1. Introduction

Integral equations of various types and kinds play an important role in many branches of linear and nonlinear function analysis and their applications in the theory of elasticity, engineering, mathematical physical and contact mixed problems. Therefor, many different methods are used to obtain the solution of the Volterra integral equation. In [1] Linz, studied analytical and numerical methods of Volterra equation. In [2], Mirzaee and Rafei used the BBM for the numerical solution of the nonlinear two-dimensional Volterra integral equations. In the references [3] - [8] the authors considered many different methods to solve linear and nonlinear system of Volterra integral equations numerically with continuous and singular kernels. In [9], Al-waqdani studied linear F-VIE with continuous kernel and solved the linear SVIEs numerically with continuous kernel.

$\mu \varphi \left(x,t\right)=f\left(x,t\right)+\lambda {\displaystyle {\int }_a^{b}K\left(x,y\right)\gamma \left(y,t,\varphi \left(y,t\right)\right)\text{d}y}+\lambda {\displaystyle {\int }_0^{t}F\left(t,\tau \right)\gamma \left(x,\tau ,\varphi \left(x,\tau \right)\right)\text{d}\tau }$ (1)

Equation (1) is called the NF-VIE in the space $L_2\left[a,b\right]\times C\left[0,T\right],T<\infty$. Here the Fredholm integral term is considered in position with a positive continuous kernel $K\left(x,y\right)$ for all $x,y\in \left[a,b\right]$, while the Volterra integral term is considered in time with a positive continuous kernel $F\left(t,\tau \right)$ for all $t,\tau \in \left[0,T\right],T<\infty$. The free term $f\left(x,t\right)$ is known continuous function in the space $L_2\left[a,b\right]\times C\left[0,T\right]$, while $\varphi \left(x,t\right)$ is unknown function representing the solution of the nonlinear integral Equation (1). The numerical coefficient $\lambda$ is called the parameter of the integral equation, may be complex, and has physical meaning, while the constant parameter $\mu$ defines the kind of the integral Equation (1).

2. Existence of Solution of NF-VIE

To prove the existence of a unique solution of Equation (1) using fixed point theorem.

We write it in the integral operator form:

$\bar{W}\varphi =f+W\varphi$ (2)

where

$W\varphi =K\varphi +F\varphi$ (3)

$K\varphi =\lambda {\displaystyle {\int }_a^{b}K\left(x,y\right)\gamma \left(y,t,\varphi \left(y,t\right)\right)\text{d}y},F\varphi =\lambda {\displaystyle {\int }_0^{t}F\left(t,\tau \right)\gamma \left(x,\tau ,\varphi \left(x,\tau \right)\right)\text{d}\tau }$ (4)

Then, we assume the following conditions:

i) The kernel of Fredholm integral term satisfies:

$\left|K\left(x,y\right)\right|\le A_1$ ( $A_1$ is a constant).

ii) The kernel of Volterra integral term satisfies:

$\left|F\left(t,\tau \right)\right|\le A_2$ ( $A_2$ is a constant).

iii) The given function $f\left(x,t\right)$ with its partial derivatives is continuous in $L_2\left[a,b\right]\times C\left[0,T\right]$ where:

$\Vert f\left(x,t\right)\Vert =\max {\displaystyle {\int }_0^t{\left({\displaystyle {\int }_a^b{\left|f\left(x,\tau \right)\right|}^2\text{d}x}\right)}^{\frac{1}{2}}\text{d}\tau }=A_3$ ( $A_3$ is a constant).

iv) The known continuous function $\gamma \left(x,t,\varphi \left(x,t\right)\right)$, for the constant $B>B_1,B>p$, the following conditions:

a) ${\left({\displaystyle {\int }_0^t{\displaystyle {\int }_a^b{\left|\gamma \left(x,t,\varphi \left(x,t\right)\right)\right|}^2\text{d}x\text{d}t}}\right)}^{\frac{1}{2}}\le B_1{\Vert \varphi \left(x,t\right)\Vert }_{L_2\left[a,b\right]\times C\left[0,T\right]}$

b) $\Vert \gamma \left(x,t,{\varphi }_1\left(x,t\right)\right)-\gamma \left(x,t,{\varphi }_2\left(x,t\right)\right)\Vert \le N\left(x,t\right)\left|{\varphi }_1\left(x,t\right)-{\varphi }_2\left(x,t\right)\right|$

where ${\Vert N\left(x,t\right)\Vert }_{L_2\left[a,b\right]\times C\left[0,T\right]}=p$.

Theorem 1:

If the condition i)-iv) are verified, then Equation (1) has unique solution in the Banach space $L_2\left[a,b\right]\times C\left[0,T\right],T<\infty$.

The provement of this theorem depends on the following two lemmas:

Lemma 1:

Under the conditions i)-iv-a), the operator $\bar{W}$ defined by (2), maps the space $L_2\left[a,b\right]\times C\left[0,T\right],T<\infty$ into itself.

