Projected Iterations of Fixed-Point Type to Solve Nonlinear Partial Volterra Integro-Differential Equations

Abstract

In this paper, we propose a method to approximate the fixed point of an operator in a Banach space.Using biorthogonal systems, this method is applied to build an approximation of the solution of a class of nonlinear partial integro-differential equations. The theoretical findings are illustrated with several numerical examples, confirming the reliability, validity and precision of the proposed method.

Taylor collocation method for solving two‐dimensional partial Volterra integro‐differential equations

“…The absolute error function is depicted in three-dimensional space as shown in Figure 3. where u(x, 𝑦) = 𝑦sin(x) as the analytic solution, and g(x, 𝑦) is calculated using the exact solution and obtained similarly as [10] g(x, 𝑦) = 𝑦cos(x) − x 2 𝑦 2 sin 2 ( x…”

“…Example In this last example, consider the two‐dimensional PVIDE discussed in Berenguer and Gamez [10] $$$$ \frac{\partial u}{\partial x}\left(x,y\right)=g\left(x,y\right)+{\int}_0^x{\int}_0^y{x}^2u\left(t,s\right) dsdt,\kern0.30em 0\le x\le 1,0\le y\le 1, $$$$ subject to initial condition $$$$ {u}_0(y)=0,0\le y\le 1, $$$$ where $$$ u\left(x,y\right)= ysin(x) $$$ as the analytic solution, and $$$ g\left(x,y\right) $$$ is calculated using the exact solution and obtained similarly as [10] $$$$ g\left(x,y\right)= ycos(x)-{x}^2{y}^2 si{n}^2\left(\frac{x}{2}\right). $$$$ The numerical results obtained in this example for $$$ N=M=10 $$$ are compared in Table 4 with the numerical results obtained by using th...…”

“…Partial integro‐differential equations arise in problems of applied sciences and engineering to model dynamical systems; they can be found in financial mathematics [1–29], biological models [17], fluid dynamics [5], and many other areas. Partial Volterra integro‐differential equations (PVIDEs) have many different types, such as elliptic [30], hyperbolic [12], and parabolic [21] in one‐ or multidimensional, that is why numerous numerical schemes have been well‐evaluated by researchers to approximate the solution of these equations.…”

“…Equations of the form () were studied by Hussain et al [19] by using an iteration variational method. Berenguer and Gamez [10] used biorthogonal systems along with fixed point theory in Banach spaces to solve these equations. Singh et al [33, 34] studied the PVIDEs of the form () with a weakly singular kernel by using an operational matrix approach based on $$$ 2D $$$ shifted Legendre polynomials, and Singh et al [28] studied Equation by using an operational matrix approach based on 2D Legendre and Chebyshev wavelets collocation method.…”

“…In this last example, consider the two-dimensional PVIDE discussed in Berenguer and Gamez[10] 𝜕u 𝜕x (x, 𝑦) = g(x, 𝑦) + ∫ u(t, s)dsdt, 0 ≤ x ≤ 1, 0 ≤ 𝑦 ≤ 1,subject to initial condition u 0 (𝑦) = 0, 0 ≤ 𝑦 ≤ 1,…”