Exact and Computational Methods for Solving some Integral and Differential Equations

Abstract

Chapter I This chapter is an introduction to the basic concepts of integral equations. It includes the classifications of integral and integro-differential equations. It also includes an introduction to the basic definitions and the necessary properties for fractional calculus such as the definitions of Riemann-Liouville, Caputo and discusses some necessary mathematical definitions that will arise in the study of these concepts. An introduction to the various methods used in this thesis to obtain the exact solutions and the numerical solutions. It also includes an introduction to some theorems and basic concepts of measure theory which will be used in this thesis. Chapter II In this chapter, we improve and extend variational iteration method (VIM) and Chebyshev spectral method to find the exact solutions and the approximate solutions for fractional differential equations, fractional integro-differential equations, nonlinear systems of fractional integro-differential equations and generalized Abel's integral equations of the second kind. Moreover, we aim to study the convergence of the VIM for fractional differential equations, fractional integro-differential equations, nonlinear systems of fractional integro-differential equations and generalized Abel's integral equations of the second kind and to address the sufficient condition for convergence. The results obtained by variational iteration method and Chebyshev spectral method in this chapter are compared with the exact solutions and with the results obtained by some other authors, this comparison shows that we obtained better results and more accurate. Chapter III In this chapter, we apply the differential transform method (DTM) and homotopy perturbation method (HPM) to solve fifth-order boundary value problem, system of second-order boundary value problem, system of Volterra integral equations, systems of linear and nonlinear integro-differential , Cauchy problem, boundary value problem of fractional order, fractional integro-differential equations and nonlinear systems of fractional integro-differential equations. In addition, we extend the modified Laplace decomposition method (mLDM) and the modified Laplace decomposition method with the Padé approximant (mLD-PA) to solve boundary value problem of fractional order and systems of linear and nonlinear fractional integro-differential equations. The results obtained by differential transform method, fractional differential transform method, homotopy perturbation method, modified homotopy perturbation method, modified Laplace decomposition method and modified Laplace decomposition method with the Padé approximant in this chapter are compared with the exact solutions and with the results obtained by some other authors, this comparison shows that we obtained better results and more accurate.