First, this paper defines a general nonlinear optimal control problem with state/control constraints and its approximation problem as the Haar wavelet Galerkin optimal control problem (HWGOCP). Then, a Haar wavelet-based Galerkin numerical method has been developed, which converts it to a nonlinear optimization problem. We theoretically prove that a Haar wavelet feasible solution of HWGOCP will exist. We also show that the approximate solutions of HWGOCP are consistent and converge to the optimal solution of the problem. A variety of application problems have been considered, which include optimal control of tumour growth using Chemotherapy drugs, optimal control of infection via the SIS model using treatment, the Brachistochrone problem in mechanics, optimal control of mold using a fungicide, optimal control of pH value of a chemical reaction to determine the quality of a product, etc.

The standard approach of first locating the equilibria, selecting the one of interest, translating it to the origin and then investigating the stability of the origin may break down when the parametric uncertainties present in the system are due to modeling inaccuracies or environmental changes. To deal with such situations, Siljak et al. recently introduced the concept of parametric stability and initiated its study, which is interesting by itself. The chapter introduces a different idea to study the stability of uncertain differential systems. Our approach exhibits moving invariant sets as the parameter changes. To investigate the stability of the moving invariant sets, the generalized derivative of the Lyapunov function must be estimated from opposite directions relative to suitable sets in phase space that depend on the moving parameter.

A study on accelerated monotone iterative methods for two systems of non-linear partial differential equations arising from catalytic converter models is presented. Two mathematical models are considered in this work. The first mathematical model consists of a semilinear parabolic partial differential equation and two integral equations. In contrast, the second mathematical model consists of a semilinear parabolic partial differential equation and an integral equation. The proposed modified monotone iterative methods converge to the unique solution of these mathematical models faster than the existing monotone iterative schemes available in the literature. Interesting theoretical justification is provided for the accelerated convergence and the monotone behaviour of the proposed iterative methods. Numerical simulation results support the theoretical claims.

We study the blow up problems for ordinary Caputo fractional differential equation and the time dependent Caputo fractional reaction diffusion equation in one dimensional space. We establish that the solution of the differential equation of the integer order which blows up in finite time can be used as a tool to construct a lower solution to the equation of the fractional order, under suitable conditions. Hence, we obtain the blow up of the solution of the differential equation of integer order implies that the blow up of the solution of the differential equation of fractional order. For that purpose, we use the known comparison results of Caputo ordinary fractional equation and Caputo fractional reaction diffusion equation. We also prove the blow up in finite time of the Caputo fractional reaction diffusion equation using a similar method which has been used to prove the blow-up of the solution of ordinary reaction diffusion equation.

We study the quenching problems of Kawarada's type for Caputo-fractional reaction diffusion equation in one dimensional space. We have developed maximum principle and comparison result relative to the linear Caputo fractional reaction diffusion equation in one dimension. In this work, we prove that the Caputo time derivative fractional reaction diffusion equation quenches by two different methods. The methods used here are by comparing the Picard's iterates of the Caputo fractional reaction diffusion equation with that of the Picard's iterates of the Kawarda's ordinary reaction reaction diffusion equation. The second method is by upper and lower solution method.

Solutions of linear sequential Caputo fractional differential equation with constant coefficients and with initial conditions of order nq; can be obtained using the Laplace transform method. Here n-􀀀1

The method of generalized quasilinearization has been extended for singular boundary value problems on unbounded domains. Natural lower and upper solutions are considered. We have obtained two sequences which converge uniformly and monotonically to the unique solution of the boundary value problems for ordinary differential equations on unbounded domains. The rate of convergence of these sequences is cubic when the nonlinearity of the iterates is quadratic.

In this paper a numerical method is developed to solve the Bagley-Torvik equation. The Bagley-Torvik equation is a fractional differential equation which occurs quite frequently in various branches of Applied Mathematics and Mechanics. The solution to this equation is proposed by using second kind Chebyshev wavelet. It finally reduces the equation in to the system of linear equations which can be easily solved. Examples are illustrated to demonstrate simplicity of proposed method. Results are also compared with those present in literature.

Consider a nonlinear differential-functional equation (1) Au+ f(x, u(x), u) = 0 ,

In the present paper, a numerical method is proposed to solve the fractional Fredholm integro-differential equation. The proposed method is based on the Chebyshev wavelet approximation. Using the approximation of an unknown function, its fractional derivative and its Integral operator in terms of Chebyshev wavelet, the fractional Fredholm integro-differential equation is ultimately reduced to a system of linear equations which can be solved easily. The test examples are given for illustration. The obtained results are compared for various number of basis functions in the Chebyshev wavelet. The proposed method is easy to understand, easy to implement and gives a very good accuracy. The errors are further measured with the help of different norms to show the good accuracy obtained.

The method of generalized quasilinearization has been extended for forced Duffing equations with integral boundary conditions. Natural lower and upper solutions have been considered. Two sequences that converge uniformly and monotonically to the unique solution of the forced Duffing equation have been obtained. The rate of convergence of these sequences is cubic when the nonlinearity of the iterates is quadratic.

We develop a generalized monotone method using coupled lower and upper solutions for Caputo fractional differential equations with periodic boundary conditions of order , where . We develop results which provide natural monotone sequences or intertwined monotone sequences which converge uniformly and monotonically to coupled minimal and maximal periodic solutions. However, these monotone iterates are solutions of linear initial value problems which are easier to compute.

In this paper, by using upper and lower solutions, we develop monotone method for the nonlinear Caputo fractional boundary value problem of order a where 1 < alpha < 2. We construct two sequences which converge uniformly and monotonically to the extremal solutions of the nonlinear Caputo fractional boundary value problem.

Monotone method combined with the method of upper and lower solutions yields monotone sequences which converge uniformly and monotonically to minimal and maximal solutions of the nonlinear systems, when the forcing function is quasi monotone nondecreasing. In this paper we develop genearalized monotone method for N system of Caputo fractional differential equations when the forcing function is the sum of an increasing and decreasing functions. In generalized monotone method we use coupled upper and lower solutions and the method yields two monotone sequences which converge uniformly and monotonically to coupled minimal and maximal solutions. This method is applicable to the Lotka-Volterra equation with Caputo fractional derivative of order q when 0 < q <= 1. This provides an opportunity to provide better results or improve on the existing results with integer derivatives. Finally, under uniqueness condition we obtain the unique solution of the Caputo fractional differential system.

In this work we develop monotone method for fractional boundary value problem of order α where 1 < α < 2. For that purpose we develop some basic fractional calculus results which are used as tools to develop comparison results relative to upper and lower solutions of Caputo fractional boundary value problems. Our results include the comparison theorem given in [2] for derivative of order α. We have also computed the Green’s function for the linear Caputo fractional boundary value problem. The integral representation of the linear fractional nonhomogeneous problem is useful in developing monotone method for the Caputo fractional boundary value problem.

Comparison results of the nonlinear scalar Riemann-Liouville fractional differential equation of order q, 0 < q <= 1, are presented without requiring Holder continuity assumption. Monotone method is developed for finite systems of fractional differential equations of order q, using coupled upper and lower solutions. Existence of minimal and maximal solutions of the nonlinear fractional differential system is proved.