}}g(t,y(t)), \end{aligned}$$

Using Eqs. (2) and (3), we have

$$\begin{aligned}&y(t)-T_{m_{Q}-1}[y;a](t)+\left( \frac{\lambda _{Q-1}}{\lambda _{Q}}\right) I_{a}^{(\alpha _{Q}-\alpha _{Q-1})}\left( y(t)-T_{m_{Q-1}-1}[y;a](t)\right) \\&\quad +\ldots +\left( \frac{\lambda _{2}}{\lambda _{Q}}\right) I_{a}^{(\alpha _{Q}-\alpha _{2})}\left( y(t) -T_{m_{2}-1}[y;a](t)\right) \\&\quad +\left( \frac{\lambda _{1}}{\lambda _{Q}}\right) I_{a}^{(\alpha _{Q}-\alpha _{1})}\left( y(t)-T_{m_{1}-1}[y;a](t)\right) =\frac{1}{\lambda _{Q}}I_{a}^{\alpha _{Q}}\left( g(t,y(t))\right) . \end{aligned}$$

Therefore,

$$\begin{aligned} y(t)=T_{m_{Q}-1}[y;a](t)-\displaystyle \sum _{i=1}^{Q-1}\frac{\lambda _{i}}{\lambda _{Q}} I^{(\alpha _{Q}-\alpha _{i})}[y(t)-T_{m_{i}-1}[y;a](t)]+\frac{1}{\lambda _{Q}}I_{a}^{\alpha _{Q}}g(t,y(t)).\nonumber \\ \end{aligned}$$
(6)

Note that

$$\begin{aligned} I_{a}^{\alpha }[T_{m-1}[y;a](t)]=I_{a}^{\alpha }\left( \displaystyle \sum ^{m-1}_{k=0}\frac{(t-a)^k}{k!}y^{(k)}(a)\right) =\displaystyle \sum ^{m-1}_{k=0}\frac{(t-a)^{k+\alpha }}{\Gamma (k+\alpha +1)}y^{(k)}(a). \end{aligned}$$
(7)

Therefore by Eq. (6), we have

$$\begin{aligned} y(t)= & {} \displaystyle \sum ^{m_{Q}-1}_{k=0}\frac{(t-a)^k}{k!}y^{(k)}(a)-\displaystyle \sum _{i=1}^{Q-1}\frac{\lambda _{i}}{\lambda _{Q}}\frac{1}{\Gamma (\alpha _{Q}-\alpha _{i})}\displaystyle \int ^{t}_{a}(t-\tau )^{\alpha _{Q}-\alpha _{i}-1}y(\tau )d\tau \nonumber \\&+\displaystyle \sum ^{Q-1}_{i=1}\frac{\lambda _{i}}{\lambda _{Q}}\displaystyle \sum ^{m_{i}-1}_{k=0}\frac{(t-a)^{k+\alpha _{Q}-\alpha _{i}}}{\Gamma (k+\alpha _{Q}-\alpha _{i}+1)}y^{(k)}(a) \nonumber \\&+\frac{1}{\lambda _{Q}}\frac{1}{\Gamma (\alpha _{Q})}\displaystyle \int ^{t}_{a}(t-\tau )^{\alpha _{Q}-1}g(\tau ,y(\tau ))\,d\tau . \end{aligned}$$
(8)

Letting \({\tilde{T}}(t)=\displaystyle \sum ^{m_{Q}-1}_{k=0}\frac{(t-a)^k}{k!}y^{(k)}(a)+\displaystyle \sum ^{Q-1}_{i=1}\frac{\lambda _{i}}{\lambda _{Q}}\displaystyle \sum ^{m_{i}-1}_{k=0}\frac{(t-a)^{k+\alpha _{Q}-\alpha _{i}}}{\Gamma (k+\alpha _{Q}-\alpha _{i}+1)}y^{(k)}(a)\) Now Eq. (8) can be rewritten as,

$$\begin{aligned} y(t)= & {} {\tilde{T}}(t)-\displaystyle \sum _{i=1}^{Q-1}\frac{\lambda _{i}}{\lambda _{Q}}\frac{1}{\Gamma (\alpha _{Q}-\alpha _{i})}\displaystyle \int ^{t}_{a}(t-\tau )^{\alpha _{Q}-\alpha _{i}-1}y(\tau )d\tau \nonumber \\&+\frac{1}{\lambda _{Q}}\frac{1}{\Gamma (\alpha _{Q})}\displaystyle \int ^{t}_{a}(t-\tau )^{\alpha _{Q}-1}g(\tau ,y(\tau ))\,d\tau . \end{aligned}$$
(9)

For solving Eqs. (4–5) on [a, T], we make partition of this interval into \({\mathfrak {q}} \in {\mathbb {N}}\) sub-parts. Here step length \(h:=\frac{T}{{\mathfrak {q}}}\) and the grid points \(t_{n}:=n \times h\) ,where \(n=0,1,2,\cdots , {\mathfrak {q}}\). Hence initial point \(t_0=a.\) We denote \(y_j\) as an approximate value of solution at the node \(t=t_j\) and \(y(t_j)\) be the true solution.

