Numerical Solutions of Fractional Order Boundary Value Problems by Weighted Approximation Method

Abstract

The major objective of this research work is to employ the weighted residual approach to numerically solve fractional order differential equations with homogeneous and non-homogeneous boundary conditions. This method uses linear combinations of several types of functions to find the approximate solutions, which must satisfy the homogeneous boundary conditions. The piecewise polynomials like the Bernstein, modified Bernoulli and modified Legendre polynomials are utilized as basis functions because these kinds of functions are easily differentiated and integrated in this study. The fractional derivatives are used in the hypothesis of Caputo sense. As a result, we provide a detailed and straightforward comprehensible matrix form of the Galerkin, Least Square and Collocation weighted residual formulation for both linear and nonlinear fractional order boundary value problems. In each chapter, few numerical examples are exhibited to illustrate the precision and usefulness of the current approach. We demonstrate that the results appear to be monotonic convergence within the approximate results and the exact solutions. The approximate results are also compared to the exact solutions along with the solutions that are currently available in the literature. Reliable accuracy is obtained in the present work; the absolute errors are presented both graphically and in tabular form. The thesis entitled Numerical Solutions of Fractional Order Boundary Value Problems by Weighted Approximation Method contains six chapters; out of these, the first chapter is confined as Introduction. In this chapter, we mention the objectives and scope of the thesis and the outline of the research work. We discuss some mathematical preliminaries that are important to establish the problems in detail, such as theorems and lemmas that are used in subsequent chapters, some special functions like Gamma and Mittag- Leffler functions and the basic concepts of fractional derivative and integration in both Riemann-Liouville and Caputo sense. The finite element method is introduced here, especially three weighted residual methods: Galerkin, Least Square and Collocation with the Bernstein, modified Legendre and modified Bernoulli polynomials and their properties. Chapter 2 is devoted to linear fractional differential equations using Bernstein, modified Legendre and modified Bernoulli polynomials as basis functions. We derive rigorous matrix formulations of the following: p(x)du dx + s(x)dαu dxα + u(x) = f(x), under the boundary conditions u(a) = a0, u(b) = b0 where α ≥ 1.5. We examine four examples of second-order linear fractional boundary value problems for the numerical solutions using the suggested formulations. It was found that there is a monotonic convergence between the approximate and exact solutions. Three weighted residual methods for solving fractional Bagley-Torvik equations are studied in chapter 3. A fractional-order differential equation arises in various engineering and physical systems, particularly in modeling viscoelastic materials and dynamic fluid systems. This work concentrates on the numerical solution of the Bagley-Torvik equation, represented as: aD2y(t) + bD3/2y(t) + cy(t) = f(t) The WRM transforms the governing equation into an approximate solution by minimizing the residual error over the problem domain, employing basis functions to represent the solution. The fractional derivative terms are discretized using suitable approximations, such as the Caputo approach, which is incorporated into the weighted residual framework. Results indicate that the weighted residual method provides flexible and efficient results for solving fractional differential equations while maintaining stability and convergence properties. The results suggest that the weighted residual method offers a robust tool for solving fractional order differential equations, making it highly applicable to a range of practical problems in engineering and applied physics. In chapter 4, the Galerkin weighted residual approach is used to quantitatively solve the fourth order fractional differential equations with homogeneous and non-homogeneous boundary conditions. The same process is also introduced to generate the approximate solutions for the two-point fourth-order linear and non-linear integro-differential problems in fractional order. Using piecewise polynomials, the matrix formulation of both scenarios is stated directly. To determine the correctness and effectiveness of the proposed method, we experiment with a variety of instances from the literature utilizing modified Bernoulli and modified Legendre polynomials as basis functions. The absolute errors are displayed in tabular form and we find that reliability has been attained in this study. In Chapter 5, the weighted residual method is used to bring out the approximate solutions for nonlinear fractional differential equations with both homogeneous and nonhomogeneous boundary conditions. We use three techniques: Galerkin, Least Square and Collocation to solve nonlinear two-point boundary value problems numerically in an efficient manner. The accuracy and reliability of the current method, which utilized the modified Legendre and modified Bernoulli polynomials as weight functions, are demonstrated by looking at few nonlinear examples to find the maximum absolute errors. The computational techniques and mathematical formulations are easier to comprehend and less difficult to understand in this literature. The last chapter entitled numerical techniques for the system of fractional differential equations is established by the method of weighted residuals such as Galerkin, Least Square and Collocation methods that are used to solve the boundary value problems (BPVs). This approach is then expanded to obtain approximate solutions of fractional order systems that use differentiable polynomials, specifically modified Legendre polynomials as basis functions. It is possible to efficiently code the algorithm for the residual formulations of matrix form. Here, the Caputo fractional derivatives interpretation is used rigorously. We have employed some examples of linear and nonlinear boundary value problems to quantitatively illustrate these techniques. The findings in absolute errors demonstrate how straightforwardly the current approach locates numerical solutions for the systems of fractional order differential equations. All numerical experiments of this thesis and its algorithm have been implemented using the frameworks of Mathematica and MATLAB, which are used to calculate scientific computations and graphical visualizations. Finally, the conclusion and the list of references are appended at the end of the dissertation.