Comparison on Solving a Class of Nonlinear Systems of Partial Differential Equations and Multiple Solutions of Second Order Differential Equations

Abstract

We apply the reproducing kernel Hilbert space method to a class of nonlinear systems of partial differential equations and to get multiple solutions of second order differential equations. We have reached meaningful results. These results have been depicted by figures. This method is a very impressive technique for solving nonlinear systems of partial differential equations and second order differential equations.

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1 Introduction

Many problems in science and engineering such as problems posed in solid state physics, fluid mechanics, chemical physics, plasma physics, optics, etc. are modelled as nonlinear partial differential equations (PDEs) or systems of nonlinear PDEs. Nonlinear systems of PDEs have taken much interest in working evolution equations. Many researchers have investigated the analytical and approximate solutions of nonlinear systems of PDEs by utilizing different techniques [7].

In this paper, a general technique is shown in the reproducing kernel space for searching the following class of nonlinear systems of PDEs:

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} A_1(f_1(\eta,\tau))&\displaystyle =&\displaystyle P_1(\eta,\tau,F(\eta,\tau))+M_1(\eta,\tau), \\ &\displaystyle &\displaystyle \ldots \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad ,\\ A_k(f_k(\eta,\tau))&\displaystyle =&\displaystyle P_k(\eta,\tau,F(\eta,\tau))+M_k(\eta,\tau), \\ (\eta,\tau) \in \varOmega&\displaystyle =&\displaystyle [0,1]\times [0,1] \end{array} \end{aligned} $$

with the initial and boundary conditions

$$\displaystyle \begin{aligned} f_i(\eta,\tau)=0, \quad \text{for} \; \tau \leq 0, \end{aligned} $$

(8.1)

$$\displaystyle \begin{aligned} f_i(0,\tau)=0, \quad f_i(1,\tau)=0, \quad \text{for} \; \tau >0. \end{aligned} $$

(8.2)

where A _i and P _i are linear and nonlinear differential operators for i = 1, 2, …, k. M _i(η, τ) are given functions and F(η, τ) = [f ₁(η, τ), f ₂(η, τ), …, f _k(η, τ)]^T is an unknown vector function to be determined. Suppose this equation is of one-order derivative in τ, and has a unique solution. We only take into consideration the homogeneous initial and boundary conditions, because the non-homogeneous initial and boundary conditions can be easily transformed to the homogeneous ones. The reproducing method has been implemented to several nonlinear problems [1]. For more details of this method, see [3, 6, 8].

We take into consideration the boundary value problems:

$$\displaystyle \begin{aligned} \begin{cases} u''(x)=\lambda\exp(\mu u(x)),\quad 0\leq x \leq 1, \\ u(0)=u(1)=0, \end{cases} \end{aligned} $$

(8.3)

and

$$\displaystyle \begin{aligned} \left(\exp(x) v'(x) \right)' + \vert \ln x \vert& =0, \forall x \in (0,\infty), x \neq 1, \varDelta v' \mid_{x_1=1}=v^2(1), \\ v(0)&=0, \quad v(\infty)=0. \end{aligned} $$

(8.4)

The problem (8.3) shows up in implementations containing the diffusion of heat produced by positive temperature-dependent sources. If μ = 1, it springs in the analysis of Joule losses in electrically conducting solids, with λ returning the square of constant current and $\exp (u)$ the temperature-dependent resistance, or frictional heating with λ projecting the square of the constant shear stress and $\exp (u)$ the temperature-dependent fluidity. In particular if λ = 1 and μ = −1 the boundary value problem (8.3) has two solutions u ₁(x) and u ₂(x). Solution u ₁(x) drops below up to − 0.14050941… and solution u ₂(x) up to − 4.0916146….

Boundary value problem (8.4) has at least two positive solutions v ₁, v ₂ satisfying $0 \leq \Vert v_1 \Vert \leq \frac {1}{2} \leq \Vert v_2 \Vert .$

This work is ordered as follows. Section 8.2 presents some useful reproducing kernel functions. The representation of solutions and a related linear operator are given in Sect. 8.3. This section shows the main results. Examples are shown in Sect. 8.4. The final section contains some conclusions.

2 Preliminaries

Definition 1

We present $G_2^1[0,1]$ by

$$\displaystyle \begin{aligned} G_2^1[0,1]=\{f \in AC[0,1]:\; f'\in L^2[0,1]\}. \end{aligned}$$

The inner product and the norm in $G_2^1[0,1]$ are defined by

$$\displaystyle \begin{aligned} \left\langle f,g \right\rangle_{G_2^1} =f(0)g(0)+\int_0^1 f'(\eta)g'(\eta)\mathrm{d}\eta, \quad f,g\in G_2^1[0,1] \end{aligned}$$

and

$$\displaystyle \begin{aligned} \left\Vert f \right\Vert_{G_2^1} =\sqrt{\left\langle f,f \right\rangle_{G_2^1}}, \quad f \in G_2^1[0,1]. \end{aligned}$$

Theorem 1

Reproducing kernel function $\widetilde {Q}_\tau$ of $G_2^1[0,1]$ is obtained as:

$$\displaystyle \begin{aligned} \widetilde{Q}_\tau(\eta )= \sum_{i=1}^2 c_i(\tau)\eta^{i-1},\quad 0 \leq \eta \leq \tau \leq 1, \sum_{i=1}^2 d_i(\tau) \eta ^{i-1},\quad 0 \leq \tau < \eta \leq 1. \end{aligned}$$

Proof

By Definition 1, we have

$$\displaystyle \begin{aligned} \left\langle u,\widetilde{Q}_\tau \right\rangle_{G_2^1} =u(0)\widetilde{Q}_\tau(0)+\int_0^1 u'(\eta )\widetilde{Q}^{\prime}_\tau(\eta)\mathrm{d}\eta , \end{aligned} $$

(8.5)

We get

$$\displaystyle \begin{aligned} \left\langle u,\widetilde{Q}_\tau \right\rangle_{G_2^1} =u(0)\widetilde{Q}_\tau(0)+u(1)\widetilde{Q}^{\prime}_\tau(1)-u(0)\widetilde{Q'}_\tau(0)-\int_0^1 u(\eta )\widetilde{Q}^{\prime \prime}_\tau(\eta )\mathrm{d}\eta, \end{aligned}$$

by integrating by parts. Note that property of the reproducing kernel is

$$\displaystyle \begin{aligned} \langle u(\eta ),\widetilde{Q}^{\prime}_\tau(\eta)\rangle_{G_2^1}=u(\tau). \end{aligned} $$

(8.6)

If

$$\displaystyle \begin{aligned} \begin{cases} \widetilde{Q}_\tau(0)-\widetilde{Q}^{\prime}_\tau(0)=0,\\ \widetilde{Q}^{\prime}_\tau(1)=0, \end{cases} \end{aligned} $$

(8.7)

then, we get

$$\displaystyle \begin{aligned} -\widetilde{Q}^{\prime \prime}_\tau(\eta )=\delta (\eta -\tau). \end{aligned}$$

If η ≠ τ, then we obtain

$$\displaystyle \begin{aligned} \widetilde{Q}^{\prime \prime}_\tau(\eta)=0. \end{aligned}$$