Corresponding numerical methods to solve the equivalent form namely Eq. (9) of MTFDE (4–5) is discussed in [6, 10]. We recall explicit product rectangle rule and implicit product trapezoidal rule to solve MTFDE below:

Explicit product rectangle rule is [10]:

$$\begin{aligned} y_n={\tilde{T}}(t)-\displaystyle \sum ^{Q-1}_{i=1}\frac{\lambda _i}{\lambda _Q}h^{\alpha _Q-\alpha _i}\displaystyle \sum _{j=0}^{n-1}b^{(\alpha _Q-\alpha _i)}_{n-j-1}y_j+\frac{h^{\alpha _{Q}}}{\lambda _Q}\displaystyle \sum _{j=0}^{n-1}b^{(\alpha _Q)}_{n-j-1}g(t_j,y_j) \end{aligned}$$
(10)

Implicit product trapezoidal rule is [10]:

$$\begin{aligned} y_n= & {} {\tilde{T}}(t)-\displaystyle \sum ^{Q-1}_{i=1}\frac{\lambda _i}{\lambda _Q}h^{\alpha _Q-\alpha _i}\left( {\tilde{a}}^{(\alpha _{Q}-\alpha _i)}_n y_0+\displaystyle \sum _{j=1}^{n}a^{(\alpha _Q-\alpha _i)}_{n-j}y_j\right) \nonumber \\&+\frac{h^{\alpha _{Q}}}{\lambda _Q}\left( {\tilde{a}}^{(\alpha _{Q})}_n g(t_0,y_0)+\displaystyle \sum _{j=1}^{n}a^{(\alpha _Q)}_{n-j}g(t_j,y_j)\right) \end{aligned}$$
(11)

with

$$\begin{aligned} b^{(\alpha )}_{n}=\displaystyle \frac{(n+1)^{\alpha }-n^{\alpha }}{\Gamma (\alpha +1)} \end{aligned}$$

and

$$\begin{aligned}&{\tilde{a}}_{n}^{(\alpha )}=\displaystyle \frac{(n-1)^{\alpha +1}-n^{\alpha }(n-\alpha -1)}{\Gamma (\alpha +2)},\,\,\,\, \\&\quad a^{(\alpha )}_{n}={\left\{ \begin{array}{ll} \displaystyle \frac{1}{\Gamma (\alpha +2)} &{}\text {if }n=0\\ \displaystyle \frac{(n-1)^{\alpha +1}-2n^{\alpha +1}+(n+1)^{\alpha +1}}{\Gamma (\alpha +2)}&{}\text {if } n=1,2,\ldots \end{array}\right. } \end{aligned}$$

Fractional Adams method (FAM) [6] is a combination of product rectangle rule (see Eq. (10)) and product trapezoidal rule (see Eq. (11)) to solve MTFDE [10]. In FAM, \(y_n\) obtained by using explicit product rectangle rule plays a role of a ‘Predictor’ which is then used in implicit product trapezoidal rule to give ‘Corrector’ for \(n=1,2,\ldots \) In this paper, we develop an alogorithm of NPCM-MT to solve MTFDE using implicit product trapezoidal rule and NIM [2]. Next section is devoted to recall NIM.

Overview of NIM

The NIM [2] is used to solve functional equations of the form:

$$\begin{aligned} v=h + L(v)+ N_1(v) \end{aligned}$$
(12)

where h is a known, L is a linear operator and \(N_1\) a non linear operator.

In this method, it is assumed that Eq. (12) has a solution of the form \(v=\displaystyle \sum ^{\infty }_{i=0}v_{i}.\) Since L is a linear operator, we have \(L\left( \displaystyle \sum ^{\infty }_{i=0}v_{i}\right) =\displaystyle \sum ^{\infty }_{i=0}L(v_{i})\) and non linear operator \(N_1\) is decomposed as

$$\begin{aligned} N_1(v)=N_1(v_{0})+[N_1(v_{0}+v_{1})-N_1(v_{0})]+[N_1(v_{0}+v_{1}+v_{2})-N_1(v_{0}+v_{1})]+\dots \end{aligned}$$

