Numerical Solution of Volterra-Hammerstein Integral Equation of the First Kind by Finite Difference Method Decomposition with Nyström Method

In this part, we’ll transmit the nonlinear integral equation Eq. (2) to a second kind integral equation defined in the interval I=[0,1]. It is necessary to highlight that the solution for ill-posed problems is generally unstable, and slight changes can make large errors. The Leibnitz theorem and the Taylor series are going to be applied in the next section:

Let $A(t)$ be a function with $\mathrm{n}$ derivatives with respect to $t$ in $I=[0,1]$ than for $0<t-\varepsilon<t<t+\varepsilon<1$, with $\varepsilon \rightarrow 0$. The Taylor series is given by

$\begin{aligned} A(t+\varepsilon) & =A(t)+\frac{\varepsilon}{1 !} \cdot \frac{\partial A(t)}{\partial t}+\frac{\varepsilon^2}{2 !} \cdot \frac{\partial^2 A(t)}{\partial t^2}+\cdots +\frac{\varepsilon^n}{n !} \cdot \frac{\partial^n A(t)}{\partial t^n}+O\left(\varepsilon^n\right), \\ A(t-\varepsilon) & =A(t)-\frac{\varepsilon}{1 !} \cdot \frac{\partial A(t)}{\partial t}+\frac{\varepsilon^2}{2 !} \cdot \frac{\partial^2 A(t)}{\partial t^2}-\cdots +(-1)^n \frac{\varepsilon^n}{n !} \cdot \frac{\partial^n A(t)}{\partial t^n}+O\left(\varepsilon^n\right),\end{aligned}$

where, $O\left(\varepsilon^n\right)$ is the approximation error term. For the derivation of higher order approximations to derivatives of any order, the Taylor expansion is a very helpful tool. If $\varepsilon$ is small, then higher order accuracy generally means higher accuracy. The first order of Taylor series is given by:

$A(t+\varepsilon)=A(t)+\frac{\varepsilon}{1 !} \cdot \frac{\partial A(t)}{\partial t}+O(\varepsilon)$          (3)

$A(t-\varepsilon)=A(t)-\frac{\varepsilon}{1 !} \cdot \frac{\partial A(t)}{\partial t}+O(\varepsilon)$          (4)

The Leibnitz rule is a famous rule utilizing for differentiation of integrals [5]. Let $A(t)$ and $\frac{\partial A(t)}{\partial t}$ be continuous in the domain $0 \leq t \leq 1$, and let:

$A(t)=\int_{g(t)}^{h(t)} H(t, x) d x$,

then differentiation of the integral in $A(t)$ exists and is given by:

$\begin{aligned} & \frac{\partial A(t)}{\partial t}=H(t, h(t)) \frac{d h(t)}{d t}-H(t, g(t)) \frac{d g(t)}{d t} +\int_{g(t)}^{h(t)} \frac{\partial H(t, x)}{\partial t} d x .\end{aligned}$

In our previous work [12, 13], we converted the Volterra-Hammerstein integral Eq. (2) to a second kind integral equation defined in the interval [0, 1] by using Taylor series of the first order Eq. (3) (This approximation is known as: "fda" the forward difference approximate of $A(t)$), we obtained this equivalent equation:

$\begin{aligned} & \int_a^{t+\varepsilon} k(t+\varepsilon, x) F\left(\phi_{\varepsilon}(x)\right) d x=f(t)+\varepsilon k(t, t) F\left(\phi_{\varepsilon}(t)\right) +\varepsilon \int_a^t \frac{\partial k}{\partial t}(t, x) F\left(\phi_{\varepsilon}(x)\right) d x=f(t+\varepsilon)\end{aligned}$,

then, we get a well-posed integral equation (VK2), which is given by:

$\begin{gathered}F\left(\phi_{\varepsilon}(t)\right)+\int_a^t \frac{k(t, x)+\varepsilon \frac{\partial k(t, x)}{\partial t}}{\varepsilon k(t, t)} F\left(\phi_{\varepsilon}(x)\right) d x=\frac{f(t+\varepsilon)}{\varepsilon k(t, t)}， \\ \varepsilon \rightarrow 0, \text { and } k(t, t) \neq 0 .\end{gathered}$        (5)