Proof:

In view of Formula (2) and (3) we get:

$\begin{array}{c}\Vert \bar{W}\varphi \left(x,t\right)\Vert \le \Vert f\left(x,t\right)\Vert +\left|\lambda \right|\Vert {\displaystyle {\int }_a^b\left|K\left(x,y\right)\right|\left|\gamma \left(y,t,\varphi \left(y,t\right)\right)\right|\text{d}y}\Vert \\ +\left|\lambda \right|\Vert {\displaystyle {\int }_0^t\left|F\left(t,\tau \right)\right|\left|\gamma \left(x,\tau ,\varphi \left(x,\tau \right)\right)\right|\text{d}\tau }\Vert \end{array}$

Using the conditions (i)-(iii), then applying Cauchy-Schwarz inequality, we have: $\begin{array}{c}\Vert \bar{W}\varphi \left(x,t\right)\Vert \le A_3+\left|\lambda \right|A_1{\left({\displaystyle {\int }_a^b{\left|\gamma \left(y,t,\varphi \left(y,t\right)\right)\right|}^2\text{d}y}\right)}^{\frac{1}{2}}\\ +\left|\lambda \right|A_2{\left({\displaystyle {\int }_0^t{\left|\gamma \left(x,\tau ,\varphi \left(x,\tau \right)\right)\right|}^2\text{d}\tau }\right)}^{\frac{1}{2}}\end{array}$

In the light of the condition (iv-a), the above inequality take the form:

$\Vert \bar{W}\varphi \left(x,t\right)\Vert \le A_3+\alpha \Vert \varphi \left(x,t\right)\Vert ,\alpha =\left|\lambda \right|B_1\left(A_1+A_2+L\right)$ (5)

The lost inequality (5) shows that, the operator $\bar{W}$ maps the ball $S_r$ into itself, where

$r=\frac{A_3}{1-\left|\lambda \right|B_1\left(A_1+A_{2}L\right)}$ (6)

Since $r>0,A_3>0$, therefore we have $\alpha <1$. Moreover, the inequality (5) involves the boundedness of the operator W of Equation (2) where:

$\Vert W\varphi \left(x,t\right)\Vert \le \alpha \Vert \varphi \left(x,t\right)\Vert$ (7)

Also, the inequalities (5) and (7) define the boundedness of the operator $\bar{W}$.

Lemma 2:

If the conditions (i),(ii) and (iv-b) are satisfied, then the operator $\bar{W}$ is contractive in space $L_2\left[a,b\right]\times C\left[0,T\right],T<\infty$.

Proof:

For two functions ${\varphi }_1\left(x,t\right)$ and ${\varphi }_2\left(x,t\right)$ in the space $L_2\left[a,b\right]\times C\left[0,T\right]$ Formula (2), (3) leads to:

$\begin{array}{c}\Vert \bar{W}{\varphi }_1\left(x,t\right)-\bar{W}{\varphi }_2\left(x,t\right)\Vert \le \left|\lambda \right|\Vert {\displaystyle {\int }_a^b\left|K\left(x,y\right)\right|\left|\gamma \left(y,t,{\varphi }_1\left(y,t\right)\right)-\gamma \left(y,t,{\varphi }_2\left(y,t\right)\right)\right|\text{d}y}\Vert \\ +\left|\lambda \right|\Vert {\displaystyle {\int }_0^t\left|F\left(t,\tau \right)\right|\left|\gamma \left(x,\tau ,{\varphi }_1\left(x,\tau \right)\right)-\gamma \left(x,\tau ,{\varphi }_2\left(x,\tau \right)\right)\right|\text{d}\tau }\Vert \end{array}$

Using the condition (iv-b), then apply Cauchy-Schwarz inequality we have:

$\begin{array}{c}\Vert \bar{W}{\varphi }_1\left(x,t\right)-\bar{W}{\varphi }_2\left(x,t\right)\Vert \le \left|\lambda \right|\left|K\left(x,y\right)\right|{\left({\displaystyle {\int }_a^b{N}^2\left(x,t\right){\left|{\varphi }_1\left(y,t\right)-{\varphi }_2\left(y,t\right)\right|}^2\text{d}y}\right)}^{\frac{1}{2}}\\ +\left|\lambda \right|\left|F\left(t,\tau \right)\right|{\left({\displaystyle {\int }_a^b{N}^2\left(x,\tau \right){\left|{\varphi }_1\left(x,\tau \right)-{\varphi }_2\left(x,\tau \right)\right|}^2\text{d}\tau }\right)}^{\frac{1}{2}}\end{array}$

Finally, with the aid of conditions (i), (ii), and (iv-b) we obtain:

$\Vert \bar{W}{\varphi }_1\left(x,t\right)-\bar{W}{\varphi }_2\left(x,t\right)\Vert \le \alpha \Vert {\varphi }_1\left(x,t\right)-{\varphi }_2\left(x,t\right)\Vert$ (8)

In equality (8) shows that, the operator $\bar{W}$ is continuous in the space $L_2\left[a,b\right]\times C\left[0,T\right],T<\infty$, then $\bar{W}$ is a contraction operator under the condition $\alpha <1$.