Note that

$$\begin{aligned}&v=v_{0}+v_{1}+v_{2}+\dots = f+L(v_0)+N_1(v_{0})+L(v_1)+[N_1(v_{0}+v_{1})\\&\quad -N_1(v_{0})]+\dots =f+L(v)+N_1(v). \end{aligned}$$

Hence v satisfies the functional Eq. (12). k-term NIM solution is given by \(v=\displaystyle \sum ^{k-1}_{i=0}v_{i}.\)

New Algorithm to Solve MTFDEs

Consider the Eq. (11) representing product trapezoidal rule. Eq. (11) can be re-written as:

$$\begin{aligned} y_n= & {} {\tilde{T}}(t)-\displaystyle \sum ^{Q-1}_{i=1}\frac{\lambda _i}{\lambda _Q}h^{\alpha _Q-\alpha _i}\left( {\tilde{a}}^{(\alpha _{Q}-\alpha _i)}_n y_0+\displaystyle \sum _{j=1}^{n-1}a^{(\alpha _Q-\alpha _i)}_{n-j}y_j+a^{(\alpha _Q-\alpha _i)}_{0}y_n\right) \nonumber \\&+\frac{h^{\alpha _{Q}}}{\lambda _Q}\left( {\tilde{a}}^{(\alpha _{Q})}_n g(t_0,y_0)+\displaystyle \sum _{j=1}^{n-1}a^{(\alpha _Q)}_{n-j}g(t_j,y_j)+a^{(\alpha _Q)}_0\,g(t_n,y_n)\right) \end{aligned}$$
(13)

Therefore,

$$\begin{aligned} y_n\left( 1+\displaystyle \sum ^{Q-1}_{i=1}\frac{\lambda _i}{\lambda _Q}h^{\alpha _Q-\alpha _i}a^{(\alpha _Q-\alpha _i)}_{0}\right)= & {} {\tilde{T}}(t)-\displaystyle \sum ^{Q-1}_{i=1}\frac{\lambda _i}{\lambda _Q}h^{\alpha _Q-\alpha _i}\left( {\tilde{a}}^{(\alpha _{Q}-\alpha _i)}_n y_0+\displaystyle \sum _{j=1}^{n-1}a^{(\alpha _Q-\alpha _i)}_{n-j}y_j\right) \\&+\frac{h^{\alpha _{Q}}}{\lambda _Q}\left( {\tilde{a}}^{(\alpha _{Q})}_n g(t_0,y_0)+\displaystyle \sum _{j=1}^{n-1}a^{(\alpha _Q)}_{n-j}g(t_j,y_j)\right) \\&\quad +\frac{h^{\alpha _{Q}}}{\lambda _Q}\,a^{(\alpha _Q)}_0\,g(t_n,y_n) \end{aligned}$$

We let \(c=1+\displaystyle \sum ^{Q-1}_{i=1}\frac{\lambda _i}{\lambda _Q}h^{\alpha _Q-\alpha _i}a^{(\alpha _Q-\alpha _i)}_{0},\) This implies that

$$\begin{aligned} y_n= & {} \frac{{\tilde{T}}(t)}{c}-\displaystyle \sum ^{Q-1}_{i=1}\frac{\lambda _i}{c \lambda _Q }h^{\alpha _Q-\alpha _i}\left( {\tilde{a}}^{(\alpha _{Q}-\alpha _i)}_n y_0+\displaystyle \sum _{j=1}^{n-1}a^{(\alpha _Q-\alpha _i)}_{n-j}y_j\right) \nonumber \\&+\frac{h^{\alpha _{Q}}}{c \lambda _Q}\left( {\tilde{a}}^{(\alpha _{Q})}_n g(t_0,y_0)+\displaystyle \sum _{j=1}^{n-1}a^{(\alpha _Q)}_{n-j}g(t_j,y_j)\right) +\frac{h^{\alpha _{Q}}}{c \lambda _Q}\,a^{(\alpha _Q)}_0\,g(t_n,y_n) \end{aligned}$$
(14)

Note that the equation (14) is of the form \(v=N_1(v)+h,\) and hence we can apply NIM to the Eq. (14), we get

$$\begin{aligned} y^{p}_{n}= & {} \frac{{\tilde{T}}(t)}{c}-\displaystyle \sum ^{Q-1}_{i=1}\frac{\lambda _i}{c \lambda _Q }h^{\alpha _Q-\alpha _i}\left( {\tilde{a}}^{(\alpha _{Q}-\alpha _i)}_n y_0+\displaystyle \sum _{j=1}^{n-1}a^{(\alpha _Q-\alpha _i)}_{n-j}y_j\right) \nonumber \\&\qquad +\frac{h^{\alpha _{Q}}}{c \lambda _Q}\left( {\tilde{a}}^{(\alpha _{Q})}_n g(t_0,y_0)+\displaystyle \sum _{j=1}^{n-1}a^{(\alpha _Q)}_{n-j}g(t_j,y_j)\right) \end{aligned}$$
(15)