Thus, we have

$$\displaystyle \begin{aligned} \widetilde{Q}_\tau(\eta )=\begin{cases} c_1(\tau)+c_2(\tau)\eta ,\quad 0 \leq \eta \leq \tau \leq 1,\\ \\ d_1(\tau)+d_2(\tau)\eta ,\quad 0 \leq \tau < \eta \leq 1. \end{cases} \end{aligned} $$

(8.8)

Since

$$\displaystyle \begin{aligned} -\widetilde{Q}^{\prime \prime}_\tau(\eta )=\delta (\eta -\tau), \end{aligned}$$

we get

$$\displaystyle \begin{aligned} \widetilde{Q}_{\tau^+}(\tau)= \widetilde{Q}_{\tau^-}(\tau) \end{aligned} $$

(8.9)

and

$$\displaystyle \begin{aligned} \widetilde{Q}^{\prime}_{\tau^+}(\tau)- \widetilde{Q}^{\prime}_{\tau^-}(\tau)=-1. \end{aligned} $$

(8.10)

The unknown coefficients c _i(τ) and d _i(τ) (i = 1, 2) can be obtained. Thus $\widetilde {Q}_\tau$ is acquired as

$$\displaystyle \begin{aligned} \widetilde{Q}_\tau(\eta )=1+\eta ,\quad 0 \leq \eta \leq \tau \leq 1, \; 1+\tau,\quad 0 \leq \tau < \eta \leq 1. \end{aligned}$$

Definition 2

We present the space $H_2^2[0,1]$ as:

$$\displaystyle \begin{aligned} H_2^2[0,1]=\{f \in AC[0,1]:\; f'\in AC[0,1],\; f'' \in L^{2}[0,1], \; f(0)=0 \}. \end{aligned}$$

The inner product and the norm in $H_2^2[0,1]$ are presented as:

$$\displaystyle \begin{aligned} \left\langle f,g \right\rangle_{H_2^2} =f(0)g(0)+f'(0)g'(0)+\int_0^1 f''(\eta)g''(\eta)\mathrm{d}\eta, \quad f,g \in H_2^2[0,1] \end{aligned}$$

and

$$\displaystyle \begin{aligned} \left\Vert f \right\Vert_{H_2^2} =\sqrt{\left\langle f,f \right\rangle_{H_2^2}}, \quad f \in H_2^2[0,1]. \end{aligned}$$

Theorem 2

Reproducing kernel function $\widetilde {T}_\tau$ of $H_2^2[0,1]$ is obtained by:

$$\displaystyle \begin{aligned} \widetilde{T}_\tau(\eta)= \sum_{i=1}^4 c_i(\tau)\eta^{i-1} ,\quad 0 \leq \eta \leq \tau \leq 1,\; \sum_{i=1}^4 d_i(\tau)\eta^{i-1} ,\quad 0 \leq \tau < \eta \leq 1.\end{aligned} $$

Proof

By Definition 2, we have

$$\displaystyle \begin{aligned} \left\langle f,\widetilde{T}_\tau \right\rangle_{H_2^2} =f(0)\widetilde{T}_\tau(0)+f'(0)\widetilde{T}^{\prime}_\tau(0)+\int_0^1 f''(\eta)\widetilde{T}^{\prime \prime}_\tau(\eta)\mathrm{d}\tau.\end{aligned} $$

(8.11)

Integrating this equation by parts two times, we get

$$\displaystyle \begin{aligned} \begin{array}{rcl} \left\langle f,\widetilde{T}_\tau \right\rangle_{H_2^2} &\displaystyle =&\displaystyle f(0)\widetilde{T}_\tau(0)+f'(0)\widetilde{T}^{\prime}_\tau(0)+f'(1)\widetilde{T}^{\prime \prime}_\tau(1)-f'(0)\widetilde{T}^{\prime \prime}_\tau(0) \\ &\displaystyle &\displaystyle -f(1)\widetilde{T}^{\prime \prime \prime}_\tau(1)+f(0)\widetilde{T}^{\prime \prime \prime}_\tau(0) +\int_0^1 f(\eta)\widetilde{T}^{(4)}_\tau(\eta)\mathrm{d}\eta.\vspace{-4pt} \end{array} \end{aligned} $$

We have

$$\displaystyle \begin{aligned} \langle f(\eta),\widetilde{T}_\tau(\eta)\rangle_{H_2^2}=f(\tau)\end{aligned} $$

(8.12)

by reproducing property. Since $\widetilde {T}_\tau \in H_2^2[0,1],$ we have

$$\displaystyle \begin{aligned} \widetilde{T}_\tau(0)=0. \end{aligned} $$

(8.13)

If

$$\displaystyle \begin{aligned} \begin{cases} \widetilde{T}^{\prime}_\tau(0)-\widetilde{T}^{\prime \prime}_\tau(0)=0,\\ \widetilde{T}^{\prime \prime}_\tau(1)=0,\\ \widetilde{T}^{\prime \prime \prime}_\tau(1)=0, \end{cases} \end{aligned} $$

(8.14)

then, we get

$$\displaystyle \begin{aligned} \widetilde{T}^{(4)}_\tau(\eta)=\delta (\eta-\tau). \end{aligned}$$

When η ≠ τ, we get

$$\displaystyle \begin{aligned} \widetilde{T}^{(4)}_\tau(\eta)=0. \end{aligned}$$

Thus

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} \widetilde{Q}_\tau(\eta)&\displaystyle =&\displaystyle c_1(\tau)+c_2(\tau)\eta+c_3(\tau)\eta^2++c_4(\tau)\eta^3,\quad 0 \leq \eta \leq \tau \leq 1,\\ \\ &\displaystyle &\displaystyle d_1(\tau)+d_2(\tau)\eta+d_3(\tau)\eta^2+d_4(\tau)\eta^3, \quad 0 \leq \tau < \eta \leq 1. \end{array} \end{aligned} $$

Since

$$\displaystyle \begin{aligned} \widetilde{T}^{(4)}_\tau(\eta)=\delta (\eta-\tau), \end{aligned}$$

we obtain

$$\displaystyle \begin{aligned} \widetilde{T}^{(k)}_{\tau^+}(\tau)= \widetilde{T}^{(k)}_{\tau^-}(\tau), \quad k=0,1,2 \end{aligned} $$

(8.15)

and

$$\displaystyle \begin{aligned} \widetilde{T}^{\prime \prime \prime}_{\tau^+}(\tau)- \widetilde{T}^{\prime \prime \prime}_{\tau^-}(\tau)=1. \end{aligned} $$

(8.16)

The unknown coefficients c _i(τ) and d _i(τ) (i = 1, 2, 3, 4) can be obtained. Thus $\widetilde {T}_\tau$ is achieved as

$$\displaystyle \begin{aligned} \begin{array}{rcl} \widetilde{T}_\tau(\eta)&\displaystyle =&\displaystyle \eta \tau+\frac{(\eta)(\tau)^2}{2} +\frac{(\tau-\eta)^3}{6}-\frac{(\tau)^3}{6} ,\quad 0 \leq \eta \leq \tau \leq 1,\;\\ &\displaystyle &\displaystyle \eta \tau+\frac{(\tau)(\eta)^2}{2} +\frac{(\eta-\tau)^3}{6}-\frac{(\tau)^3}{6} ,\quad 0 \leq \tau < \eta \leq 1. \end{array} \end{aligned} $$