Eq. (5) can be rewritten as follows:

$\Phi_{\varepsilon}(t)+\int_0^t K(t, x) \Phi_{\varepsilon}(x) d x=f_{\varepsilon}(t)$       (6)

where,

$\begin{gathered}K(t, x)=\frac{k(t, x)+\varepsilon \frac{\partial k(t, x)}{\partial t}}{\varepsilon k(t, t)}, f_{\varepsilon}(t)=\frac{f(t+\varepsilon)}{\varepsilon k(t, t)} \\ \Phi_{\varepsilon}=F\left(\phi_{\varepsilon}(t)\right)=\Phi \text { if } \varepsilon \rightarrow 0\end{gathered}$.

and $\phi_{\varepsilon}(t)=F^{-1}\left(\Phi_{\varepsilon}(t)\right)=\phi(t)$ if $\varepsilon \rightarrow 0$. Substituting $t=0$ into Eq. (6) gives the initial condition $\Phi_{\varepsilon}(0)=\Phi_0$. For more details, refer to references [11-13].

If we use the Taylor series of the first order Eq. (4) (the first order backward difference approximation "bda"), and Leibnitz rule, we get a well-posed integral equation (VK2), which is given by:

$\begin{gathered} & F\left(\phi_{\varepsilon}(t)\right)+\int_a^t \frac{-k(t, x)+\varepsilon \frac{\partial k(t, x)}{\partial t}}{\varepsilon k(t, t)} F\left(\phi_{\varepsilon}(x)\right) d x =\frac{-f(t-\varepsilon)}{\varepsilon k(t, t)}, \\ &\varepsilon \rightarrow 0, \text { and } k(t, t) \neq 0 .\end{gathered}$         (7)

Eq. (7) can be simplified as follows:

$\Phi_{\varepsilon}(t)+\int_0^t K(t, x) \Phi_{\varepsilon}(x) d x=f_{\varepsilon}(t)$,

where

$\begin{aligned} & K(t, x)=\frac{k(t, x)+\varepsilon \frac{\partial k(t, x)}{\partial t}}{\varepsilon k(t, t)} \\ & f_{\varepsilon}(t)=\frac{-f(t-\varepsilon)}{\varepsilon k(t, t)} \\ & \Phi_{\varepsilon}=F\left(\phi_{\varepsilon}(t)\right)=\Phi \text { if } \varepsilon \rightarrow 0\end{aligned}$

Now, if we use Eq. (3) and Eq. (4) (the central difference approximation "cda") of the flowing form:

$A(t+\varepsilon)-A(t-\varepsilon) \approx 2 \frac{\varepsilon}{1 !} A^{\prime}(t)+O(\varepsilon), \quad \varepsilon \rightarrow 0$

and Leibnitz rule, we get a well-posed integral equation (VK2), which is given by:

$\begin{gathered} & F\left(\phi_{\varepsilon}(t)\right)+\int_a^t \frac{k^{\prime}(t, x)}{k(t, t)} F\left(\phi_{\varepsilon}(x)\right) d x =\frac{f(t+\varepsilon)-f(t-\varepsilon)}{2 \varepsilon k(t, t)}, \\ &\varepsilon \rightarrow 0, \text { and } k(t, t) \neq 0 .\end{gathered}$

Eq. (8) can be rewritten in the form:

$\Phi_{\varepsilon}(t)+\int_0^t K(t, x) \Phi_{\varepsilon}(x) d x=f_{\varepsilon}(t)$,

where,

$\begin{aligned} & K(t, x)=\frac{k^{\prime}(t, x)}{k(t, t)}, f_{\varepsilon}(t)=\frac{f(t+\varepsilon)-f(t-\varepsilon)}{2 \varepsilon k(t, t)} \\ & \Phi_{\varepsilon}=F\left(\phi_{\varepsilon}(t)\right)=\Phi \text { if } \varepsilon \rightarrow 0 .\end{aligned}$

Now, we state the theorem of the existence and uniqueness of the solution to Volterra integral equation of the first kind [14, 15]:

Theorem [14]: Assume that,

1. $k(t, x)$ and $\frac{\partial k(t, x)}{\partial t}$ are continuos in $0 \leq x \leq t \leq T$,
2. $k(t, t)$ does not vanish anywhere in $0 \leq x \leq t \leq T$,
3. $f(0)=0$,
4. $f(t)$ and $f^{\prime}(t)$ are continuos in $0 \leq x \leq t \leq T$.

