Abstract
Many different problems in mathematics, physics, engineering can be expressed in the form of integral equations. Among these are diffraction problems, population growth, heat transfer, particle transport problems, electrical engineering, elasticity, control, elastic waves, diffusion problems, quantum mechanics, heat radiation, electrostatics and contact problems. Therefore, the solutions which are obtained by the mathematical methods play an important role in these fields. The most two basic types of integral equations are called Fredholm (FIEs) and Volterra (VIEs). In many instances, the ordinary and partial differential equations can be converted into Fredhom and Volterra integral equations that are solved more effectively. We aim through this research to present an improved Adomian decomposition method based on modified Bernstein polynomials (ADM-MBP) to solve nonlinear integral equations of the second kind. We introduced efficient method, constructed on modified Bernstein polynomials. The formulation is developed to solve nonlinear Fredholm and Volterra integral equations of second kind. This method is tested for some examples from nonlinear integral equations. Maple software was used to obtain the solutions of these examples. The results demonstrate reliability of the proposed method. Generally, the proposed method is very convenient to apply to find the solutions of Fredholm and Volterra integral equations of second kind.
Highlights
To introduce our research we will need the following basic definitions, concepts and results of integralFredholm and Volterra family of integral equations play equations and Bernstein polynomials.Mathematics and Statistics 8(3): 278-285, 2020The integral equation in g x is of the form: h(x)g x f x λ k x, t g t dt, (1)u(x) where u(x) and h(x) may be both fixed, variables, or mixed, is a constant, k x,t is the kernel, f (x) is given function and g(x) is unknown
The Adomian Decomposition Method (ADM)-Modified Bernstein Polynomials (MBP) solution can be obtained as gm x gj x
We improved ADM based on MBP in order to solve nonlinear Fredholm and Volterra integral equations
Summary
To introduce our research we will need the following basic definitions, concepts and results of integral. Fredholm and Volterra family of integral equations play equations and Bernstein polynomials. U(x) where u(x) and h(x) may be both fixed, variables, or mixed, is a constant, k x,t is the kernel, f (x) is given function and g(x) is unknown. If u(x) and h(x) are fixed, it is called a Fredholm. If at least one limit is a variable, it is called a Volterra
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