3. The SVIEs

Consider:

$\begin{array}{l}\varphi \left(x,t\right)=f\left(x,t\right)+\lambda {\displaystyle \underset{a}{\overset{b}{\int }}K\left(x,y\right)\gamma \left(y,t,\varphi \left(y,t\right)\right)\text{d}y}\\ +\lambda {\displaystyle \underset{0}{\overset{t}{\int }}F\left(t,\tau \right)\gamma \left(x,\tau ,\varphi \left(x,\tau \right)\right)\text{d}\tau },\left(\mu =1\right)\end{array}$ (9)

when $a=b=0$, Equation (9) becomes:

$\varphi \left(0,t\right)=f\left(0,t\right)+\lambda {\displaystyle {\int }_0^{t}F\left(t,\tau \right)\gamma \left(0,\tau ,\varphi \left(0,\tau \right)\right)\text{d}\tau }$

Then,

${\varphi }_0\left(t\right)=f_0\left(t\right)+\lambda {\displaystyle \underset{0}{\overset{t}{\int }}F\left(t,\tau \right)\gamma \left(\tau ,{\varphi }_0\left(\tau \right)\right)\text{d}\tau }$ (10)

where $u_0\left(t\right)=u\left(0,t\right),f_0\left(t\right)=f\left(0,t\right)$.

Formula (10) represents Volterra integral equation of the second kind at $a=b=0$. For representing (9) as a VIEs, we use the numerical method. Divide the interval $\left[a,b\right]$ as $a=x_0\le x_1\le \cdots \le x_N=b$. Using the quadrature formula, Equation (9) becomes:

$\begin{array}{l}{\displaystyle {\int }_a^{b}K\left(x,y\right)\gamma \left(y,t,\varphi \left(y,t\right)\right)\text{d}y}\\ ={\displaystyle \underset{m=0}{\overset{n}{\sum }}{u}_{m}K\left(x_n,x_m\right)\gamma \left(x_n,t,\varphi \left(x_n,t\right)\right)}\end{array}$ (11)

where $n=1,2,\cdots ,N$, and $u_0=\frac{1}{2}{h}_0,u_n=\frac{1}{2}{h}_n,u_i=h_i,\left(i\ne 0,n\right)$.

Using (11) in (9), we have:

$\begin{array}{c}\varphi \left(x,t\right)=f\left(x,t\right)+\lambda {\displaystyle {\int }_0^{t}F\left(t,\tau \right)\gamma \left(x,\tau ,\varphi \left(x,\tau \right)\right)\text{d}\tau }\\ +{\displaystyle \underset{m=0}{\overset{n}{\sum }}{u}_{m}K\left(x_n,x_m\right)\gamma \left(x_n,t,\varphi \left(x_n,t\right)\right)}\end{array}$ (12)

Then:

${\varphi }_n=W_n\left(t\right)+\lambda {\displaystyle \underset{0}{\overset{t}{\int }}F\left(t,\tau \right)\gamma \left(x_n,\tau ,{\varphi }_n\left(\tau \right)\right)\text{d}\tau }$ (13)

where $W_n\left(t\right)=f_n\left(t\right)+\lambda {\displaystyle \underset{m=0}{\overset{n-1}{\sum }}{u}_{m}K\left(x_n,x_m\right)\gamma \left(x_m,t,{\varphi }_m\left(t\right)\right)}$, $n=0,1,\cdots ,N$. Formula (13) represents a NSVIEs of the second kind, and we have N unknown functions ${\varphi }_n\left(t\right)$ corresponding to time interval $\left[0,T\right]$.

4. Some Numerical Methods for Solving SVIEs

4.1. RKM

In this section, the RKM is used to solve NF-VIE of the second kind. By divide the interval $\left[a,b\right]$ as $a=x_0\le x_1\le \cdots \le x_i\le \cdots \le x_N=b$, $i=0,1,\cdots ,N$ and using the quadrature formula, the integral Equation (1) represent a NSVIEs as:

${\varphi }_n\left(t\right)=W_n\left(t\right)+{\displaystyle \underset{0}{\overset{t}{\int }}F\left(t,\tau \right)\gamma \left(\tau ,{\varphi }_n\left(\tau \right)\right)\text{d}\tau }$

where $W_n\left(t\right)=f_i\left(t\right)+{\displaystyle \underset{i=0}{\overset{n}{\sum }}{u}_i{k}_{ni}\gamma \left(t,{\varphi }_i\left(t\right)\right)}$.

