Application of hybrid functions operational matrices in the numerical solution of two-dimensional nonlinear integral equations
This paper is devoted to the two-dimensional nonlinear Volterra integral equations of the second kind \[ f(x,t)=g(x,t)+\int\limits_{0}^{t}\int\limits_{0}^{x}k(x,t,y,z)H(y,z,f(y,z))dydz, \; (x,t)\in\Omega=[0,1)\times [0,1) \] with respect to the unknown function $f(x,t)$ defined on $\Omega$. \par An effective numerical technique based on the two-dimensional hybrid of Block-pulse functions and Legendre polynomials is presented to approximate the solution of a class of two-dimensional nonlinear Volterra integral equations of the second kind. \par Two-dimensional hybrid of Block-pulse functions and Legendre polynomials on the $\Omega$ can be defined as follows \[ \psi_{i_1,j_1,i_2,j_2}(x,t)=\begin{cases} L_{j_1}(2Nx-2i_1+1)L_{j_2}(2Nt-2i_2+1), & (x,t)\in [\frac{i_1-1}{N},\frac{i_1}{N})\times [\frac{i_2-1}{N},\frac{i_2}{N}), \\ 0, & \text{otherwise,} \end{cases} \] where $i_1,i_2=\overline{1,N}$, $j_1,j_2=\overline{1,M-1}$, and $L_{j_i}, \; i=1,2,$ are the well-known Legendre polynomials of order $j_i$. \par The main idea of this work is to develop the operational matrices of hybrid functions into two-dimensional modes. These operational matrices are then applied to convert the two-dimensional nonlinear integral equations into a system of nonlinear algebraic equations that can be solved numerically by Newton's method. An approximation of the error bound for the proposed method is presented. Finally, some numerical examples are considered to confirm the applicability and the efficiency of the method, especially in cases where the solution is not sufficiently smooth.
scientific article; zbMATH DE number 6995986
scientific article