Definition 3

We give $W_2^3[0,1]$ as:

$$\displaystyle \begin{aligned} \begin{array}{rcl} W_2^3[0,1]&\displaystyle =&\displaystyle \{f \in AC[0,1]:\; f',f''\in AC[0,1], \; f^{(3)}\in L^{2}[0,1],\; \\ f(0)&\displaystyle =&\displaystyle f(1)=0\}. \end{array} \end{aligned} $$

The inner product and the norm in $W_2^3[0,1]$ are defined by

$$\displaystyle \begin{aligned} \left\langle f,g \right\rangle_{W_2^3} =\sum_{i=0}^2f^{(i)}(0)g^{(i)}(0) +\int_{0}^1f^{(3)}(\eta)g^{(3)}(\eta)\mathrm{d}\eta, \quad f,g \in W_2^3[0,1] \end{aligned}$$

and

$$\displaystyle \begin{aligned} \left\Vert f \right\Vert_{W_2^3} =\sqrt{\left\langle f,f \right\rangle_{W_2^3}}, \quad f \in W_2^3[0,1]. \end{aligned}$$

Theorem 3

Reproducing kernel function R _τ of $W_2^3[0,1]$ is obtained as:

$$\displaystyle \begin{aligned} R_\tau(\eta)=\begin{cases} \sum_{i=1}^5c_i(\tau) \eta^i,\quad 0 \leq \eta \leq \tau \leq 1,\\ \\ \sum_{i=0}^5d_i(\tau) \eta^i,\quad 0 \leq \tau < \eta \leq 1, \end{cases} \end{aligned} $$

(8.17)

where

$$\displaystyle \begin{aligned} \begin{array}{rcl} c_1(\tau)&\displaystyle =&\displaystyle -\frac{1}{156}\tau^{5}+\frac{5}{156}\tau^{4}-\frac{5}{78}\tau^{3} -\frac{5}{26}\tau^{2}+\frac{3}{13}\tau,\\ c_2(\tau)&\displaystyle =&\displaystyle -\frac{1}{624}\tau^{5}+\frac{5}{624}\tau^{4}-\frac{5}{312}\tau^{3} +\frac{21}{104}\tau^{2}-\frac{5}{26}\tau,\\ c_3(\tau)&\displaystyle =&\displaystyle -\frac{1}{1872}\tau^{5}+\frac{5}{1872}\tau^{4}-\frac{5}{936}\tau^{3} +\frac{7}{104}\tau^{2}-\frac{5}{78}\tau,\\ c_4(\tau)&\displaystyle =&\displaystyle \frac{1}{3744}\tau^{5}-\frac{5}{3744}\tau^{4}+\frac{5}{1872}\tau^{3} +\frac{5}{624}\tau^{2}-\frac{1}{104}\tau,\\ c_5(\tau)&\displaystyle =&\displaystyle -\frac{1}{18720}\tau^{5}+\frac{1}{3744}\tau^{4} -\frac{1}{1872}\tau^{3} -\frac{1}{624}\tau^{2} -\frac{1}{156}\tau+\frac{1}{120},\\ d_0(\tau)&\displaystyle =&\displaystyle \frac{1}{120}\tau^{5},\\ d_1(\tau)&\displaystyle =&\displaystyle -\frac{1}{156}\tau^{5}-\frac{1}{104}\tau^{4}-\frac{5}{78}\tau^{3} -\frac{5}{26}\tau^{2}+\frac{3}{13}\tau,\\ d_2(\tau)&\displaystyle =&\displaystyle -\frac{1}{624}\tau^{5}+\frac{5}{624}\tau^{4}+\frac{7}{104}\tau^{3} +\frac{21}{104}\tau^{2}-\frac{5}{26}\tau,\\ d_3(\tau)&\displaystyle =&\displaystyle -\frac{1}{1872}\tau^{5}+\frac{5}{1872}\tau^{4}-\frac{5}{936}\tau^{3} -\frac{5}{312}\tau^{2}-\frac{5}{78}\tau,\\ d_4(\tau)&\displaystyle =&\displaystyle \frac{1}{3744}\tau^{5}-\frac{5}{3744}\tau^{4}+\frac{5}{1872}\tau^{3} +\frac{5}{624}\tau^{2}+\frac{5}{156}\tau,\\ d_5(\tau)&\displaystyle =&\displaystyle -\frac{1}{18720}\tau^{5}+\frac{1}{3744}\tau^{4}-\frac{1}{1872}\tau^{3} -\frac{1}{624}\tau^{2}-\frac{1}{156}\tau. \end{array} \end{aligned} $$

Proof

Let $f \in W_2^3[0,1]$ and 0 ≤ τ ≤ 1. Note that

$$\displaystyle \begin{aligned} R^{\prime}_\tau(\eta)=\begin{cases} \sum_{i=0}^4 (i+1) c_{i+1}(\tau) \eta^i,\quad 0 \leq \eta < \tau \leq 1,\\ \\ \sum_{i=0}^4 (i+1) d_{i+1}(\tau) \eta^i,\quad 0 \leq \tau < \eta \leq 1, \end{cases} \end{aligned}$$

$$\displaystyle \begin{aligned} R^{\prime \prime}_\tau(\eta)=\begin{cases} \sum_{i=0}^3 (i+1)(i+2) c_{i+2}(\tau) \eta^i,\quad 0 \leq \eta < \tau \leq 1,\\ \\ \sum_{i=0}^3 (i+1)(i+2) d_{i+2}(\tau) \eta^i,\quad 0 \leq \tau < \eta \leq 1, \end{cases} \end{aligned}$$

$$\displaystyle \begin{aligned} R^{(3)}_\tau(\eta)=\begin{cases} \sum_{i=0}^2 (i+1)(i+2)(i+3) c_{i+3}(\tau) \eta^i,\quad 0 \leq \eta < \tau \leq 1,\\ \\ \sum_{i=0}^2 (i+1)(i+2)(i+3) d_{i+3}(\tau) \eta^i,\quad 0 \leq \tau < \eta \leq 1, \end{cases} \end{aligned}$$

$$\displaystyle \begin{aligned} R^{(4)}_\tau(\eta)=\begin{cases} \sum_{i=0}^1 (i+1)(i+2)(i+3)(i+4) c_{i+4}(\tau) \eta^i,\quad 0 \leq \eta < \tau \leq 1,\\ \\ \sum_{i=0}^1 (i+1)(i+2)(i+3)(i+4) d_{i+4}(\tau) \eta^i,\quad 0 \leq \tau < \eta \leq 1, \end{cases} \end{aligned}$$

and

$$\displaystyle \begin{aligned} R^{(5)}_\tau(\eta)=\begin{cases} 120c_5(\tau),\quad 0 \leq \eta < \tau \leq 1,\\ \\ 120d_5(\tau),\quad 0 \leq \tau < \eta \leq 1. \end{cases} \end{aligned}$$