To solve the NSVIEs:

$\phi \left(t\right)={\displaystyle \underset{0}{\overset{t}{\int }}F\left(t,\tau \right)\gamma \left(\tau ,\phi \left(\tau \right)\right)\text{d}\tau }+F\left(t\right)+{\displaystyle \underset{i=0}{\overset{n}{\sum }}{u}_i{k}_{ni}\gamma \left(t,\phi \left(t\right)\right)},n=0,1,\cdots ,N$ (14)

where

$\begin{array}{l}\phi \left(t\right)={\left({\varphi }_1\left(t\right),{\varphi }_2\left(t\right),\cdots ,{\varphi }_l\left(t\right)\right)}^{\text{T}},\\ \gamma \left(\tau ,\phi \left(\tau \right)\right)=\left(\gamma \left(\tau ,{\phi }_1\left(\tau \right)\right),\gamma \left(\tau ,{\phi }_2\left(\tau \right)\right),\cdots ,\gamma \left(\tau ,{\phi }_l\left(\tau \right)\right)\right),\\ F\left(t\right)={\left(f_1\left(t\right),f_2\left(t\right),\cdots ,f_l\left(t\right)\right)}^{\text{T}}\end{array}$

and

$\begin{array}{l}F\left(t,\tau \right)\gamma \left(\tau ,\phi \left(\tau \right)\right)\\ =\left[\begin{array}{cccc}{F}_{1,1}\left(t,\tau \right)\gamma \left(\tau ,\phi \left(\tau \right)\right)& F_{1,2}\left(t,\tau \right)\gamma \left(\tau ,\phi \left(\tau \right)\right)& \cdots & F_{1,s}\left(t,\tau \right)\gamma \left(\tau ,\phi \left(\tau \right)\right)\\ F_{2,1}\left(t,\tau \right)\gamma \left(\tau ,\phi \left(\tau \right)\right)& F_{2,2}\left(t,\tau \right)\gamma \left(\tau ,\phi \left(\tau \right)\right)& \cdots & F_{2,s}\left(t,\tau \right)\gamma \left(\tau ,\phi \left(\tau \right)\right)\\ ⋮& ⋮& \ddots & ⋮\\ F_{s,1}\left(t,\tau \right)\gamma \left(\tau ,\phi \left(\tau \right)\right)& F_{s,2}\left(t,\tau \right)\gamma \left(\tau ,\phi \left(\tau \right)\right)& \cdots & F_{s,s}\left(t,\tau \right)\gamma \left(\tau ,\phi \left(\tau \right)\right)\end{array}\right]\end{array}$

Then, we get

${\varphi }_n\left(t\right)=W_n\left(t\right)+{\displaystyle \underset{0}{\overset{t}{\int }}F\left(t,\tau \right)\gamma \left(\tau ,\phi \left(\tau \right)\right)\text{d}\tau },n=0,1,\cdots ,N$ (15)

Now, applying the RKM for solve (15):

Suppose that:

$F\left(t,\tau \right)={\displaystyle \underset{s}{\sum }{\psi }_s\left(t\right){\chi }_s\left(\tau \right)}$ (16)

Substituting from (16) into (15),

${\varphi }_n\left(t\right)=W_n\left(t\right)+{\displaystyle \underset{0}{\overset{t}{\int }}{\displaystyle \underset{s}{\overset{}{\sum }}{\psi }_s\left(t\right){\chi }_s\left(\tau \right)\gamma \left(\tau ,\phi \left(\tau \right)\right)\text{d}\tau }}$

${\varphi }_n\left(t\right)=W_n\left(t\right)+{\displaystyle \sum {\psi }_s\left(t\right){\displaystyle {\int }_0^t{\chi }_s\left(\tau \right)\gamma \left(\tau ,\phi \left(\tau \right)\right)\text{d}\tau }}$

Then, we have,

${\varphi }_n\left(t\right)=W_n\left(t\right)+{\displaystyle \underset{s}{\sum }{\psi }_s\left(t\right)z_s}\left(t\right),t>0$ (17)

where

$z_s\left(t\right)={\displaystyle \underset{0}{\overset{t}{\int }}{\chi }_s\left(\tau \right)\gamma \left(\tau ,\phi \left(\tau \right)\right)d\tau }$ (18)

By derivative (18), we have,

${z^{\prime }}_s\left(t\right)={\chi }_s\left(\tau \right)\gamma \left(\tau ,\phi \left(\tau \right)\right),z_s\left(0\right)=0$

