Effective Technique for Converting Ill-Posed Volterra Equation to Integro-Differential Equation and Solving It

We will transfer the integral Eq. (1) to an integro-differential equation of the second kind defined in [0,1]. It is important to note that the solution for an ill-posed problem may not exist, and if it does exist it may not be unique. We will use in this section the Taylor series and Leibnitz rule.

Let A(t) be a function with derivatives of all orders with respect to t in an interval [0,1] than for 0<t-ε<t<t+ε<1, with ε→0. The Taylor series is given by:

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

where, O(εⁿ) is an unknown error term of approximation.

In last our work [8], we transformed the Volterra integral Eq. (1) to an equivalent integral equation of the second kind defined in the interval [0,1] by using Taylor series of the first order, and we found this equivalent equation:

$\phi_{\varepsilon}(t)+\int_0^t K(t, x) \phi_{\varepsilon}(x) d x=f_{\varepsilon}(t)$                (3)

where, $K(t, x)=\frac{k(t, x)+\varepsilon \frac{\partial k(t, x)}{\partial t}}{\varepsilon k(t, t)}$, $f_{\varepsilon}(t)=\frac{f(t+\varepsilon)}{\varepsilon k(t, t)}$ and ϕ_ε=ϕ if ε→0.

For more information, see researches [9].

Now, by using the Taylor series of the second order Eq. (2) and Leibnitz rule for equation of first kind Eq. (1), we find:

$\begin{array}{r}A(t+\varepsilon)=A(t)+\varepsilon \int_0^t \frac{\partial k(t, x)}{\partial t} \varphi(x) d x+\frac{\varepsilon^2}{2 !} \frac{\partial k(t, x)}{\partial t} \varphi(t)+ \\ +\frac{\varepsilon^2}{2 !} k(t, t) \varphi^{\prime}(t)+\frac{\varepsilon^2}{2 !} \int_0^t \frac{\partial^2 k(t, x)}{\partial t^2} \varphi(x) d x+\varepsilon k(t, t) \varphi(t)+O\left(\varepsilon^2\right),\end{array}$

Then, the equivalent equation of Volterra equation of the first kind Eq. (1) is given by:

$\begin{aligned} \frac{\varepsilon^2}{2 !} k(t, t) \varphi^{\prime}(t)= & -\int_0^t\left(k(t, x)+\varepsilon \frac{\partial k(t, x)}{\partial t}+\frac{\varepsilon^2}{2 !} \frac{\partial^2 k(t, x)}{\partial t^2}\right) \varphi(x) d x -\left(\varepsilon k(t, t)+\frac{\varepsilon^2}{2 !} \frac{\partial k(t, t)}{\partial t}\right) \varphi(t)+f(t+\varepsilon)+O\left(\varepsilon^2\right),\end{aligned}$

after simplify, we obtain:

$\begin{aligned} \varphi^{\prime}(t)= & \frac{-2}{\varepsilon^2 k(t, t)} \int_0^t\left(k(t, x)+\varepsilon \frac{\partial k(t, x)}{\partial t}+\frac{\varepsilon^2}{2 !} \frac{\partial^2 k(t, x)}{\partial t^2}\right) \varphi(x) d x -\frac{1}{k(t, t)}\left(\frac{2}{\varepsilon}+\frac{\partial k(t, t)}{\partial t}\right) \varphi(t)+\frac{2}{\varepsilon^2 k(t, t)} f(t+\varepsilon),\end{aligned}$

where, k(t,t)≠0, $\frac{\partial k(t, x)}{\partial t} \neq 0$ for t$\in$[0,1] and ε→0, we obtain the Volterra integro-differential equation of the second kind of the following form:

$\phi^{\prime}(t)=K_1(t) \phi_{\varepsilon}(t)+\int_0^t K(t, x) \phi_{\varepsilon}(x) d x+F_{\varepsilon}(t)$            (4)

where,

$\begin{aligned} K(t, x) & =\frac{-2}{\varepsilon^2 k(t, t)}\left(k(t, x)+\varepsilon \frac{\partial k(t, x)}{\partial t}+\frac{\varepsilon^2}{2 !} \frac{\partial^2 k(t, x)}{\partial t^2}\right), \\ K_1(t) & =\frac{-1}{k(t, t)}\left(\frac{2}{\varepsilon}+\frac{\partial k(t, t)}{\partial t}\right), \\ F_{\varepsilon}(t) & =\frac{2}{\varepsilon^2 k(t, t)} f(t+\varepsilon),\end{aligned}$

and, ϕ_ε(t)=ϕ(t) if ε→0. Substituting t=0 into Eq. (3) gives the initial condition ϕ_ε(0)=ϕ₀.

Now, we apply modified Simpson method [10] for Eq. (4) and take ϕ_ε(t)=ϕ(t). Consider let:

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

be an equidistant subdivision of a step h=t_2j+1-t_2j for j=0,1,2,…,n. Our objective then, it's to approximate the solutions of the equivalent Eq. (4) to the nodes of even indices (at the point t_2j), then the modified Simpson have the form:

$\int_{t_{2 j}}^{t_{2 j+2}} f(t) d t=\frac{h}{3}\left[f\left(t_{2 j}\right)+4 f\left(t_{2 j+1}\right)+f\left(t_{2 j+2}\right)\right]$            (5)

with the error of integration is:

$E(h)=-2 \frac{(h / 2)^5}{90}(f(\zeta))^{(4)}$

Now, by the numerical integration formulas of modified Simpson method Eq. (5) and finite difference formulation for the integro-differential Eq. (4), we obtain the following iteration formula:

$\begin{aligned} \phi^{\prime}\left(t_{2 j}\right)= & \frac{h}{3} \sum_{i=0}^{j-1}\left[\begin{array}{l}K\left(t_{2 j}, x_{2 i}\right) \phi\left(t_{2 i}\right) +4 K\left(t_{2 j}, x_{2 i+1}\right) \phi\left(t_{2 i+1}\right)+K\left(t_{2 j}, x_{2 i+2}\right) \phi\left(t_{2 i+2}\right)\end{array}\right] +K_1\left(t_{2 j}\right) \phi\left(t_{2 j}\right)+F\left(t_{2 j}\right)\end{aligned}$            (6)

We approximate $\varphi_{2 j}^{\prime}$ and $\varphi_{2 i+1}$ by $\frac{\varphi_{2 j+2}-\varphi_{2 j}}{2 h}$ and $\frac{\varphi_{2 i}+\varphi_{2 i+2}}{2}$ respectively. The Eq. (6) becomes:

$\begin{aligned} \phi_{2 j+2}=\frac{2 h^2}{3}\left(\sum_{i=0}^{j-1}\right. & \left(K_{2 j, 2 i}+2 K_{2 j, 2 i+1}\right) \phi_{2 i} \left.+\sum_{i=0}^{j-1} \left(2 K_{2 j, 2 i+1}+K_{2 j, 2 i+2}\right) \phi_{2 i+2}\right) +\left(2 h.\ K_1+1\right) \phi_{2 j}+2 h. \ F_{2 j} .\end{aligned}$

By recurrence, we can to calculate the approximation solutions ϕ of the Eq. (4) in all points t_2j for j=0, 1, …, n. Cleary from Eq. (3) the initial value of e is ϕ(0)=ϕ₀=f_ε(0).