MULTI POINT BOUNDARY VALUE PROBLEMS FOR SYSTEMS OF FRACTIONAL DIFFERENTIAL EQUATIONS: EXISTENCE THEORY AND NUMERICAL SIMULATIONS

Abstract

In recent years, it has been proved that differential equations of fractional order and their systems are the best tools for modeling various phenomena of chemical and physical as well as biological sciences. Besides from these, it is also proved that the aforementioned area has also important applications in various field of engineering and technology. Therefore, considerable attentions were given to study the subject of fractional order differential equa- tions in last few decades. This thesis is concerned with a detailed study of the existence theory and numerical solutions of multi-point boundary value problems of fractional order differential equations. For such study, first we review some useful notations, definitions, re- sults from fractional calculus, functional analysis and fixed point theory. Also for numerical solutions, we use some orthogonal polynomials like Legendre, Bernstein and Jacobi polyno- mials. We begin our work with the study of existence and uniqueness of positive solutions for simple multi point boundary value problem. Then, we obtain necessary and sufficient conditions for existence of at least three positive solutions for the considered problems in chapter 3. Then another class with nonlocal boundary conditions is studied by topological degree method for existence and uniqueness of positive solutions. While a class of fractional order differential equations, where nonlinear function involved in it depending on fractional derivative involve in it with multi point boundary conditions is also studied for existence of solutions. These conditions are developed by using some classical fixed point theorems and results of functional analysis. Existence and uniqueness of positive solution for multi point boundary value problems for coupled systems are studied in chapter 4. Sufficient conditions for existence and uniqueness results of multi point boundary value problems for coupled systems are established with the help of fixed point theorems such as Banach, Gue- Krasnoselskii’s fixed point theorem of cone expansion and compression, Schauder’s fixed point theorem and Perov’s fixed point theorem etc. Some multiplicity results for existence of solutions to the nonlocal boundary value problem are discussed in chapter 5. For every differential equation or system of differential equations of classical or arbitrary order it is nor compulsory that it has a solution. Therefore conditions for nonexistence are developed in a part of the same chapter 5. Moreover, for multiplicity of positive solutions, the nec- essary and sufficient conditions are developed by means of monotone iterative technique together with the method of upper and lower solutions in chapter 6. It is very difficult to find exact solution for each and every problem of fractional order differential equations due to the complexity of fractional order differential and integral operator involved in the system. Therefore, there exists a strong motivation to develop numerical schemes which are easily understandable and easily computable as well as efficient and reliable. By means of some orthogonal polynomials such as Shifted Legendre, Bernstein and Shifted Jacobi poly- nomials, we are developed some operational matrices of integrations and differentiations for numerical solutions of boundary value problems for both ordinary and partial fractional order differential equations in chapter 7. With the help of these operational matrices, we convert the problems under consideration to algebraic equations which are easily soluble for unknown coefficient matrices. The obtain coefficient matrices are used to fined numerical solutions for the concerned problems. The method is extended to solve coupled systems of boundary value problems of fractional order differential equations. To perform the com- putations, we use Matlab and Maple software. The efficiency of the numerical methods is checked by solving several examples, and the comparison of exact and numerical solutions will also be demonstrated.