By Definition 3 and integrating by parts, we obtain

$$\displaystyle \begin{aligned} \begin{array}{rcl} \left\langle f,R_\tau \right\rangle_{W_2^3} &\displaystyle =&\displaystyle \sum_{i=0}^2f^{(i)}(0)R_\tau^{(i)}(0) +\int_{0}^1f^{(3)}(\eta)R_\tau^{(3)}(\eta)\mathrm{d}\eta \\ &\displaystyle =&\displaystyle f'(0)R_\tau'(0)+f''(0)R_\tau''(0)+f''(1)R_\tau^{(3)}(1)-f''(0)R_\tau^{(3)}(0)\\ &\displaystyle &\displaystyle -f'(1)R_\tau^{(4)}(1)+f'(0)R_\tau^{(4)}(0)+\int_{0}^{1}f'(\eta)R_\tau^{(5)}(\eta)\mathrm{d}\eta \\ &\displaystyle =&\displaystyle c_1(\tau)f'(0)+2c_2(y)f''(0)\\ &\displaystyle &\displaystyle +6(d_3(\tau)+4d_4(\tau)+10d_5(\tau))f''(1)-6c_3(\tau)f''(0)\\ &\displaystyle &\displaystyle -24(d_4(\tau)+5d_5(\tau))f'(1)+24c_4(\tau)f'(0)\\ &\displaystyle &\displaystyle +\int_{0}^{\tau}120c_5(\tau)f'(\eta)\mathrm{d}\eta +\int_{\tau}^{1}120d_5(\tau)f'(\eta)\mathrm{d}\tau \\ &\displaystyle =&\displaystyle (c_1(\tau)+24c_4(\tau))f'(0)+2(c_2(\tau)-3c_3(\tau))f''(0)\\ &\displaystyle &\displaystyle +6(d_3(\tau)+4d_4(\tau)+10d_5(\tau))f''(1)-24(d_4(\tau)+5d_5(\tau))f'(1) \\ &\displaystyle &\displaystyle +120(c_5(\tau)-d_5(\tau))f(\tau)\\ &\displaystyle =&\displaystyle f(\tau). \end{array} \end{aligned} $$

Definition 4

We give the binary space W(Ω) as:

$$\displaystyle \begin{aligned} \begin{array}{rcl} W(\varOmega)&\displaystyle =&\displaystyle \left\{f: \; \frac{\partial^{3}f}{\partial \eta^2 \partial t} \in CC(\varOmega),\quad \frac{\partial^5 f}{\partial \eta^3\partial t^2}\in L^2(\varOmega), \;\right.\\ &\displaystyle &\displaystyle \left. f(\eta,0)=f(0,t)=f(1,t)=0 \right\}, \end{array} \end{aligned} $$

where CC denotes the space of completely continuous functions. The inner product and the norm in W(Ω) are obtained as:

$$\displaystyle \begin{aligned} \begin{array}{rcl} \left\langle f,g \right\rangle_W &\displaystyle =&\displaystyle \sum_{i=0}^1\int_0^1 \left[\frac{\partial^2}{\partial t^2}\frac{\partial^i}{\partial \eta^i}u(0,t) \frac{\partial^2}{\partial t^2}\frac{\partial^i}{\partial \eta^i}g(0,t)\right]\mathrm{d} t\\ &\displaystyle &\displaystyle +\sum_{j=0}^1 \left\langle\frac{\partial^j}{\partial t^j}f(\cdot,0), \frac{\partial^j}{\partial t^j}g(\cdot,0)\right\rangle_{W_2^3}\\ &\displaystyle &\displaystyle +\int_0^1\int_0^1 \left[\frac{\partial^3}{\partial \eta^3}\frac{\partial^2}{\partial t^2}f(\eta,t) \frac{\partial^3}{\partial \eta^3}\frac{\partial^2}{\partial t^2}g(\eta,t)\right]\mathrm{d}t\mathrm{d}\eta, \quad f,g \in W(\varOmega) \end{array} \end{aligned} $$

and

$$\displaystyle \begin{aligned} \left\Vert g \right\Vert_W =\sqrt{\left\langle g,g \right\rangle_W}, \quad g \in W(\varOmega). \end{aligned}$$

Lemma 1 (See [4, page 148])

Reproducing kernel function K _(τ,s) of W(Ω) is given by:

$$\displaystyle \begin{aligned} K_{(\tau,s)}=R_{\tau}r_{s}. \end{aligned}$$

Definition 5

We define the binary space $\widehat W(\varOmega )$ by

$$\displaystyle \begin{aligned} \begin{array}{rcl} \widehat W(\varOmega)=\left\{f \in CC(\varOmega): \quad \frac{\partial^2 f}{\partial \eta \partial t} \in L^2(\varOmega)\right\}. \end{array} \end{aligned} $$

The inner product and the norm in $\widehat W(\varOmega )$ are obtained as:

$$\displaystyle \begin{aligned} \begin{array}{rcl} \left\langle f,g \right\rangle_{\widehat W} &\displaystyle =&\displaystyle \int_0^1\left[\frac{\partial}{\partial t}f(0,t) \frac{\partial}{\partial t}g(0,t)\right]\mathrm{d}t +\left\langle f(\cdot,0),g(\cdot,0) \right\rangle_{G_2^1} \\ &\displaystyle &\displaystyle +\int_0^1\int_0^1 \left[\frac{\partial}{\partial \eta} \frac{\partial}{\partial t}f(\eta,t) \frac{\partial}{\partial \eta} \frac{\partial}{\partial t}g(\eta,t)\right]\mathrm{d}t\mathrm{d}\eta, \quad f,g \in \widehat W(\varOmega) \end{array} \end{aligned} $$

and

$$\displaystyle \begin{aligned} \left\Vert g \right\Vert_{\widehat W} =\sqrt{\left\langle g,g \right\rangle_{\widehat W}}, \quad g \in \widehat W(\varOmega). \end{aligned}$$

Lemma 2 (See [4, page 23])

Reproducing kernel function G _{( τ, s)} of $\widehat W(\varOmega )$ is given as:

$$\displaystyle \begin{aligned} G_{(\tau,s) }=(\widetilde{Q}_\tau)^2. \end{aligned}$$

Definition 6

We define the space $W_2^1[0,1]$ by

$$\displaystyle \begin{aligned} W_2^1[0,1]=\{u \in AC[0,1]:\; u'\in L^2[0,1]\}. \end{aligned}$$

The inner product and the norm in $W_2^1[0,1]$ are given as:

$$\displaystyle \begin{aligned} \left\langle u,g\right\rangle_{W_2^1} =\int_0^1 u(x)g(x)+ u'(x)g'(x)\mathrm{d}x, \quad u,g\in G_2^1[0,1] \end{aligned} $$

(8.18)

and

$$\displaystyle \begin{aligned} \left\Vert u\right\Vert_{W_2^1} =\sqrt{\left\langle u,u\right\rangle_{W_2^1}}, \quad u\in W_2^1[0,1]. \end{aligned} $$

(8.19)

The space $W_2^1[0,1]$ is a reproducing kernel space, and its reproducing kernel function T _x is obtained as [4]

$$\displaystyle \begin{aligned} T_x(y){=\frac{1}{2\sinh (1)}\left[\cosh(x+y-1)+\cosh (\left\vert x-y\right\vert -1)\right]}. \end{aligned} $$

