An Algorithm for the Solution of Nonlinear Volterra–Fredholm Integral Equations with a Singular Kernel

Abstract

The nonlinear Volterra–Fredholm integral Equation (NVFIE) with a singular kernel is discussed such that the kernel of position can take the Hilbert kernel form, Carleman function, logarithmic form, or Cauchy kernel. Using the quadrature method, the NVFIE with a singular kernel leads to a system of nonlinear integral equations. The existence and unique numerical solution of this system is discussed, as is the truncation error of the numerical solution. The solution of the nonlinear integral equation system is obtained using the spectral relations and techniques of the Chebyshev polynomial method. Finally, we will discuss examples of when the kernel takes various forms to demonstrate this technique’s high accuracy and simplicity. Some numerical results and estimating errors are calculated and plotted using the program Wolfram Mathematica 10.

Keywords

• singular nonlinear Volterra–Fredholm integral equation
• algebraic system
• Banach fixed point theorem
• singular kernel
• Chebyshev polynomials