Generalization of Fixed Point Theorems, Existence Results and Analytical Solutions of Fractional Differential Equations

Abstract

Fractional calculus is a generalization of classical calculus. Due to the fractional order modeling in different physical fields, scientists have shown their interests in the exploration of its different features. The demand of fractional calculus is increasing day by day due to accuracy in modeling of hereditary problems in physics and engineering in the last two decades. Some of the recently well considered aspects are including; existence of positive solutions, analytical solutions numerical solutions of frac- tional differential equations involving integral boundary conditions, local boundary conditions, non local boundary conditions, periodic boundary conditions, anti boundary conditions and multi points boundary conditions of fractional order. In the existence results of fractional differential equations, scientists are utilizing various types of fixed point approaches. In the early time, scientists have studied fixed point results for contractions over Banach spaces. Later on, a trend was established for the generalization of fixed point theorems in literature. The generalizations were made with the help of contractions or spaces or generalizing both the spaces and contractions. Once it is ensured that a solution exists then the researchers consider analytical and numerical studies about the problems. In this dissertation, we are concerned to generalize fixed point theorems in multiplicative metric spaces. For this aspect of our dissertation, we take a help of integral inequalities, E.A property, Weakly compatibility, CLR property and provide applications of our new fixed point theorems for the existence results of functional equations. After this work, a study of existence results, our targets are to get ex- istence and uniqueness of solutions for coupled fractional differential equations, fractional differential equations involving nonlinear operator ϕp, coupled fractional differential equations involving ϕp with two types of fractional derivatives that is the Riemann-Liouville and Caputo. Then the analytical solu- tions of fractional order models are considered by the help of Sumudu Adomian decomposition method. For this we apply the proposed method to Burger equation and Fisher equation. Finally, exact analyti- cal solutions of wave equations will be considered by the use of double Laplace transform method. In chapter one we have given introduction of the dissertation. In chapter two we have given definitions, theorems and some important results which we will use throughout the dissertation. In chapter three, we have given new definition and generalization of fixed point theorems. In chapter four, we have given EUS of fractional order DEs by the use of FPTs and stability solutions by the use of Hyers-Ulam. In chapter five, we have given analytical solutions of fractional order Burger’s model and Fisher Model by the use of Sumudu adomian decomposition method and also provided the analytical solutions of Time-fractional wave model by double Laplace method and the models were expressed by examples.