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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.
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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. |
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