Now, apply the RKM to this system of equations to give,

$\tilde{z}\left(v_{p}h\right)=h{\displaystyle \underset{q=0}{\overset{p-1}{\sum }}{A}_{pq}{\chi }_s\left(v_{q}h\right)\gamma \left(v_{q}h,\tilde{\phi }\left(v_{q}h\right)\right)},p=1,2,\cdots ,m$

which lead to,

$\begin{array}{c}{\tilde{\varphi }}_n\left(v_{p}h\right)=W_n\left(v_{p}h\right)+{\displaystyle \underset{s}{\sum }{\tilde{z}}_s\left(v_{p}h\right){\psi }_s\left(v_{p}h\right)}\\ =W_n\left(v_{p}h\right)+{\displaystyle \underset{s}{\sum }{\psi }_s\left(v_{p}h\right)h{\displaystyle \underset{q=0}{\overset{p-1}{\sum }}{A}_{pq}{\chi }_s\left(v_{q}h\right)\gamma \left(v_{q}h,\tilde{\phi }\left(v_{q}h\right)\right)}}\\ =W_n\left(v_{p}h\right)+h{\displaystyle \underset{q=0}{\overset{p-1}{\sum }}{A}_{pq}\gamma \left(v_{q}h,\tilde{\phi }\left(v_{q}h\right)\right){\displaystyle \underset{s}{\sum }{\psi }_s\left(v_{p}h\right){\chi }_s\left(v_{q}h\right)}}\end{array}$

By using Equation (16) to give,

${\tilde{\varphi }}_n\left(v_{p}h\right)=W_n\left(v_{p}h\right)+h{\displaystyle \underset{q=0}{\overset{p-1}{\sum }}{A}_{pq}F\left(v_{p}h,v_{q}h\right)\gamma \left(v_{q}h,\tilde{\phi }\left(v_{q}h\right)\right)},p=1,2,\cdots ,m$ (19)

which is approximate solution for Equation (15).

Now, if $m=4$ consider the Pouzet’s derivation, we define:

$\begin{array}{l}{p}_{i,j}\left(t\right)={\varphi }_{i,j-1}\left(t\right)\\ q_{i,j}\left(t\right)=G_{i,j}\left(t_{j+\frac{1}{2}}\right)+\frac{h}{2}F\left(t_{j+\frac{1}{2}},t_j\right)p_{i,j}\\ r_{i,j}\left(t\right)=G_{i,j}\left(t_{j+\frac{1}{2}}\right)+\frac{h}{2}F\left(t_{j+\frac{1}{2}},t_{j+\frac{1}{2}}\right)q_{i,j}\\ s_{i,j}\left(t\right)=G_{i,j}\left(t_{j+1}\right)+hF\left(t_{j+1},t_{j+1}\right)r_{i,j}\end{array}$ (20)

${\varphi }_{i,j}\left(t\right)=G_{i,j}\left(t_{j+1}\right)+\frac{h}{6}\left[F\left(t_{j+1},t_j\right)p_{i,j}+2F\left(t_{j+1},t_{j+\frac{1}{2}}\right)\left[q_{i,j}+r_{i,j}\right]+F\left(t_{j+1},t_{j+1}\right)s_{i,j}\right]$

The function ${\varphi }_{i,j}\left(t\right)$ is unknown function, such that

$G_{i,j}\left(t\right)=W_i\left(t\right)+\frac{h}{6}{\displaystyle \underset{n=0}{\overset{j-1}{\sum }}\left[F\left(t,t_n\right)p_{i,n}+2F\left(t,t_{n+\frac{1}{2}}\right)\left[q_{i,n}+r_{i,n}\right]+F\left(t,t_{n+1}\right)s_{i,n}\right]}$ (21)

where $G_{i,0}\left(t\right)=W_i\left(t\right)$, where $j=2,3,\cdots ,N-1$.

when $j=1$, Equation (10) becomes:

${\varphi }_{i,1}\left(t\right)=G_{i,1}\left(t_2\right)+\frac{h}{6}\left[F\left(t_2,t_1\right)p_{i,1}+2F\left(t_2,t_{1+\frac{1}{2}}\right)\left[q_{i,1}+r_{i,1}\right]+F\left(t_2,t_2\right)s_{i,1}\right]$

such that $G_{i,1}=W\left(t_2\right)$ and

$p_{i,1}\left(t\right)=W_i(\; t\; 1\; )$

$\begin{array}{l}{q}_{i,1}\left(t\right)=G_{i,1}\left(t_{1+\frac{1}{2}}\right)+\frac{h}{2}F\left(t_{1+\frac{1}{2}},t_1\right)p_{i,1}\\ r_{i,1}\left(t\right)=G_{i,1}\left(t_{1+\frac{1}{2}}\right)+\frac{h}{2}F\left(t_{1+\frac{1}{2}},t_{1+\frac{1}{2}}\right)q_{i,1}\\ s_{i,1}\left(t\right)=G_{i,1}\left(t_2\right)+hF\left(t_2,t_2\right)r_{i,1}\end{array}$

Since the function ${\varphi }_{i,j}\left(t\right)={\varphi }_{i,j}$ is the approximate solution at $\left(x_i,t_j\right)$ for Equation (1).