Note that \(y^{p}_{n}\) is the known part. Now denote \(N_1(y_n)=\frac{h^{\alpha _{Q}}}{c \lambda _Q}\,a^{(\alpha _Q)}_0\,g(t_n,y_n).\) Set

$$\begin{aligned} z^{p}_{n}=\frac{h^{\alpha _{Q}}}{c \lambda _Q}\,a^{(\alpha _Q)}_0\,g(t_n,y^{p}_n)\end{aligned}$$
(16)

Using two predictors namely \(y^{p}_{n}\) and \(z^{p}_{n}\) we get, the corrector for the solution as

$$\begin{aligned} y^{c}_{n}=y^{p}_{n}+\frac{h^{\alpha _{Q}}}{c \lambda _Q}\,a^{(\alpha _Q)}_0\,g(t_n,y^{p}_n+z^{p}_{n})\end{aligned}$$
(17)

This is the required NPCM-MT method to solve multi-term fractional differential equations.

Convergence Analysis

In this section, we review some important theorems, which are used later in the section 6.2 for error analysis of NPCM-MT.

Preliminaries

Theorem 6.1

[6] If \(y(t) \in C^2[0,T],\) then \(\exists \) a constant \(C_{\alpha _Q-\alpha _{i}}\) depending on \(\alpha _Q-\alpha _{i},\,\,\,\forall i=1,2,\ldots Q\) such that

$$\begin{aligned}&\left| \displaystyle \int ^{t_{k+1}}_{a}(t_{k+1}-t)^{\alpha _{Q}-\alpha _{i}-1}y(t)\,dt-h^{\alpha _{Q}-\alpha _i}\Gamma {(\alpha _{Q}-\alpha _i)} \displaystyle \sum ^{k+1}_{j=0} a^{(\alpha _Q-\alpha _i)}_{k-j+1}y(t_j)\right| \nonumber \\&\quad \le C_{\alpha _Q-\alpha _{i}} ||y''||_{\infty }t^{\alpha _Q-\alpha _i}_{k+1}h^2. \end{aligned}$$
(18)

Theorem 6.2

[6] If \(y(t) \in C^1[0,T],\) and \(y'\) satisfies Lipschitz condition of order \(\nu \) where \(0< \nu <1.\) With above conditions, there exist \(M_{y,\nu }\) depending only on y and \(\nu \), and \(B_{(\alpha _Q-\alpha _{i}), \nu }\) (depending on \(\alpha _Q-\alpha _{i} ,\,\,\,\forall i=1,2,\ldots Q \) and on \(\nu \)) such that

$$\begin{aligned}&\left| \displaystyle \int ^{t_{k+1}}_{a}(t_{k+1}-t)^{\alpha _{Q}-\alpha _{i}-1}y(t)\,dt-h^{\alpha _{Q}-\alpha _i}\Gamma {(\alpha _{Q}-\alpha _i)} \displaystyle \sum ^{k+1}_{j=0} a^{(\alpha _Q-\alpha _i)}_{k-j+1}y(t_j)\right| \nonumber \\&\quad \le B_{(\alpha _Q-\alpha _{i}), \nu } M_{y,\nu } t^{\alpha _Q-\alpha _i}_{k+1}h^{1+\nu }. \end{aligned}$$
(19)

Theorem 6.3

[6] If \(y(t)=t^p,\) for some \(0< p <2\) and let q be the minimum of 2 and \(p+1\) then

$$\begin{aligned}&\left| \displaystyle \int ^{t_{k+1}}_{a}(t_{k+1}-t)^{\alpha _{Q}-\alpha _{i}-1}y(t)\,dt-h^{\alpha _{Q}-\alpha _i}\Gamma {(\alpha _{Q}-\alpha _i)} \displaystyle \sum ^{k+1}_{j=0} a^{(\alpha _Q-\alpha _i)}_{k-j+1}y(t_j)\right| \nonumber \\&\quad \le A_{(\alpha _Q-\alpha _{i}),p} t^{\alpha _Q-\alpha _i+p-q}_{k+1}h^{q};\, \end{aligned}$$
(20)

where \(A_{(\alpha _Q-\alpha _{i}),p} \) is a constant dependent on \(\alpha _Q-\alpha _i\) and \(p\, \forall i=1,2,\ldots Q.\)

Based on above error estimates given in [6], further we present a general convergence analysis for NPCM-MT.

