On the Solution of Quadratic Nonlinear Integral Equation with Different Singular Kernels

Abstract

All the previous authors discussed the quadratic equation only with continuous kernels by different methods. In this paper, we introduce a mixed nonlinear quadratic integral equation (MQNLIE) with singular kernel in a logarithmic form and Carleman type. An existence and uniqueness of MQNLIE are discussed. A quadrature method is applied to obtain a system of nonlinear integral equation (NLIE), and then the Toeplitz matrix method (TMM) and Nystrom method are used to have a nonlinear algebraic system (NLAS). The Newton–Raphson method is applied to solve the obtained NLAS. Some numerical examples are considered, and its estimated errors are computed, in each method, by using Maple 18 software.

On a Solution of a Third Kind Mixed Integro-Differential Equation with Singular Kernel Using Orthogonal Polynomial Method

“…In thermoelasticity, Abdou and Basseem in [2] derived a closed form of Gaursat functions for first and second fundamental problems with variant time. Basseem and Alalyani in [3] used Toeplitz and Nystrom methods on the solution of quadratic nonlinear integral equation with different singular kernels. The theory of singular integral equations, developed by the scholar Muskhelishvili in [4], has assumed different techniques and increased important applications in different areas of science.…”

Solution of nonlinear mixed integral equation via collocation method basing on orthogonal polynomials

“…While, in [18] Almousa and Ismail used the same two numerical methods to obtain the numerical solution of Volterra- Fredholm integral equation in two- dimensional. For quadratic integral equations, Basseem and Alalyani in [19] , used Chebyshev polynomials method for solving a quadratic integral equation with a continuous kernel. While, in [20] Abdou, et al used the Chebyshev polynomial method to solve numerically, quadratic integral equation with discontinuous continuous.…”

A new Model for Solving Three Mixed Integral Equations with Continuous and Discontinuous Kernels

“…For example, the radiation transfer process is described using the Chandrasekhar H-function, the biological populations are evolving in a way that obeys the delayed Volterra integro differential equations. Also, there are the integral equations with discontinuous kernels that occur in elasticity and solid mechanics during the formulation of the boundary value problem of the mixed type, see [1][2][3][4][5][6][7][8] and the references therein. The importance of the mixed integral equations (MIEs), in position and time, and the contact problems came from the work of Abdou [1].…”