Then:

$\int_0^t k(t, x) \Phi(x) d x=f(t), \quad 0 \leq x \leq t \leq T$

has a unique continuous solution. This solution is identical with the continuous solution of

$k(t, x) \Phi(x)+\int_a^t \frac{\partial k(t, x)}{\partial t} \Phi(x) d x=f^{\prime}(t)$.

There are different iterative methods available for resolving nonlinear Volterra integral equations such as variational iteration method, Adomian's decomposition method, and homotopy perturbation method [16-18]. In our previous research [12], we applied the variational iteration method with Taylor series for solving the above ill-posed problem Eq. (2) which is equivalent to the well-posed problems Eq. (5), Eq. (7) and Eq. (8). Having converted the above integral equation of the first kind to the linear Volterra integral equation of the second kind, we then can use any numerical method like Nyström methods; trapezoidal method, Simpson method, modified Simpson method.

In this section, we shall describe the quadrature or Nyström method (Trapezoidal rule and Simpson’s rule….), for the approximate solution of linear Volterra integral equations of the second kind with continuous kernels.

The quadrature methods are intended to estimate the definite integral of $f(t)$ over the interval $I=[a, b]$ by evaluating $f(t)$ at a finite number of sample points.

Assume that:

$a=t_1^{(n)}<t_2^{(n)}<\cdots<t_n^{(n)}=b$.

A form of formula:

$Q_n[f]=\sum_{i=1}^n w_i^{(n)} f\left(t_i^{(n)}\right)$,

with a property that

$\int_a^b f(t) d t=Q_n[f]+E[f]$,

The term $E[f]$ is called the truncation error for integration. The values $\left\{t_i^{(n)}\right\}_{i=1}^n$ are called the quadrature nodes and $\left\{w_i^{(n)}\right\}_{i=1}^n$ are called the weights. A sequence $Q[f]$ of quadrature formulas is called convergent if $Q_n[f] \rightarrow Q[f]$, $n \rightarrow \infty$, for all $f \in C(I) [16]$.

Corollary [19] (Trapezoidal rule: Error analysis)

Suppose that $[a, b]$ is subdivided into $\mathrm{n}$ subintervals $\left[t_i, t_{i+1}\right]$ of width $h=(b-a) / n$. The composite trapezoidal rule:

$T(f, h)=\frac{h}{2}(f(a)+f(b))+h \sum_{i=1}^{n-1} f\left(x_i\right)$

is an approximation to the integral

$\int_a^b f(x) d x=T(f, h)+E_T(f, h)$.

Furthermore, if $f \in C^2[a, b]$, there exists a value $c$ with $a<c<b$ so that the error term $E_T(f, h)$ has the form:

$E_T(f, h)=\frac{-(b-a) f^{(2)}(c) h^2}{12}=O\left(h^2\right)$.

The linear integral of the second kind can be mathematically approximated using any quadrature rule, as shown below:

$\int_I k(t, x,) \phi(x) d x \approx \sum_{j=1}^n w_j k\left(t, x_j,\right) \phi\left(x_j\right)$,

with quadrature points (nodes) $\left\{x_j\right\}_{j=1}^n$ contained in $I$ and real quadrature weights $\left\{w_j\right\}_{j=1}^n$.

$\phi_n(t)+\sum_{j=1}^n w_j k\left(t, x_j\right) \phi_n\left(x_j\right)=f(t)$,

where $\phi_n(t)$ is an approximation to $\phi(t)$.