Main Results

Theorem 6.4

Let \(g(t,y(t)) \le M'\) where \(M'\) is a positive constant and assume that the solution of IVP (4-5) namely \(y(t) \in C^2[0,T]\) , then for given \(\alpha _Q\) and \(m_Q,\) and for suitably choosen \(T>0,\) we have

$$\begin{aligned} \max |y(t_{n+1})-y_{n+1}| \le kh^{\delta }, \end{aligned}$$

where k is a positive constant and \(\delta =\min \{m_Q-1,2,\alpha _Q\};\,\,\forall n \in {\mathbb {N}}.\) In this case, \(y(t_j):\) analytic solution of IVP (4-5) and \(y_j:\) solution of IVP (4-5) yielded by NPCM-MT.

Proof

At \((n+1)\)th step, \(|y(t_{n+1})-y^{p}_{n+1}|=\)

$$\begin{aligned}&\left| {\tilde{T}}(t_{n+1}) -\displaystyle \sum _{i=1}^{Q-1}\frac{\lambda _{i}}{\lambda _{Q}}\frac{1}{\Gamma (\alpha _{Q} -\alpha _{i})}\displaystyle \int ^{t_{n+1}}_{a}(t-\tau )^{\alpha _{Q} -\alpha _{i}-1}y(\tau )d\tau +\frac{1}{\lambda _{Q}}\frac{1}{\Gamma (\alpha _{Q})} \right. \nonumber \\&\quad \displaystyle \int ^{t_{n+1}}_{a}(t-\tau )^{\alpha _{Q}-1}g(\tau ,y(\tau ))\, d\tau -\frac{{\tilde{T}}(t_{n+1})}{c}\nonumber \\&\quad +\displaystyle \sum ^{Q-1}_{i=1}\frac{\lambda _i}{c \lambda _Q }h^{\alpha _Q-\alpha _i}\left( {\tilde{a}}^{(\alpha _{Q}-\alpha _i)}_{n+1} y_0+\displaystyle \sum _{j=1}^{n+1}a^{(\alpha _Q-\alpha _i)}_{n-j+1}y_j\right) \nonumber \\&\quad \left. -\frac{h^{\alpha _{Q}}}{c \lambda _Q}\left( {\tilde{a}}^{(\alpha _{Q})}_{n+1} g(t_0,y_0)+\displaystyle \sum _{j=1}^{n+1}a^{(\alpha _Q)}_{n-j+1}g(t_j,y_j)\right) \right| \end{aligned}$$
(21)
$$\begin{aligned}&\quad =\left| \left( 1-\frac{1}{c}\right) {\tilde{T}}(t_{n+1}) -\left( \displaystyle \sum ^{Q-1}_{i=1}\frac{\lambda _{i}}{\lambda _{Q}}\left( 1-\frac{1}{c}\right) \frac{1}{\Gamma (\alpha _{Q}-\alpha _{i})}\right. \right. \nonumber \\&\left. \left( \displaystyle \int ^{t_{n+1}}_{a}(t-\tau )^{\alpha _{Q} -\alpha _{i}-1}y(\tau )d\tau -h^{\alpha _Q-\alpha _i}\Gamma (\alpha _{Q}\!-\!\alpha _{i}) \sum _{j=0}^{n+1}a^{(\alpha _Q-\alpha _i)}_{n-j+1}y_j\!\right) \right) \!+\! \frac{(1-1/c)1/\lambda _Q}{\Gamma {\alpha _Q}} \nonumber \\&\left. \left( \displaystyle \int ^{t_{n+1}}_{a}(t-\tau )^{\alpha _{Q}-1}g(\tau ,y(\tau ))d\tau -h^{\alpha _Q} \Gamma (\alpha _{Q})\sum _{j=0}^{n+1}a^{(\alpha _Q)}_{n-j+1}g(t_j,y_j)\right) \right| \end{aligned}$$
(22)
Table 1 Example (7.1) Note: Error obtained by NPCM-MT is less than error obtained by FAM
Full size table

Let \((1-1/c)=c'\) and note that \({\tilde{T}}(t)\le c'' h^{m_Q-1},\) Then by Theorem 6.1,

$$\begin{aligned}&\left| y(t_{n+1})-y^{p}_{n+1}) \right| \le c'\,c''h^{m_Q-1}+\displaystyle \sum ^{Q-1}_{i=1}\frac{\lambda _{i}}{\lambda _{Q}} c'\frac{1}{\Gamma (\alpha _{Q}-\alpha _{i})}C_{\alpha _Q-\alpha _{i}} ||y''||_{\infty }t^{\alpha _Q-\alpha _i}_{k+1}h^2\nonumber \\&\quad +\frac{c'}{\lambda _Q\,\Gamma {(\alpha _Q)}} C_{\alpha _Q} ||g''||_{\infty }t^{\alpha _Q}_{k+1}h^2 \le C''h^{m_Q-1}+c'''h^2. \end{aligned}$$
(23)