(8.20)

3 Analytical and Approximate Solutions

We consider

$$\displaystyle \begin{aligned} A_1(f_1(\eta,t))=P_1(\eta,t,F(\eta,t))+M_1(\eta,t), \end{aligned} $$

(8.21)

where $A_1: W(\varOmega ) \rightarrow \widehat {W}(\varOmega )$ is a bounded linear operator, P ₁ is a nonlinear operator, M ₁(η, t) is an arbitrary function, and F(η, t) = [f ₁(η, t), f ₂(η, t), …, f _k(η, t)]^T. The spaces W(Ω) and $\widehat {W}(\varOmega )$ are reproducing kernel spaces which are defined according to the highest derivatives. We pick a countable dense subset $\{(\eta _j,t_j) \}_{j=1}^\infty$ in Ω, and describe $\rho _j(\eta ,t)=G_{(\eta _j,t_j)}(\eta ,t),$ $\vartheta _{j_1}(\eta ,t)=A^*_1\rho _j(\eta ,t),$ where $A^*_1$ is the adjoint operator of A ₁. It is simple to show that [2]

$$\displaystyle \begin{aligned} \vartheta_{j_1}(\eta,t)=A_1 K_{(\tau,s)}(\eta,t). \end{aligned}$$

The solutions of (8.3) and (8.4) are considered in the reproducing kernel space $W_2^3[0,1]$. On defining the linear operator $L:W_2^3[0,1]\rightarrow W_2^1[0,1]$ as

$$\displaystyle \begin{aligned} \mathrm{Lu}(x) =u''(x) \end{aligned} $$

(8.22)

the problem changes the form:

$$\displaystyle \begin{aligned} \begin{cases} \mathrm{Lu}=f(x,u),\quad x\in [0,1], \\ u(0)=u(1)=0, \end{cases} \end{aligned} $$

(8.23)

where $f(x,u)=\lambda \exp (\mu u(x)).$

In Eq. (8.23) since u(x) is sufficiently smooth $L:W_2^3[0,1]\rightarrow W_2^1[0,1]$ is a bounded linear operator. For model problem (8.4) similar things can be done.

Theorem 4

Assume that $\{(\eta _j,t_j) \}_{j=1}^\infty$ is dense in Ω, then the solution of (8.21) can be shown as

$$\displaystyle \begin{aligned} f_1(\eta,t)=\sum_{j=1}^\infty \sigma_{j_1}\vartheta_{j_1}(\eta,t),\end{aligned} $$

(8.24)

where the $\sigma _{j_1}$ are found by

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} Z_1 \times \sigma_{j_1} &\displaystyle =&\displaystyle [P_1(\eta_1,t_1,F(\eta_1,t_1))+M_1(\eta_1,t_1),P_1(\eta_2,t_2,F(\eta_2,t_2))\\ &\displaystyle &\displaystyle +M_1(\eta_2,t_2), \ldots]^T,\quad \vspace{-4pt} \end{array} \end{aligned} $$

(8.25)

$$\displaystyle \begin{aligned} \begin{array}{rcl} Z_1=[A_1 \vartheta_{j_1}(\eta,t)\mid_{(\eta,t)=(\eta_i,t_i)}]_{i,j=1,2,\ldots}, \quad \sigma_{j_1} = [\sigma_{11}, \sigma_{21}, \ldots]^T. \end{array} \end{aligned} $$

Proof

$\{(\eta _j,t_j) \}_{j=1}^\infty$ is dense in Ω. Therefore, $\vartheta _{j_1}(\eta ,t)$ is complete system in W(Ω) [2]. We get

$$\displaystyle \begin{aligned} \begin{array}{rcl} \left \langle \vartheta_{i_1}, \vartheta_{j_1} \right \rangle_{W(\varOmega)} &\displaystyle =&\displaystyle \left \langle A^*_1\rho_i(\eta,t), \vartheta_{j_1} \right \rangle_{W(\varOmega)}=\left \langle \rho_i(\eta,t), A_1\vartheta_{j_1} \right \rangle_{\widehat{W} (\varOmega)} \\ &\displaystyle =&\displaystyle A_1 \vartheta_{j_1}(\eta,t)= \left \langle f_1(\eta,t), \vartheta_{i_1} \right \rangle_{W(\varOmega)}= \left \langle f_1(\eta,t),A^*_1 \rho_{i_1} \right \rangle_{W(\varOmega)} \\ &\displaystyle =&\displaystyle \left \langle A_1 f_1(\eta,t),\rho_{i_1} \right \rangle_{\widehat{W}(\varOmega)} = P_1(\eta_i,t_i, F(\eta_i,t_i))+M_1(\eta_i,t_i). \end{array} \end{aligned} $$

This completes the proof.

Remark 1

If P _η(η, t, F(η, t)) = 0 for η = 1, 2, …, k, then the analytical solution of each equation can be achieved and the approximate solution of each equation is the m-term intercept of the analytical solution which can be obtained by solving an m × m system of linear equations. If P _η(η, t, F(η, t)) ≠ 0, then we need to construct an iterative method. We select the number of points m, the number of iterations n and put the initial vector function F _0,m(η, t) = [0, 0, …, 0]^T. Then the approximate solution is presented as:

$$\displaystyle \begin{aligned} F_{n,m}(\eta,t)=[f_{n,m,1}(\eta,t),f_{n,m,2}(\eta,t),\ldots,f_{n,m,k}(\eta,t)]^T, \end{aligned}$$

where

$$\displaystyle \begin{aligned} \begin{cases} f_{n,m,1}(\eta,t)=\sum_{j=1}^m \sigma_{j_1}\vartheta_{j_1}(\eta,t), \\ \\ \ldots \\ \\ f_{n,m,k}(\eta,t)=\sum_{j=1}^m \sigma_{j_k}\vartheta_{j_k}(\eta,t). \end{cases} \end{aligned}$$

Theorem 5

Suppose that $\{(\eta _j,t_j) \}_{j=1}^\infty$ is dense in Ω. Then the approximate solution F _n,m(η, t) converges to the analytical solution F(η, t).

Proof

We have

$$\displaystyle \begin{aligned} A_\eta(f_{n,m,\eta}(\eta_j,t_j))=P_\eta(\eta_j,t_j, F_{n-1,m}(\eta_j,t_j))+M_\eta(\eta_j,t_j) \end{aligned} $$

(8.26)

for

$$\displaystyle \begin{aligned} \eta=1,2,\ldots,k, \quad j=1,2,\ldots,m, \quad \text{and} \; n=1,2,\ldots. \end{aligned}$$

There exists a convergent subsequence $\{f_{n_\epsilon ,m,\eta }(\eta ,t) \}_{\epsilon =1}^\infty$ of $\{f_{n,m,\eta }(\eta ,t) \}_{n=1}^\infty$ such that $f_{n_\epsilon ,m,\eta }(\eta ,t) \rightarrow u_\varpi (\eta ,t) \; \text{as} \; \epsilon \rightarrow \infty , \; m \rightarrow \infty , \; \text{for} \; \varpi =1,2,\ldots ,k.$ Then, we acquire

$$\displaystyle \begin{aligned} A_\varpi(f_{n_\epsilon,m,\varpi}(\eta_j,t_j))=P_\varpi(\eta_j,t_j,F_{n_\epsilon-1,m}(\eta_j,t_j))+M_\varpi(\eta_j,t_j). \end{aligned} $$

(8.27)

The operators A _ϖ and P _ϖ are both continuous. Therefore it can be concluded that F(η, t) = [f ₁(η, t), …, f _k(η, t)]^T is the analytical solution of (8.21) and $F_{n_\epsilon ,m}(\eta ,t)=[f_{n_\epsilon ,m,1}(\eta ,t),\ldots , f_{n_\epsilon ,m,k}(\eta ,t)]^T$ is the approximate solution of (8.21) after taking limit from both sides. This completes the proof.

