Solutions to Uncertain Volterra Integral Equations by Fitted Reproducing Kernel Hilbert Space Method

Ghaleb Gumah,Khaled Moaddy,Mohammed Al-Smadi,Ishak Hashim

doi:10.1155/2016/2920463

Abstract

We present an efficient modern strategy for solving some well-known classes of uncertain integral equations arising in engineering and physics fields. The solution methodology is based on generating an orthogonal basis upon the obtained kernel function in the Hilbert spaceW21a,bin order to formulate the analytical solutions in a rapidly convergent series form in terms of theirα-cut representation. The approximation solution is expressed byn-term summation of reproducing kernel functions and it is convergent to the analytical solution. Our investigations indicate that there is excellent agreement between the numerical results and the RKHS method, which is applied to some computational experiments to demonstrate the validity, performance, and superiority of the method. The present work shows the potential of the RKHS technique in solving such uncertain integral equations.

Highlights

Building mathematical model of a specific phenomenon under uncertainty is essentially important for a large number of applications in economics, medicine, mathematics, physics, and engineering [1,2,3,4,5]
We study the fuzzy IEs using the concept of Riemann integrability in which the FIEs are converted into equivalent system of crisp integral equations (CIEs)
In the process of computation, all symbolic and numerical computations are performed by Mathematica 7.0 software package

Summary

Introduction

Building mathematical model of a specific phenomenon under uncertainty is essentially important for a large number of applications in economics, medicine, mathematics, physics, and engineering [1,2,3,4,5]. The uncertain integral equations are powerful tools to introduce uncertain parameters and to deal with their dynamical systems in natural fuzzy environments They are of great importance in the fuzzy analysis theory and its application in fuzzy control models, atmosphere, artificial intelligence, measure theory, quantum optics, and so forth [6,7,8,9,10]. The experts in such areas extensively use these equations to make the uncertain problems, which are usually too complex to be defined in precise terms, more understandable. It is immensely important to develop appropriate and applicable strategy to accomplish the mathematical construction that would appropriately treat uncertain problems and solve them

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