Now consider,

$$\begin{aligned}&\left| y(t_{n+1})-(y^{p}_{n+1}+z^{p}_{n+1})\right| \le \left| y(t_{n+1})-y^{p}_{n+1}\right| +\left| z^{p}_{n+1}\right| \le C''h^{m_Q-1}+c'''h^2 \nonumber \\&+\frac{h^{\alpha _Q}}{c \lambda _Q}a^{(\alpha _Q)}_0 M'. \end{aligned}$$
(24)

Further consider,

$$\begin{aligned}&\left| y(t_{n+1})-y^{c}_{n+1}\right| =\left| y(t_{n+1})-y^{p}_{n+1}-\frac{h^{\alpha _Q}}{c \lambda _Q}a^{(\alpha _Q)}_{0}g(t_{n+1},y^{p}_{n+1}+z^{p}_{n+1})\right| \nonumber \\&\quad \le |y(t_{n+1})-y^{p}_{n+1}|+\frac{h^{\alpha _Q}}{c \lambda _Q}a^{(\alpha _Q)}_{0} M'\le C''h^{m_Q-1}+c'''h^2+c^{(iv)} h^{\alpha _Q}\le k'h^{\delta }.\qquad \end{aligned}$$
(25)

Where, \(\delta =\min \{m_Q-1, 2, \alpha _Q\}.\) Hence the proof. \(\square \)

Theorem 6.5

Let \(g(t,y(t)) \le M'\) where \(M'\) is a positive constant and assume that the solution \(y(t)=t^p,\,0< p < 2\) and \(q=\min (2,p+1)\) of IVP (4-5), then for appropriately chosen \(T>0,\,\,\exists \,\, \text {a constant}\,\, A_{(\alpha _Q-\alpha _i)} \) such that

$$\begin{aligned} \max |y(t_{n+1})-y_{n+1}| \le k^{(iv)}h^{\delta _4}, \end{aligned}$$

where \(k^{(iv)}\) is a positive constant and \(\delta _4=\min \{q,m_Q-1,\alpha _Q\}\,\,\forall n \in N.\)

Proof

At \((n+1)\)th step,

$$\begin{aligned}&|y(t_{n+1})-y^{p}_{n+1}| \le c''h^{m_Q-1}+c^{(v)}B_{(\alpha _Q-\alpha _i),\nu }M_{y,\nu } t_{k+1}^{\alpha _Q-\alpha _i}h^{1+\nu }\\&\quad +c^{(vi)}B_{(\alpha _Q),\nu }M_{g,\nu } t_{k+1}^{\alpha _Q}h^{1+\nu }\le k'' h^{\delta _1} \end{aligned}$$

Where \(\delta _1=\min (1+\nu ,m_Q-1).\) and

$$\begin{aligned} |y(t_{n+1})-y^{c}_{n+1}| \le k'' h^{\delta _1}+\frac{h^{\alpha _Q}}{c \lambda _Q}a^{\alpha _Q}_0 M' \le k''' h^{\delta _2}\end{aligned}$$

, where \(\delta _2=\min \{\delta _1, \alpha _Q\}.\) \(\square \)

Theorem 6.6

Let \(g(t,y(t)) \le M'\) where \(M'\) is a positive constant and assume that the solution \(y(t)=t^p,\) \(0<p<2\) and \(q=\min \left\{ 2,p+1\right\} \) of IVP (4-5), then for suitably choosen \(T>0,\,\,\exists \,\, \text {a constant}\,\, A_{(\alpha _Q-\alpha _i)} \) such that

$$\begin{aligned} \max |y(t_{n+1})-y_{n+1}| \le k^{(iv)}h^{\delta _4}, \end{aligned}$$

where \(k^{(iv)}\) is a positive constant and \(\delta _4=\min \{q,m_Q-1,\alpha _Q\}\,\,\forall n \in N.\)

Proof

At \((n+1)\)th step,

$$\begin{aligned}&|y(t_{n+1})-y^{p}_{n+1}| \le c''h^{m_Q-1}+c^{(vii)}A_{(\alpha _Q-\alpha _i)} t^{\alpha _Q-\alpha _i+p-q}h^{q}\\&\quad +c^{(viii)}A_{(\alpha _Q)} t^{\alpha _Q+p-q}h^{q}\le k''' h^{\delta _3}\end{aligned}$$