It is obvious that $\ L:W_2^3[0,1]\rightarrow W_2^1[0,1]$ is a bounded linear operator. Put $\varphi _i(x)=T_{x_i}(x)$ and ψ _i(x) = L^∗ φ _i(x), where L^∗ is conjugate operator of L. The orthonormal system $\left \{\widehat {\varPsi }_i(x)\right \}_{i=1}^{\infty }$ of $W_2^3[0,1]$ can be obtained from Gram-Schmidt orthogonalization process of $\{\psi _i(x)\}_{i=1}^{\infty }$,

$$\displaystyle \begin{aligned} \widehat{\psi}_{i}(x)=\sum_{k=1}^{i}\beta _{ik}\psi_{k}(x),\quad (\beta _{ii}>0,\quad i=1,2,\ldots) \end{aligned} $$

(8.28)

Lemma 3 (See [5])

Let $\left \{ x_{i}\right \}_{i=1}^{\infty }$ be dense in [0, 1] and $\psi _{i}(x)=\left . L_{y}R_{x}(y)\right \vert _{y=x_{i}}$ . Then the sequence $\left \{ \psi _{i}(x)\right \} _{i=1}^{\infty }$ is a complete system in $W_{2}^{3}[0,1]$.

Theorem 6

If u ₁ and u ₂ are the exact solutions of (8.3), then

$$\displaystyle \begin{aligned} u_1(x)=\sum_{i=1}^{\infty}\sum_{k=1}^{i}\beta_{ik} f(x_k,u_{1_k})\widehat{\varPsi}_i(x) \end{aligned} $$

(8.29)

and

$$\displaystyle \begin{aligned} u_2(x)=\sum_{i=1}^{\infty}\sum_{j=1}^{i}\gamma_{ij} f(x_j,u_{2_j})\widehat{\varPsi}_i(x), \end{aligned} $$

(8.30)

where $\{(x_i)\}_{i=1}^{\infty }$ is dense in [0, 1].

Proof

We have

$$\displaystyle \begin{aligned} \begin{array}{rcl} u_1(x) &\displaystyle =&\displaystyle \sum_{i=1}^{\infty}\left\langle u_1(x), \widehat{\varPsi}_i(x)\right\rangle_{W_2^3} \widehat{\varPsi}_i(x) \\ &\displaystyle =&\displaystyle \sum_{i=1}^{\infty}\sum_{k=1}^i\beta_{ik}\left\langle u_1(x), \varPsi_k(x) \right\rangle_{W_2^3} \widehat{\varPsi}_i(x) \\ &\displaystyle =&\displaystyle \sum_{i=1}^{\infty}\sum_{k=1}^i\beta_{ik}\left\langle u_1(x),L^{\ast}\varphi_k(x)\right\rangle_{W_2^3} \widehat{\varPsi}_i(x) \\ &\displaystyle =&\displaystyle \sum_{i=1}^{\infty }\sum_{k=1}^{i}\beta_{ik}\left\langle \mathrm{Lu}_1(x), \varphi_k(x) \right\rangle_{W_2^1} \widehat{\varPsi}_i(x) \\ &\displaystyle =&\displaystyle \sum_{i=1}^{\infty }\sum_{k=1}^{i}\beta_{ik}\left\langle f(x,u_1),T_{x_k}\right\rangle_{W_2^1}\widehat{\varPsi}_i(x) \\ &\displaystyle =&\displaystyle \sum_{i=1}^{\infty}\sum_{k=1}^{i}\beta_{ik}f(x_k,u_{1_k}) \widehat{\varPsi}_i(x). \end{array} \end{aligned} $$

Similar things can be done for u ₂.

The approximate solutions u _n(x) and u _m(x) can be acquired from the n and m terms truncation of the exact solutions u ₁ and u ₂ as

$$\displaystyle \begin{aligned} u_n(x)=\sum_{i=1}^n\sum_{k=1}^i\beta_{ik}f(x_k,u_{1_k})\widehat{\varPsi}_i(x), \end{aligned} $$

(8.31)

and

$$\displaystyle \begin{aligned} u_m(x)=\sum_{i=1}^m \sum_{j=1}^i\gamma_{ij}f(x_k,u_{2_j})\widehat{\varPsi}_i(x). \end{aligned} $$

(8.32)

Theorem 7

For any fixed $u_{1_{0}}(x)\in W_2^3[0,1]$ assume that the following conditions are hold:

• (i)
  $$\displaystyle \begin{aligned} u_{n}(x)=\sum_{i=1}^{n}A_{i}\widehat{\psi }_{i}(x), \end{aligned} $$

  (8.33)
  $$\displaystyle \begin{aligned} A_{i}=\sum_{k=1}^{i}\beta _{ik}f(x_{k},u_{1_{k-1}}(x_{k})), \end{aligned} $$

  (8.34)

• (ii) $\left \Vert u_n \right \Vert _{W_2^3}$ is bounded;

• (iii) $\left \{x_{i}\right \}_{i=1}^{\infty }$ is dense in [0, 1];

• (iv) f(x, u ₁) ∈ $W_2^1[0,1]$ for any $u_1(x)\in W_2^3[0,1]$.

Then u _n(x) converges to the exact solution of (8.3) in $W_2^3[0,1]$ and

$$\displaystyle \begin{aligned} u_1(x)=\sum_{i=1}^{\infty }A_{i}\widehat{\psi }_{i}(x), \end{aligned}$$

where A _i is given by (8.34).