4.2. BBM

In this section, we use the BBM for solving the NF-VIE of the second kind. The interval $\left[a,b\right]$ is divided into steps of width h, $t_j=jh,j=0,1,\cdots ,n$ and $h=\left(b-a\right)/n$. the approximate solution of ${\varphi }_i\left(t\right)$ will be define at mesh-points $t_j$ and denoted by ${\varphi }_{ij},j=0,1,\cdots ,n$ such as ${\varphi }_{ij}$ is an approximation to ${\varphi }_i\left(t_j\right)$.

To solve the NSVIEs:

$\phi \left(t\right)=F\left(t\right)+\lambda {\displaystyle \underset{j=0}{\overset{i}{\sum }}{u}_j{K}_{nj}\gamma \left(t,\varphi \left(t\right)\right)}+\lambda {\displaystyle \underset{0}{\overset{t}{\int }}F\left(t,\tau \right)\gamma \left(\tau ,\phi \left(\tau \right)\right)\text{d}\tau },n=0,1,\cdots ,N$ (22)

where

$\begin{array}{l}\phi \left(t\right)={\left({\varphi }_1\left(t\right),{\varphi }_2\left(t\right),\cdots ,{\varphi }_l\left(t\right)\right)}^{\text{T}},\\ \gamma \left(\tau ,\phi \left(\tau \right)\right)=\left(\gamma \left(\tau ,{\phi }_1\left(\tau \right)\right),\gamma \left(\tau ,{\phi }_2\left(\tau \right)\right),\cdots ,\gamma \left(\tau ,{\phi }_l\left(\tau \right)\right)\right),\\ F\left(t\right)={\left(f_1\left(t\right),f_2\left(t\right),\cdots ,f_l\left(t\right)\right)}^{\text{T}}\end{array}$

and

$\begin{array}{l}F\left(t,\tau \right)\gamma \left(\tau ,\phi \left(\tau \right)\right)\\ =\left[\begin{array}{cccc}{F}_{1,1}\left(t,\tau \right)\gamma \left(\tau ,\phi \left(\tau \right)\right)& F_{1,2}\left(t,\tau \right)\gamma \left(\tau ,\phi \left(\tau \right)\right)& \cdots & F_{1,s}\left(t,\tau \right)\gamma \left(\tau ,\phi \left(\tau \right)\right)\\ F_{2,1}\left(t,\tau \right)\gamma \left(\tau ,\phi \left(\tau \right)\right)& F_{2,2}\left(t,\tau \right)\gamma \left(\tau ,\phi \left(\tau \right)\right)& \cdots & F_{2,s}\left(t,\tau \right)\gamma \left(\tau ,\phi \left(\tau \right)\right)\\ ⋮& ⋮& \ddots & ⋮\\ F_{s,1}\left(t,\tau \right)\gamma \left(\tau ,\phi \left(\tau \right)\right)& F_{s,2}\left(t,\tau \right)\gamma \left(\tau ,\phi \left(\tau \right)\right)& \cdots & F_{s,s}\left(t,\tau \right)\gamma \left(\tau ,\phi \left(\tau \right)\right)\end{array}\right]\end{array}$

Then, we get

${\varphi }_n\left(t\right)=W_n\left(t\right)+\lambda {\displaystyle \underset{0}{\overset{t}{\int }}F\left(t,\tau \right)\gamma \left(\tau ,\phi \left(\tau \right)\right)\text{d}\tau },n=0,1,\cdots ,N$ (23)

Rewrite Equation (23) as follows:

${\varphi }_i\left(t_k\right)=W_i\left(t_k\right)+\lambda {\displaystyle \underset{0}{\overset{t_{pm}}{\int }}{F}_{i,s}\left(t_k,\tau \right)\gamma \left(\tau ,\phi \left(\tau \right)\right)\text{d}\tau }+\lambda {\displaystyle \underset{t_{pm}}{\overset{t_n}{\int }}{F}_{i,s}\left(t_k,\tau \right)\gamma \left(\tau ,\phi \left(\tau \right)\right)\text{d}\tau }$

$\begin{array}{c}{\varphi }_i\left(t_k\right)=f_i\left(t_k\right)+\lambda {\displaystyle \underset{n=0}{\overset{i}{\sum }}{u}_n{k}_{in}\gamma \left(t_k,{\varphi }_n\left(t_k\right)\right)}+\lambda {\displaystyle \underset{0}{\overset{t_{pm}}{\int }}{F}_{i,s}\left(t_k,\tau \right)\gamma \left(\tau ,\phi \left(\tau \right)\right)\text{d}\tau }\\ +\lambda {\displaystyle \underset{t_{pm}}{\overset{t_n}{\int }}{F}_{i,s}\left(t_k,\tau \right)\gamma \left(\tau ,\phi \left(\tau \right)\right)\text{d}\tau }\end{array}$ (24)

where $s=1,2,\cdots$.

