An effective direct method to determine the numerical solution of the specific nonlinear Volterra-Fredholm integro-differential equations is proposed. The method is based on new vector forms for representation of triangular functions and its operational matrix. This approach needs no integration, so all calculations can be easily implemented. Some numerical examples are provided to illustrate the accuracy and computational efficiency of the method.

Numerical solution of nonlinear Volterra-Fredholm integro-differential equations via direct method using triangular functions

Two-dimensional orthogonal triangular functions (2D-TFs) are presented as a new set of basis functions for expanding 2D functions. Their properties are determined and an operational matrix for integration obtained. Furthermore, 2D-TFs are used to approximate solutions of nonlinear two-dimensional integral equations by a direct method. Since this approach does not need integration, all calculations can be easily implemented, and several advantages in reducing computational burdens arise. Finally, the efficiency of this method will be shown by comparison with some numerical results.

Two-dimensional triangular functions and their applications to nonlinear 2D Volterra-Fredholm integral equations

https://www.cs.ubc.ca/~arnaud/doucet_johansen_tadic_MCMCfredholm.pdf

On solving integral equations using Markov chain Monte Carlo methods

Applications of two-dimensional triangular functions for solving nonlinear class of mixed Volterra–Fredholm integral equations

E. Babolian a, Z. Masouri b , S. Hatamzadeh-Varmazyar c (a) Department of Mathematics, Teacher Training University, Tehran, Iran (b) Department of Mathematics, Khorramabad Branch, Islamic Azad University, Khorramabad, Iran (c) Department of Electrical Engineering, Science and Research Branch, Islamic Azad University, Tehran, Iran |||||||||||||||||||||||||||||||Abstract A practical direct method to compute numerical solutions of the linear Volterra and Fredholm integral equations system is proposed. This approach is based on vector forms of triangular functions and its operational matrices and without any integration reduces an integral equations system to a system of algebraic equations. Numerical results of some examples show that the method is practical and has high accuracy.

https://ijim.srbiau.ac.ir/article_2018_4c7a85b99689a4b31008d117da60348d.pdf

M. Salmani

Papers

Using triangular orthogonal functions for solving Fredholm integral equations of the second kind

Using direct method for solving variational problems via triangular orthogonal functions