Proof

We will show the convergence of u _n(x). We get

$$\displaystyle \begin{aligned} u_{n+1}(x)=u_n(x)+A_{n+1}\widehat{\varPsi}_{n+1}(x), \end{aligned} $$

(8.35)

from the orthonormality of $\{\widehat {\varPsi }_i\}_{i=1}^{\infty }$, it follows that

$$\displaystyle \begin{aligned} \left\Vert u_{n+1}\right\Vert ^2=\left\Vert u_n\right\Vert ^2+A_{n+1}^2=\left\Vert u_{n-1}\right\Vert ^2+A_n^2+A_{n+1}^2=\ldots=\sum_{i=1}^{n+1}A_i^2, \end{aligned} $$

(8.36)

from boundedness of $\left \Vert u_n\right \Vert _{W_2^3}$, we obtain

$$\displaystyle \begin{aligned} \sum_{i=1}^{\infty}A_i^2<\infty, \end{aligned}$$

i.e.,

$$\displaystyle \begin{aligned} \left\{ A_{i}\right\} \in l^{2}\quad (i=1,2,\ldots). \end{aligned}$$

Let p > n, in view of $\left (u_p-u_{p-1}\right ) \perp \left (u_{p-1}-u_{p-2}\right ) \perp \ldots \perp \left ( u_{n+1}-u_{n}\right )$, it follows that

$$\displaystyle \begin{aligned} \begin{array}{rcl} \left\Vert u_p-u_n\right\Vert_{W_2^3}^2 &\displaystyle =&\displaystyle \left\Vert u_p-u_{p-1}+u_{p-1}-u_{p-2}+\ldots+u_{n+1}-u_n\right\Vert_{W_2^3}^2 \\ &\displaystyle \leq &\displaystyle \left\Vert u_p-u_{p-1}\right\Vert_{W_2^3}^2+\ldots +\left\Vert u_{n+1}-u_n\right\Vert_{W_2^3}^2 \\ &\displaystyle =&\displaystyle \sum_{i=n+1}^p A_i^2\rightarrow 0, \quad p,n\rightarrow \infty. \end{array} \end{aligned} $$

Considering the completeness of $W_2^3[0,1]$, there exists $u_1(x)\in W_2^3[0,1]$, such that

$$\displaystyle \begin{aligned} u_n(x){\rightarrow }u_1(x)\quad as\ n\rightarrow \infty .\end{aligned} $$

(ii) Taking limits,

$$\displaystyle \begin{aligned} u_1(x)=\sum_{i=1}^{\infty }A_{i}\widehat{\psi }_{i}(x).\end{aligned} $$

Since

$$\displaystyle \begin{aligned} \begin{array}{rcl} \left(\mathrm{Lu}_1\right) \left( x_{j}\right) &\displaystyle =&\displaystyle \sum_{i=1}^{\infty}A_{i}\left\langle L\widehat{\psi}_{i}(x),\varphi _{j}(x)\right\rangle _{W_2^1}=\sum_{i=1}^{\infty }A_{i}\left\langle \widehat{\psi}_{i}(x),L^{\ast }\varphi_{j}(x)\right\rangle _{W_2^{3}}\\ &\displaystyle =&\displaystyle \sum_{i=1}^{\infty }A_{i}\left\langle \widehat{\psi}_{i}(x),{\psi}_j(x)\right\rangle _{W_{2}^{3}}, \end{array} \end{aligned} $$

it follows that

$$\displaystyle \begin{aligned} \begin{array}{rcl} \sum_{j=1}^{n}\beta _{nj}(\mathrm{Lu}_1)(x_{j})&\displaystyle =&\displaystyle \sum_{i=1}^{\infty}A_{i}\left\langle \widehat{\psi }_{i}(x),\sum_{j=1}^{n}\beta _{nj} {\psi}_j(x)\right\rangle_{W_2^3}\\ &\displaystyle =&\displaystyle \sum_{i=1}^{\infty}A_i \left\langle \widehat{\psi}_i(x),\widehat{\psi}_n(x)\right\rangle_{W_2^3}=A_n. \end{array} \end{aligned} $$

If n = 1, then

$$\displaystyle \begin{aligned} \mathrm{Lu}_1(x_1)=f(x_1,u_{1_{0}}(x_1)). \end{aligned} $$

(8.37)

If n = 2, then

$$\displaystyle \begin{aligned} \beta_{21}(\mathrm{Lu}_1)(x_1) +\beta_{22}(\mathrm{Lu}_1)(x_2)=\beta_{21}f(x_1,u_{1_0}(x_1))+\beta_{22}f(x_2,u_{1_1}(x_{2})). \end{aligned} $$

(8.38)

From (8.37) and (8.38), we have

$$\displaystyle \begin{aligned} (\mathrm{Lu}_1)(x_2)=f(x_2,u_{1_1}(x_2)). \end{aligned}$$

We get

$$\displaystyle \begin{aligned} (\mathrm{Lu}_1)(x_j)=f(x_j,u_{1_{j-1}}(x_j)), \end{aligned} $$

(8.39)

by induction. By the convergence of u _n(x) we get

$$\displaystyle \begin{aligned} (\mathrm{Lu}_1)(y)=f(y,u_1(y)), \end{aligned}$$

that is, $u_1\left ( x\right )$ is the solution of (8.3) and

$$\displaystyle \begin{aligned} u_1(x)=\sum_{i=1}^{\infty }A_{i}\widehat{\psi }_{i}, \end{aligned}$$

where A _i are given by (8.34). It can be shown in a similar way that u ₂(x) is a solution of (8.4).

Theorem 8

If $u_1 \in W_2^3[0,1]$ , then

$$\displaystyle \begin{aligned} \left\Vert u_n-u_1\right\Vert_{W_2^3} \rightarrow 0, \quad n\rightarrow \infty. \end{aligned}$$

Moreover a sequence $\left \Vert u_n-u_1 \right \Vert _{W_2^3}$ is monotonically decreasing in n.

Proof

We have

$$\displaystyle \begin{aligned} \left\Vert u_n-u_1\right\Vert_{W_2^3} = \left\Vert \sum_{i=n+1}^{\infty}\sum_{k=1}^{i}\beta_{ik} f(x_k,u_{1_k})\widehat{\varPsi}_i \right\Vert_{W_2^3}. \end{aligned}$$

Thus

$$\displaystyle \begin{aligned} \left\Vert u_n-u_1\right\Vert_{W_2^3} \rightarrow 0, \quad n\rightarrow \infty. \end{aligned}$$

In addition

$$\displaystyle \begin{aligned} \begin{array}{rcl} \left\Vert u_n-u_1\right\Vert_{W_2^3}^2 &\displaystyle =&\displaystyle \left\Vert \sum_{i=n+1}^{\infty}\sum_{k=1}^{i}\beta_{ik} f(x_k,u_{1_k})\widehat{\varPsi}_i \right\Vert_{W_2^3}^2 \\ &\displaystyle =&\displaystyle \sum_{i=n+1}^{\infty}\left(\sum_{k=1}^{i}\beta_{ik} f(x_k,u_{1_k})\widehat{\varPsi}_i\right)^2. \end{array} \end{aligned} $$

Clearly, $\left \Vert u_n-u_1\right \Vert _{W_2^3}$ is monotonically decreasing in n. In a similar way $\left \Vert u_m-u_2\right \Vert _{W_2^3}$ is monotonically decreasing in m. This completes the proof.

Remark 2

Let us consider countable dense set {x ₁, x ₂, …}∈ [0, 1] and define

$$\displaystyle \begin{aligned} \varphi_i=T_{x_i}, \quad \varPsi_i=L^{*}\varphi_i, \quad \widehat{\varPsi}_i=\sum_{k=1}^1 \beta_{ik} \varPsi_k. \end{aligned}$$

Then β _ik coefficients can be found by

$$\displaystyle \begin{aligned} \beta_{11}=\frac{1}{\Vert \varPsi_1 \Vert}, \quad \beta_{ii}=\frac{1}{\sqrt{\Vert \varPsi_i \Vert^2}-\sum_{k=1}^{i-1}c^2_{ik}}, \end{aligned}$$

$$\displaystyle \begin{aligned} \beta_{ij}=\frac{-\sum_{k=j}^{i-1} c_{ik}\beta_{kj}}{\sqrt{\Vert \varPsi_i \Vert^2}-\sum_{k=1}^{i-1}c^2_{ik}}, \quad c_{ik}=\left \langle \varPsi_i, \widehat{\varPsi}_k \right \rangle. \end{aligned}$$

In a similar way γ _ij can be defined by using $Q_{x_i}$ .