Where \(\delta _3=\min (q,m_Q-1).\) and

$$\begin{aligned} |y(t_{n+1})-y^{c}_{n+1}| \le k''' h^{\delta _3}+\frac{h^{\alpha _Q}}{c \lambda _Q}a^{\alpha _Q}_0 M' \le k^{(iv)} h^{\delta _4}\end{aligned}$$

, where \(\delta _4=\min \{q,m_Q-1, \alpha _Q\}.\)

Next We let the operator

$$\begin{aligned} D^{*}=\lambda _{Q}\,^{c}D_{a}^{\alpha _{Q}}+\lambda _{Q-1}\,^{c}D_{a}^{\alpha _{Q-1}}+\ldots +\lambda _{2}\,^{c}D_{a}^{\alpha _{2}}+\lambda _{1}\,^{c}D_{a}^{\alpha _{1}} \end{aligned}$$

Using Theorem 5.4 from [3], we have following result for error bound for NPCM-MT to solve MTFDEs. \(\square \)

Theorem 6.7

Suppose that the solution y(t) of the initial value problem (4), satisfies following conditions:

$$\begin{aligned} \left| \displaystyle \int ^{t_{k+1}}_{a}(t_{k+1}-t)^{\alpha _{Q}-1}\,D^{*}y(t)\,dt-h^{\alpha _{Q}}\,\,\Gamma (\alpha _Q) \displaystyle \sum ^{k}_{j=0} a_{k-j+1}^{(\alpha _Q)}D^{*}y(t_j)\,dt\right| \le C_Q t^{\gamma _Q}_{k+1}h^{\delta _Q}. \end{aligned}$$

for some \(\gamma _Q \ge 0\) and \(\delta _Q > 0\) and g(t, y(t)) satisfies Lipschitz condition in the second variable with Lipschitz constant L. Then for \(T>0,\) we have \(\displaystyle \max _{0 \le j \le N}|y(t_j)-y_j| \le C h^{\delta _Q},\) where \(N=\frac{T}{h}\) and C is an arbitrary positive constant. In this case \(y(t_j):\) analytic solution of the IVP (4-5) and \(y_j:\) solution of the IVP (4 - 5) yielded by NPCM-MT.

Theorem 6.8

Let \(D^{*}y(t) \in C^{2}[0, T],\,\,T>0,\) then \( \displaystyle \max _{0 \le j \le N} \, |y(t_j)-y_j|=O({h^2}).\)

Proof: Proof follows immediately from Theorem (6.1) and Theorem (6.7).

Fig. 1
figure 1

a Solid graph: Solution by FAM, Dashed graph: Exact solution, Dotted graph: solution by NPCM-MT. It is noted that in a exact solution and approximate solutions by the NPCM-MT and by FAM overlaps on each other. In b, error by FAM and error by NPCM-MT is plotted. Thick error graph: error by FAM and Thin error graph: error by NPCM-MT. It has been observed that error obtained by NPCM-MT is less than error in FAM. Step length is taken as \(h=0.01\)

Full size image

Illustrative Examples

We solve some MTFDEs using FAM and using NPCM-MT and compare results obtained by both methods. Following illustrative examples shows that new method viz. NPCM-MT is more accurate than FAM.

Example 7.1

Consider the multi-term fractional differential equation

$$\begin{aligned} y'''(t)+\,^{c}D_{0}^{\frac{5}{2}}y(t)+y''(t)+4y'(t)+\,^{c}D_{0}^{\frac{1}{2}}y(t)+4y(t)= & {} 6 \cos (t),\nonumber \\ \, y_0 = y^{(1)}_0=1,\,y^{(2)}_{0}=-1. \end{aligned}$$
(26)

Exact solution of this MTFDE is \(y(t)=\sqrt{2}\sin (t+\frac{\pi }{4}).\) Table 1 shows the absolute error between the exact solution and solutions obtained by FAM and NPCM-MT methods. Figure 1 shows the plot of the exact solution and solutions obtained by FAM and NPCM-MT along with their corresponding errors.

The CPU-time required for solving Example 7.1 by FAM (\(T_1\)) and by NPCM-MT (\(T_2\)) is compared in the Table 2. For this calculation step length \(h=0.01\).

From the observations in Table 2, we conclude that NPCM-MT is more time efficient than FAM.

Table 3 below shows absolute errors by FAM and NPCM-MT obtained while solving Example 7.1. These errors are compared against various values of h. In these calculations, \(t=1\).

From the observations of Table 3 it is noteworthy that absolute error in NPCM-MT is significantly smaller than error in FAM for all values of h. Further it is clear that as h reduces, absolute error also reduces.

