Using of Bernstein spectral Galerkin method for solving of weakly singular Volterra–Fredholm integral equations

Abstract

In this work, we applied a new method for solving the linear weakly singular mixed Volterra-Fredholm integral equations. We now begin the theoretical study with acquirement of the variational form; in addition, we are using Bernstein spectral Galerkin method to be approximate to my problems. We estimate the error of the method by proved some theorems. Moreover, in the final section, we solved some numerical examples.

Investigation approach for a nonlinear singular Fredholm integro-differential equation

“…Also, other equation's type as fuzzy or fractional Fredholm integro-differential equations can be investigated using our numerical method. Last, but not least, the author plans to explore more venues: Similar ideas to the ones described in the paper of [5] can be applied to an integrodifferential equation.…”

Numerical Solutions for Systems of Fractional and Classical Integro-Differential Equations via Finite Integration Method Based on Shifted Chebyshev Polynomials

“…Thus, we achieve the operational matrix of Riemann-Liouville fractional integral. Note that, the elements contained in J α can be computed by (12). However, we observe that for the order α = 1, we have J α = S which is defined in Section 2.…”

“…However, the fractional order derivative of FIDEs can be reduced to a positive integer order. Then, it is called the classical integro-differential equation (CIDE) which is frequently used to describe many applications which can be seen in [11][12][13] for details of applications. Actually, many problems of both FIDE and CIDE are often constructed to be a system.…”

A numerical approach for a nonhomogeneous differential equation with variable delays

“…a. Delay differential equations of the type 1 arise in a variety of applications including control systems, electrodynamics, mixing liquids, neutron transportation, population models, physiological processes and conditions including production of blood cells [1,14,23,25,27,28,34,37].…”