4 Numerical Results

We conceive the following nonlinear system of partial differential equations by RKM:

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} \frac{\partial f}{\partial t} &\displaystyle =&\displaystyle \alpha +\frac{1}{Re}\frac{\partial}{\partial \tau} \left(\mu(T) \frac{\partial f}{\partial \tau} \right)-\frac{Ha^2}{Re}f-\frac{R}{Re}(f-f_p), \\ \\ \frac{\partial f_p}{\partial t}&\displaystyle =&\displaystyle \frac{1}{ReG}(f-f_p), \\ \\ \frac{\partial T}{\partial t}&\displaystyle =&\displaystyle \frac{1}{RePr}\frac{\partial}{\partial \tau} \left(K(T) \frac{\partial T}{\partial \tau} \right)+\frac{Ec}{Re}\mu(T)\left(\frac{\partial f}{\partial \tau} \right)^2 \\ &\displaystyle &\displaystyle +\frac{Ec}{Re}Ha^2u^2+\frac{2R}{3Pr}(T_p-T),\\ \\ \frac{\partial T_p}{\partial t}&\displaystyle =&\displaystyle -L(T_p-T),\\ \\ f(\tau,t)&\displaystyle =&\displaystyle f_p(\tau,t)=T(\tau,t)=T_p(\tau,t)=0, \quad \text{for} \quad t \leq 0,\\ f_p(0,t)&\displaystyle =&\displaystyle f_p(1,t)=T_p(0,t)=T(0,t)=0, \quad \text{for} \quad t > 0,\\ \\ \beta \frac{\partial f}{\partial \tau}&\displaystyle =&\displaystyle f, \quad \text{for} \; \tau=0,1, \; t>0,\\ \\ T_p(1,t)&\displaystyle =&\displaystyle T(1,t)=1, \quad \text{for} \; t>0. \end{array} \end{aligned} $$

where

$$\displaystyle \begin{aligned} \begin{array}{rcl} \mu(T)&\displaystyle =&\displaystyle \exp(-aT), \quad K(T)=\exp(bT), \\ R&\displaystyle =&\displaystyle 0.5, \; G=0.8, \; \alpha=1, \; a=1, \; b=0.01, \; Pr=7.1,\\ Ec&\displaystyle =&\displaystyle 0.2, \; Re=1, \; L=0.7, \; \beta=1, \; t=10. \end{array} \end{aligned} $$

We obtained the numerical results and demonstrated them in Figs. 8.1, 8.2, 8.3, 8.4, 8.5, 8.6, 8.7, 8.8, 8.9, 8.10, 8.11, 8.12, 8.13, 8.14, 8.15, 8.16, 8.17, and 8.18.

Fig. 8.1

Approximate solutions of u _p = f _p for various β

Fig. 8.2

Approximate solutions of u _p = f _p for various t

Fig. 8.3

Approximate solutions of u _p = f _p for various a

Fig. 8.4

Comparison of approximate solutions of u = f and u _p = f _p

Fig. 8.5

Approximate solutions of u = f for different values of β

Fig. 8.6

Approximate solutions of u = f for various t

Fig. 8.7

Approximate solutions of u = f for various a

Fig. 8.8

Approximate solutions of u = f for various b

Fig. 8.9

Approximate solutions of u _p = f _p for various b

Fig. 8.10

Approximate solutions of u = f for various Ha

Fig. 8.11

Approximate solutions of u _p = f _p for various Ha

Fig. 8.12

Approximate solutions of T for various β

Fig. 8.13

Approximate solutions of T _p for various β

Fig. 8.14

Approximate solutions of T _p for various t

Fig. 8.15

Approximate solutions of T for various a

Fig. 8.16

Approximate solutions of T _p for various a

Fig. 8.17

Approximate solutions of T for various b

Fig. 8.18

Approximate solutions of T _p for various b

Example 1

We now consider (8.3). If μλ < 0, the problem (8.3) has as many solutions as the number of roots of the equation

$$\displaystyle \begin{aligned} \theta= \sqrt{2 \vert \mu \lambda \vert} \cosh\left(\frac{\theta}{4} \right), \end{aligned}$$

also for each such θ _i

$$\displaystyle \begin{aligned} u_i(x)=-\frac{2}{\mu}\left\{\ln\left[\cosh\left(\frac{\theta_i}{2} \left(x-\frac{1}{2} \right) \right) \right]-\ln\left[\cosh\left(\frac{\theta_i}{4} \right) \right] \right\}. \end{aligned}$$

We obtain Tables 8.1, 8.2, 8.3, 8.4, 8.5, and 8.6 by RKM.

Table 8.1 The numerical results of Example 1 for first solutions when θ = 1.51716, t ₁ = 1.389, t ₂ = 8.362, λ = 1, and μ = −1

Table 8.2 Absolute errors of Example 1 for first solutions when θ = 1.51716, t ₁ = 1.389, t ₂ = 8.362, λ = 1, and μ = −1

Table 8.3 Relative errors of Example 1 for first solutions when θ = 1.51716, t ₁ = 1.389, t ₂ = 8.362, λ = 1, and μ = −1

Table 8.4 The numerical results of Example 1 for second solutions when θ = 10.9387, t ₁ = 2.247, t ₂ = 8.034, λ = 1, and μ = −1

Table 8.5 Absolute error of Example 1 for second solutions when θ = 10.9387, t ₁ = 2.247, t ₂ = 8.034, λ = 1, and μ = −1

Table 8.6 Relative error of Example 1 for second solutions when θ = 10.9387, t ₁ = 2.247, t ₂ = 8.034, λ = 1, and μ = −1

Example 2

We consider (8.4) for the second example. We obtain Tables 8.7, 8.8, and 8.9 by RKM.

Table 8.7 The first approximate solutions of Example 2 when t ₁ = 3.026, t ₂ = 17.629

Table 8.8 The first approximate solutions of Example 2 when t ₁ = 3.026, t ₂ = 17.629

Table 8.9 The second approximate solutions of Example 2 when t ₁ = 2.246, t ₂ = 8.642

5 Conclusion

We studied approximate solutions of nonlinear systems of partial differential equations and multiple solutions of differential equations in the reproducing kernel space in this paper. We demonstrated our results with Tables 8.1, 8.2, 8.3, 8.4, 8.5, 8.6, 8.7, 8.8, and 8.9 and Figs. 8.1, 8.2, 8.3, 8.4, 8.5, 8.6, 8.7, 8.8, 8.9, 8.10, 8.11, 8.12, 8.13, 8.14, 8.15, 8.16, 8.17, and 8.18. We proved that the reproducing kernel method is an accurate technique for solving nonlinear systems of partial differential equations and second order differential equations.