Existense and Approximation of Solutions of Differential Equations

Abstract

Nonlinear coupled boundary value problems (BVPs) have very important and interesting aspects in the kingdom of Nonlinear Analysis due to not only the theoretical aspects but also the applications which they have in almost every eld of science. Problems with coupled boundary conditions (BCs) appear while studying mathematical biology, Sturm-Liouville problems, reaction di usion phenomena, chemical systems, and Lotka-Volterra models. This thesis has two parts. In the rst part, the existence results are established for the rst{order and the second{order nonlinear coupled BVPs subject to nonlinear coupled BCs. Also in the same part, the existence results are established for the second{order nonlinear coupled BVPs when the nonlinear functions have dependence on the rst-order derivative. Multiple approaches are available in the literature to investigate the existence of solutions of nonlinear BVPs, but lower and upper solutions (LUSs) approach is one of the strongest. In this approach the original problem is modi ed logically to a new problem, known as the modi ed problem, then the theory of di erential inequalities with the combination of well-known existence results are applied to establish the existence of solution of the modi ed problem. Finally the solution of the modi ed problem leads to the solution of the original problem. Moreover in the rst part of the thesis the treatment of the many di erent rst-order and the second-order nonlinear BVPs are uni ed by developing the idea of coupled LUSs. Under this idea, some monotonicity assumptions are imposed on the arguments of the nonlinear BCs in the presence of the existence of a lower solution and an upper solution to unify the classical existence results for very important types of BVPs, like periodic, anti-periodic, Dirichlet, and Neumann. Several examples are discussed to support the theoretical results. The subject fractional calculus being a generalization of integer-order calculus has numerous applications in almost every eld of science. Due to the intensive use of fractional order di erential problems (FODPs) in almost every eld of science including, but not limited to, uid dynamics, physics, aerodynamics, chemistry, mathematical biology, image processing, and psychology, there is a strong motivation for the researchers to develop reliable and e cient numerical methods to nd the approximate solutions of FODPs. xii xiii In the second part of the thesis, we consider a generalized class of multi{terms fractional order partial di erential equations (FOPDEs) and their coupled systems. We develop a new numerical method and generalize the corresponding Jacobi operational matrices of integrals and derivatives considered on a rectangular plane. By means of the operational matrices, the considered problem of fractional order is reduced to an algebraic one. Being easily solvable, the associated algebraic system leads to nding the solution of the considered problem of fractional order. Validity of the method is established by comparing our simulation results obtained by using MATLAB softwares with the exact solutions in the literature yielding negligible errors.