If the values ${\varphi }_{i0},{\varphi }_{i1},\cdots ,{\varphi }_{i,pm}$ are known, then the first integral can be approximated by standard quadrature methods, and the second integral is obtain by a quadrature rule using values of the integrand at $\tau =t_{pm},t_{pm+1},\cdots ,t_{p\left(m+1\right)}$.

Since the values of ${\varphi }_i$ at these points are unknown, we have a system of $l_p$ nonlinear equations by applied the BBM:

$\begin{array}{c}{\varphi }_{ik}=f_i\left(t_k\right)+\lambda {\displaystyle \underset{n=0}{\overset{i}{\sum }}{u}_n{K}_{in}\gamma \left(t_k,{\varphi }_n\left(t_k\right)\right)}\\ +\lambda \left[h{\displaystyle \underset{j=0}{\overset{mp}{\sum }}{v}_{kj}{F}_{i,s}\left(t_k,{\tau }_j\right)\left(\gamma \left({\tau }_j,{\varphi }_{ij}\right),\cdots ,\gamma \left({\tau }_j,{\varphi }_{lj}\right)\right)}\right]\\ +\lambda \left[h{\displaystyle \underset{j=mp}{\overset{\left(m+1\right)p}{\sum }}{v}_{kj}^{*}{F}_{i,s}\left(t_k,{\tau }_{mp+j}\right)\left(\gamma \left({\tau }_{mp+j},{\varphi }_{1,mp+j}\right),\cdots ,\gamma \left({\tau }_{mp+j},{\varphi }_{l,mp+j}\right)\right)}\right]\end{array}$ (25)

For $n=mp+1,mp+2,\cdots ,\left(m+1\right)p$, $m=0,1,\cdots ,\left(N-1\right)$, where $v_{kj},v_{kj}^{*}$ depend on the quadrature rule used.

Now, for the Modified method of two Blocks we take $p=2$, this integration over $\left[0,t_{2m}\right]$ can be accomplished by Simpson’s rule, and the integral over $\left[t_{2m},t_n\right]$ by using a quadratic interpolation of the integrand at the point $t_{2m},t_{2m+1},t_{2m+2}$, then Equation (23) becomes:

${\varphi }_{i,2m+1}=W_i\left(t_{2m+1}\right)+\lambda {\displaystyle \underset{0}{\overset{\left(2m+1\right)h}{\int }}{F}_{i,s}\left(t_{2m+1},\tau \right)\gamma \left(\tau ,\phi \left(\tau \right)\right)\text{d}\tau }$ (26)

and

$\begin{array}{l}{\varphi }_{i,2m+2}\\ =W_i\left(t_{2m+2}\right)+\lambda {\displaystyle \underset{0}{\overset{\left(2m+2\right)h}{\int }}{F}_{i,s}\left(t_{2m+2},\tau \right)\gamma \left(\tau ,\phi \left(\tau \right)\right)\text{d}\tau }\end{array}$ (27)

where $i=1,2,\cdots ,l;m=0,1,\cdots$.

Therefore, by Equation (25) the approximate solution is computed by:

$\begin{array}{c}{\varphi }_{i,2m+1}=W_i\left(t_{2m+1}\right)+\lambda \left[\frac{h}{3}{\displaystyle \underset{j=0}{\overset{2m}{\sum }}{v}_j{F}_{i,s}\left(t_{2m+1},{\tau }_j\right)}\left(\gamma \left({\tau }_j,{\varphi }_{1j}\right),\cdots ,\gamma \left({\tau }_j,{\varphi }_{lj}\right)\right)\right]\\ +\lambda [\frac{h}{12}[5F_{i,s}\left(t_{2m+1},{\tau }_{2m}\right)\left(\gamma \left({\tau }_{2m},{\varphi }_{1,2m}\right),\cdots ,\gamma \left({\tau }_{2m},{\varphi }_{l,2m}\right)\right)\\ +8F_{i,s}\left(t_{2m+1},{\tau }_{2m+1}\right)\left(\gamma \left({\tau }_{2m+1},{\varphi }_{1,2m+1}\right),\cdots ,\gamma \left({\tau }_{2m+1},{\varphi }_{l,2m+1}\right)\right)\\ \begin{array}{c}\\ \end{array}-F\left(t_{2m+1},{\tau }_{2m+2}\right)\left(\gamma \left({\tau }_{2m+2},{\varphi }_{1,2m+2}\right),\cdots ,\gamma \left({\tau }_{2m+2},{\varphi }_{l,2m+2}\right)\right)]]\end{array}$ (28)

$\begin{array}{l}{\varphi }_{i,2m+2}\\ =W_i\left(t_{2m+2}\right)+\lambda \frac{h}{3}{\displaystyle \underset{j=0}{\overset{2m+2}{\sum }}{v^{\prime }}_j{F}_{i,s}\left(t_{2m+2},{\tau }_j\right)}\left(\gamma \left({\tau }_j,{\varphi }_{1j}\right),\cdots ,\gamma \left({\tau }_j,{\varphi }_{lj}\right)\right)\end{array}$ (29)

where $u_{i,0}=g_i\left(t_0\right)$.

