Ordinary Differential Equations and Applications II: with Maple Illustrations

This chapter starts with the revision of basic concepts in real, complex, and functional analyses. Vector-valued differential equations are formulated and conditions for generating solution bases for the differential equations are stated.

Fundamental concepts in normed spaces are elucidated and linear systems are considered and applied to some problems.

 In differential equations, one of the cornerstones of most theorems and their framework (hypotheses) is the existence and uniqueness theorem. We consider theorems that guarantee the existence and uniqueness of solutions of differential equations. We consider the Carathedory theorem, Peano existence theorem, PicardLinderlof existence and uniqueness theorem, Brower and Schauder fixed point theorems. Picard successive approximation method is applied to establish the existence and uniqueness of solution to some initial value problems. Moreover, the conditions for the continuation of solutions from a given interval to an extended interval are also derived for ordinary differential equations.

In this chapter, we will consider methods for estimating the norm of a matrix and matrix exponents. The conditions for the existence and uniqueness of solutions are considered for ordinary differential equations using the Lipchitz conditions. Adjoint systems are revisited together with the application of the Carathedory theorem to some selected problems.

We consider canonical transformation for transforming scalar differential equations to matrix differential equations. We determine conditions for linear independence of solutions using the Wronskian method and use the Jordan canonical form to find bounds for solutions of ODES. Also considered are: the generalized eigenvectors method for obtaining matrix solutions to ODES and corresponding bounds for the autonomous differential equations, upper and lower bounds for solutions. Conditions for continuous dependence of solutions on initial data are formulated. Periodic systems are studied too with the application of the Floquet rule to finding solutions to some linear periodic systems. The Theorem on how to construct monodromy matrices is presented for the linear periodic systems together with some examples.

Stability including its characterizations is considered using quantitative and qualitative theories. Stability criteria are discussed using the Routh-Hurwitz criterion and fundamental matrices. Stability is investigated for nonlinear systems through linearization and Lyaponov’s methods. Applications are made to single and multispecies population models.

The stability property is investigated for perturbed linear autonomous and non-linear systems using linearization and Lyapunov’s methods. Some examples are given on the stability of some nonlinear systems through eigenvalues of the linearized systems and coupled with the estimation of the norm of the error of approximation

The stability property of equilibrium points of some neural network models is investigated. We have introduced different types of Lyapunov functions to carry out the investigation. The models considered are: Grossberg, Hopfield, Fitz-Nagomo and Fitzhugh models, respectively. The equilibrium points and stability conditions are obtained for the Avian influenza infection. The conditions for bio economic equilibrium points for the fish model were also obtained.

Many complex nonlinear problems in science, economics, and engineering require computers to solve and simulate mathematical models describing them. Hence, it becomes extremely necessary to apply numerical methods to solve such problems. In this chapter, numerical methods for solving initial value problems are considered. Taylor series, Euler, Modified Euler’s, Runge-Kutta, Adams-Bash forth-Moulton and Milne numerical methods are considered together with some Maple examples given. Numerical simulations are designed and implemented using Maple software for HIV/AIDS, Fitzhugh, and Fitzhugh-Nagumo, sickle cell anemia, zooplankton-fish, Gompertz tumor and neural firing models. The Explore facility in Maple 2022 is utilized to design sliders for investigating the behaviors of solutions of some ordinary differential equations subject to parameter change.