By the numerical integration formulas of Trapezium rule, so we get

$\phi\left(t_j\right)=f\left(t_j\right)+\frac{h}{2}\left[\begin{array}{l}k\left(t_j, t_1\right) \phi\left(t_1\right)+2 \sum_{i=2}^{j-1} k\left(t_j, t_i\right) \phi\left(t_i\right) \\ +k\left(t_j, t_J\right) \phi\left(t_J\right)\end{array}\right]$

We will now apply modified Simpson method and take $\Phi_{\varepsilon}(t)=\Phi(t)$. Consider let

$t_0=0<t_1<\cdots<t_{2 j}<\cdots<t_{2 n}=1$,

be a step's equidistant subdivision $h=t_{2 j+1}-t_{2 j}$ for $j=0,1, \ldots, n$. The goal is to approximate the solutions of the approximation of the second kind equation Eq. (5), Eq. (7) or Eq. (8) to the all nodes of 2j indices (at the point $t_{2 j}$), then the form of modified Simpson method is:

$\int_{t_{2 j}}^{t_{2 j+2}} g(t) d t=\frac{h}{3}\left[g\left(t_{2 j}\right)+4 g\left(t_{2 j+1}\right)+g\left(t_{2 j+2}\right)\right]$,

where the integration error is $O(h)=-2 \frac{(h / 2)^5}{90}(g(\zeta))^{(4)}$. This method has been used by Nadir and Rahmoune [20].

By using this method can be written the equation Eq. (5), Eq. (7) and Eq. (8) of the second kind in the algorithm of the following form:

$\begin{aligned} \Phi\left(t_{2 j}\right)= & \frac{h}{3} \sum_{i=0}^{j-1}\left[\begin{array}{l}K\left(t_{2 j}, x_{2 i}\right) \Phi\left(t_{2 i}\right)+4 K\left(t_{2 j}, x_{2 i+1}\right) \Phi\left(t_{2 i+1}\right) \\ +K\left(t_{2 j}, x_{2 i+2}\right) \Phi\left(t_{2 i+2}\right)\end{array}\right] +f\left(t_{2 j}\right)\end{aligned}$

We approximate $\Phi_{2 i+1}$ by $\frac{\Phi_{2 i}+\Phi_{2 i+2}}{2}$, the Eq. (5) becomes:

$\begin{aligned} & \Phi_{2 j}\left(1-\frac{h}{3}\left(2 K_{2 j, 2 j-1}+K_{2 j, 2 j}\right)\right) = f_{2 j}+\left(\frac{h}{3}\left(K_{2 j, 0}+2 K_{2 j, 1}\right)\right) \Phi_0 \\ & +\frac{2 h^2}{3} \sum_{i=0}^{j-1}\left(K_{2 j, 2 i}+2 K_{2 j, 2 i+1}\right) \Phi_{2 i} +\frac{2 h^2}{3} \sum_{i=0}^{j-1}\left(2 K_{2 j, 2 i+1}+K_{2 j, 2 i+2}\right) \Phi_{2 i+2} .\end{aligned}$

We are able to calculate the approximate solutions $\Phi$ of the equations Eq. (5), Eq. (7) and Eq. (8) by using recurrence. in all points $t_{2 j}$ for $j=0,1, \ldots, n$. It is evident that the initial value of $\Phi$ is $\Phi(0)=\Phi_0=f_{\varepsilon}(0)$. Suppose that $F\left(\phi_{\varepsilon}(t)\right)$ is invertible. After that, we will be able to set:

$\phi_{\varepsilon}=F^{-1}(\Phi(t))$

In this paper, we gave an approach technique that uses Taylor series and Nyström methods for approximating solution of first kind nonlinear Volterra problems, the effectiveness of this above technique was tested by utilizing three distinct examples. It has been observed that all equivalent equations converge and the absolute error is near which was proved that numeric results were accepted for all types of the first kind Volterra-Hammerstein integral equation. Then, the most accurate approximation by Taylor series is the central difference approximation "cda".

In our future project, we will apply the techniques that have been proposed to general equations and systems of all ill-posed problems.