Thus, replace the second term in Equation (28) by using integration formulas, then we get:

$\begin{array}{c}{\varphi }_{i,2m+1}=W_i\left(t_{2m+1}\right)+\lambda \left[\frac{h}{3}{\displaystyle \underset{j=0}{\overset{2m}{\sum }}{v}_j{F}_{i,s}\left(t_{2m+1},{\tau }_j\right)}\left(\gamma \left({\tau }_j,{\varphi }_{1j}\right),\cdots ,\gamma \left({\tau }_j,{\varphi }_{lj}\right)\right)\right]\\ +\lambda [\frac{h}{6}[F_{i,s}\left(t_{2m+1},{\tau }_{2m}\right)\left(\gamma \left({\tau }_{2m},{\varphi }_{1,2m}\right),\cdots ,\gamma \left({\tau }_{2m},{\varphi }_{l,2m}\right)\right)\\ +4F_{i,s}\left(t_{2m+1},{\tau }_{2m+\frac{1}{2}}\right)(\left(\frac{3}{8}{\varphi }_{1,2m}+\frac{3}{4}{\varphi }_{1,2m+1}-\frac{1}{8}{\varphi }_{1,2m+2}\right),\cdots ,\\ \left(\frac{3}{8}{\varphi }_{l,2m}+\frac{3}{4}{\varphi }_{l,2m+1}-\frac{1}{8}{\varphi }_{l,2m+2}\right))\\ \begin{array}{c}\\ \end{array}+F_{l,s}\left(t_{2m+1},{\tau }_{2m+1}\right)\left(\gamma \left({\tau }_{2m+1},{\varphi }_{1,2m+1}\right),\cdots ,\gamma \left({\tau }_{2m+1},{\varphi }_{l,2m+1}\right)\right)]]\end{array}$ (30)

$\begin{array}{c}{\varphi }_{i,2m+2}=W_i\left(t_{2m+2}\right)+\lambda [\frac{h}{3}{F}_{i,s}\left(t_{2m+2},{\tau }_0\right)\left(\gamma \left({\tau }_0,{\varphi }_{10}\right),\cdots ,\gamma \left({\tau }_0,{\varphi }_{l0}\right)\right)\\ +4F_{i,s}\left(t_{2m+2},{\tau }_1\right)\left(\gamma \left({\tau }_1,{\varphi }_{11}\right),\cdots ,\gamma \left({\tau }_1,{\varphi }_{l1}\right)\right)+\cdots \\ \begin{array}{c}\\ \end{array}+F_{i,s}\left(t_{2m+2},{\tau }_{2m+2}\right)\left(u_{1,2m+2},\cdots ,u_{l,2m+2}\right)]\end{array}$ (31)

where

$\begin{array}{l}{v}_0=v_{2m}=1,v_j=3-{\left(-1\right)}^j,j=1,2,\cdots ,2m-1\\ {v^{\prime }}_0={v^{\prime }}_{2m+2},{v^{\prime }}_j=3-{\left(-1\right)}^j,j=1,2,\cdots ,2m+1\end{array}$

Finally, we construct $2l$ nonlinear equations from (30) and (31) to find the unknown functions ${\varphi }_{i,2m+1},{\varphi }_{i,2m+2}$. The resulting system is solved by using modified Newton-Raphson method.

5. Numerical Examples

We solve two examples by RKM and BBMat $N=20,50$, $T=0.01,0.1,0.3$, $\lambda =1$ and $\mu =1$.

In Tables 1-6: f_Exact®Exact solution, f_R.K.®approximate solution of RKM, E_R.K.®the absolute error of RKM, f_B.B.®approximate solution of BBM, E_B.B.®the absolute error of BBM.

Example 1

Consider: $\varphi \left(x,t\right)=xt-\frac{x^2{t}^2}{5}-\frac{x^2{t}^6}{5}+{\displaystyle {\int }_0^1{x}^2{y}^2{\varphi }^2\left(y,t\right)\text{d}y}+{\displaystyle {\int }_0^{t}t\tau {\varphi }^2\left(x,\tau \right)\text{d}\tau }$

Exact solution $\varphi \left(x,t\right)=xt$

Case 1: $N=20$

Case 2: $N=50$

6. Conclusions

From the previous discussions we conclude the following:

1) As N is increasing the errors are decreasing.

2) As x and t are increasing in $\left[0,1\right]\times \left[0,1\right]$, the errors due to RKM and BBM are also increasing.

3) The errors due to the BBM are less than the errors due to RKM (i.e. BBM the better than RKM to solve NF-VIE with continuous kernel).