Analysis and Differential Equations

The following sections are included:

• Preface
• Contents

This chapter introduces the ordinary and partial differential equations formaly and through working examples. We present the implicit equations related to differential equations and solutions of second order differential equations and system of partial differential equations, with the extensions developed later.

In this chapter, we present methods of expansions of functions by projections in bases of squared integrable functions with respect to real measures. We define Laguerre, Hermite and Legendre orthonormal polynomials and recursive methods for their calculus, with their graphs. The conditions for the convergence of the expansions of functions in polynomial series depend on the properties of the measures defining their orthogonality. Several generalizations to the real line and the plane and their differential equations are presented.

We present conditions for the differentiability and the existence of a Taylor expansion in complex spaces, the conditions for the existence of an extremum for an implicit function and the Euler-Lagrange conditions for the minimization of the integral of a function in ℝ₊ as x_t and its derivative belong to metric spaces. We generalize them to a function including more derivatives. The applications concern the optimality of areas or volumes, the calculus of parametric integrals as solutions of differential equations, and the elliptic functions.

This chapter presents methods for the resolution of linear differential equations and special functions defined by second order linear differential equations. The solutions of the homogeneous equations with constant coefficients are defined by boundary conditions, otherwise the solutions are generally not unique. The other methods use the expansion in an orthonormal basis for inhomogeneous differential equations and reparametrizations. We present equivalences between linear and nonlinear second order equations and between classes of first order nonlinear equations such as Bernoulli's and Riccati's equations.

This chapter studies first and second order spatial and spatio-temporal differential equations. We give exact and implicit solutions of first order equations, Laplace's equation, heat conduction, wave, potential and elasticity equations. Several solutions of the equations with constant coefficient are presented and the differential equations with functional coefficients are also studied.

This chapter focuses on systems of partial differential equations (PDEs), and their solutions. Their equivalence to implicit functions leads to conditions for the existence of exact solutions, for first and second order partial differential equations, and we define classes of solutions for the several equations. By reparametrizations, some systems of time varying PDEs are equivalent to ordinary differential equations where the time variable has disappeared. We present models for population dynamics and their solutions.

Several Eulerian integral functions have been defined by Legendre, Riemann, Jacobi and other mathematicians, they are related to the function Gamma and their recursive properties rely on integrations. Some of them are detailed in the first section which originates from Legendre (1825), he defined the functions I and L and calculated tables of the numerical values of the function Beta. Here we present them with simple proofs and different methods to get their numerical values. Expansions of other functions solutions of second order differential equations are explicited, in particular the Airy, Bessel, Hermite and Laguerre functions. Byerly (1893) proved result about Bessel's functions and the results about the other equations are new.

In this chapter the title of the sections are those of the chapters where the exercises were given and the references of the exercises are the same.

The following sections are included:

• Bibliography
• Index