Example 7.2

Consider the Bogley-Torvik equation [5] given below:

$$\begin{aligned} Ay''(t)+B\,^{c}D_{0}^{3/2}y(t)+Cy(t)= & {} t+1,\nonumber \\\, y_0 = y^{(1)}_0=1. \end{aligned}$$
(27)

For any choice of \(A,\, B,\, C\), exact solution is \(y(t)=t+1.\) Here we choose \(A=B=C=1, \) and solve the equation on the time interval [0, 5] with step length \(h=0.01.\) (Fig. 2).

Fig. 2
figure 2

a Solid graph: Solution by FAM, Dashed graph: Exact solution, Dotted graph: solution by NPCM-MT. It is noted that in a exact solution and approximate solutions by the NPCM-MT and by FAM overlaps on each other. In b, error by FAM and error by NPCM-MT is plotted. Thick error graph: error by FAM and Thin error graph: error by NPCM-MT. It has been observed that error obtained by NPCM-MT ( is almost zero as graph of error obtained by NPCM-MT is coinciding with X-axis) is less than error in FAM. Step length is taken as \(h=0.01\)

Full size image
Fig. 3
figure 3

a Solid graph: Solution by FAM, Dashed graph: Exact solution, Dotted graph: solution by NPCM-MT. It is noted that in a exact solution and approximate solutions by the NPCM-MT and by FAM overlaps on each other. In b, error by FAM and error by NPCM-MT is plotted. Thick error graph: error by FAM and Thin error graph: error by NPCM-MT. It has been observed that error obtained by NPCM-MT ( is almost zero ) is less than error in FAM. Step length is taken as \(h=0.01\)

Full size image
Fig. 4
figure 4

a Solid graph: Solution by FAM, Dashed graph: Exact solution, Dotted graph: solution by NPCM-MT. It is noted that in a exact solution and approximate solutions by the NPCM-MT and by FAM overlaps on each other. In b, error by FAM and error by NPCM-MT is plotted. Thick error graph: error by FAM and Thin error graph: error by NPCM-MT. It has been observed that error obtained by NPCM-MT (is almost zero as graph of error obtained by NPCM-MT is coinciding with X-axis) is less than error in FAM. Step length is taken as \(h=0.01\)

Full size image

Example 7.3

Consider the MTFDE given below:

$$\begin{aligned}&^{c}D_{0}^{5/2}y(t)+\,^{c}D_{0}^{2}y(t)-2\sqrt{\pi }\,^{c}D_{0}^{1/2}y(t)+4y(t)= g(t),\nonumber \\&\quad \, y_0 = y^{(1)}_0= y^{(2)}_0=0\nonumber \\&\quad \text {R.H.S function}\,\, g(t)=4t^9-\frac{131072}{12155}t^{\frac{17}{2}}+72\,t^7+\frac{49152}{143\sqrt{\pi }}t^{\frac{13}{2}}. \end{aligned}$$
(28)

Exact solution is \(y(t)=t^9.\) We solve the equation on the time interval [0, 1] with step length \(h=0.01.\) Figure 3 shows the plot of exact solution and solutions obtained by FAM and NPCM-MT along with their corresponding errors.

Example 7.4

Consider the MTFDE given below:

$$\begin{aligned}&\displaystyle ^{c}D_{0}^{2}y(t)+\,^{c}D_{0}^{1/2}y(t)+y(t)= g(t),\nonumber \\&\displaystyle y_0 = y^{(1)}_0=0\nonumber \\&\displaystyle \text {R.H.S function}\,\, g(t)=t^3+6t+\frac{3.2\,t^{2.5}}{\Gamma {(0.5)}}. \end{aligned}$$
(29)

Exact solution is \(y(t)=t^3.\) We solve the equation on the time interval [0, 1] with step length \(h=0.01.\) Figure 4 shows the plot of exact solution and solutions obtained by FAM and NPCM-MT along with their corresponding errors.

Table 2 Example (7.1) CPU-time comparison between FAM (\(T_1\)) and NPCM-MT (\(T_2\))
Full size table
Table 3 Example (7.1) Absolute error comparison between FAM and NPCM-MT
Full size table

Conclusions

A generalized version of ‘New predictor corrector method (NPCM-MT)’ is developed for solving linear multi-term fractional differential equations. NPCM-MT is then compared with existing method viz. ‘Fractional Adams method’ with respect to accuracy and time-efficiency. NPCM-MT is found to be more accurate and time-efficient than existing method. Convergence analysis of NCPM-MT is done and numerous illustrative examples are solved. As a conclusion we found that, NPCM-MT is more accurate than the existing method. Mathematica 12 has been used for numerical computations and graphical representations in this article.