Abstract
Volterra integral equations are a special type of integrative equations; they are divided into two categories referred to as the first and second type. This paper will deal with the second type which has wide range of the applications in physics and engineering problems. Spline functions are piece-wise polynomials of degree <i>n</i> joined together at the break points with <i>n</i>-1 continuous derivatives. The break points of splines are called Knot, spline function can be integrated and differentiated due to being piece wise polynomials and can easily store and implemented on digital computer, non-polynomial spline function apiece wise is a blend of trigonometric, as well as, polynomial basis function, which form a complete extended Chebyshev space. Matlab is a powerful computing system for handling the calculations involved scientific and engineering problems. The aim of this paper is to compare between Adomain decomposition method and numerical solution to solve Volterra Integral Equations of second kind using the fifth order non-polynomial Spline functions by Matlab. We followed the applied mathematical method numerically by Matlab. Numerical examples are presented to illustrate the applications of this methods and to compare the computed results with analytical solutions. Finally by comparison of numerical results, Simplicity and efficiency of this method be shown.
Highlights
IntroductionThere is increase in concern by integrated equations, Volterra integral equations has wide range of the applications in physics and engineering such as potential theory, Dirichlet problems, electrostatics, the particle transport problems of astrophysics, reactor problems, contact problems, diffusion problems and heat transfer problems(for more details see [4])
There is increase in concern by integrated equations, Volterra integral equations has wide range of the applications in physics and engineering such as potential theory, Dirichlet problems, electrostatics, the particle transport problems of astrophysics, reactor problems, contact problems, diffusion problems and heat transfer problems(for more details see [4]).In the mid of 1960s the numerical solution of integral equations was begun when the Kernel is unwell In the beginning of 1980s the numerical methods began to take more importance when there is no analytical solution
In 2015, Harbis, S, Murad, M and Majed, S [3] presented a numerical solution for linear Voltera integral equation from second kind by using non-polynomial spline function from the third degree
Summary
There is increase in concern by integrated equations, Volterra integral equations has wide range of the applications in physics and engineering such as potential theory, Dirichlet problems, electrostatics, the particle transport problems of astrophysics, reactor problems, contact problems, diffusion problems and heat transfer problems(for more details see [4]). 69 Elgaili Abdalla Elhassan Ibrahim et al.: Comparison Between Adomain Decomposition Method and Numerical Solutions of Linear Volterra Integral Equations of the Second Kind by Using the Fifth Order of Non-Polynomial Spline Functions the Spline function. In 2011 Majeed,S [8] presented a numerical solution to the Volterra differential integrative equation of the second kind using non-polynomial Spline functions. In 2015, Harbis, S, Murad, M and Majed, S [3] presented a numerical solution for linear Voltera integral equation from second kind by using non-polynomial spline function from the third degree. In 2017, Najwa, S and Mohammed, S [7] presented a numerical solution for linear Voltera integral equations with weakly singular kernel by using non-polynomial spline function from the fifth degree. We try to solve equation (22) using fifth order nonpolynomial Spline functions
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