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Historically, partial differential equations originated from the study of surfaces in geometry and a wide variety of problems in mechanics. During the second half of the nineteenth century, a large number of famous mathematicians became actively involved in the investigation of numerous problems presented by partial differential equations. The primary reason for this research was that partial differential equations both express many fundamental laws of nature and frequently arise in the mathematical analysis of diverse problems in science and engineering.

Many problems in mathematical, physical, and engineering sciences deal with the formulation and the solution of first-order partial differential equations. From a mathematical point of view, first-order equations have the advantage of providing a conceptual basis that can be utilized for second-, third-, and higher-order equations.

Partial differential equations arise frequently in formulating fundamental laws of nature and in the study of a wide variety of physical, chemical, and biological models. We start with a special type of second-order linear partial differential equation for the following reasons. First, second-order linear equations arise more frequently in a wide variety of applications. Second, their mathematical treatment is simpler and easier to understand than that of first-order equations in general. Usually, in almost all physical phenomena (or physical processes), the dependent variable u = u ( x, y, z, t ) is a function of three space variables, x, y, z and time variable t .

The general linear second-order partial differential equation in one dependent variable u may be written as (4.1.1) $$ \sum\limits_{i,j = 1}^n {A_{ij} u_{x_i x_j } } + \sum\limits_{i = 1}^n {B_i u_{x_i } } + Fu = G, $$ in which we assume A _ij = A _ji and A _ij, B _i, F , and G are real-valued functions defined in some region of the space ( x _1, x _2, . . ., x _n).

In the theory of ordinary differential equations, by the initial-value problem we mean the problem of finding the solutions of a given differential equation with the appropriate number of initial conditions prescribed at an initial point. For example, the second-order ordinary differential equation $$ \frac{{d^2 u}} {{dt^2 }} = f\left( {t,u\frac{{du}} {{dt}}} \right) $$ and the initial conditions $$ u\left( {t_0 } \right) = \alpha , \left( {\frac{{du}} {{dt}}} \right)\left( {t_0 } \right) = \beta , $$ constitute an initial-value problem.

This chapter is devoted to the theory of Fourier series and integrals. Although the treatment can be extensive, the exposition of the theory here will be concise, but sufficient for its application to many problems of applied mathematics and mathematical physics.

The method of separation of variables combined with the principle of superposition is widely used to solve initial boundary-value problems involving linear partial differential equations. Usually, the dependent variable u ( x, y ) is expressed in the separable form u ( x, y ) = X ( x ) Y ( y ), where X and Y are functions of x and y respectively. In many cases, the partial differential equation reduces to two ordinary differential equations for X and Y . A similar treatment can be applied to equations in three or more independent variables. However, the question of separability of a partial differential equation into two or more ordinary differential equations is by no means a trivial one. In spite of this question, the method is widely used in finding solutions of a large class of initial boundary-value problems. This method of solution is also known as the Fourier method (or the method of eigenfunction expansion ). Thus, the procedure outlined above leads to the important ideas of eigenvalues, eigenfunctions, and orthogonality, all of which are very general and powerful for dealing with linear problems. The following examples illustrate the general nature of this method of solution.

In the preceding chapter, we determined the solutions of partial differential equations by the method of separation of variables. In this chapter, we generalize the method of separation of variables and the associated eigenvalue problems. This generalization, usually known as the Sturm-Liouville theory , greatly extends the scope of the method of separation of variables.

In the preceding chapters, we have treated the initial-value and initial boundary-value problems. In this chapter, we shall be concerned with boundary-value problems. Mathematically, a boundary-value problem is finding a function which satisfies a given partial differential equation and particular boundary conditions. Physically speaking, the problem is independent of time, involving only space coordinates. Just as initial-value problems are associated with hyperbolic partial differential equations, boundary-value problems are associated with partial differential equations of elliptic type. In marked contrast to initial-value problems, boundary-value problems are considerably more difficult to solve. This is due to the physical requirement that solutions must hold in the large unlike the case of initial-value problems, where solutions in the small, say over a short interval of time, may still be of physical interest.

The treatment of problems in more than two space variables is much more involved than problems in two space variables. Here a number of multidimensional problems involving the Laplace equation, wave and heat equations with various boundary conditions will be presented. Included are the Dirichlet problem for a cube, for a cylinder and for a sphere, wave and heat equations in three dimensional rectangular, cylindrical polar and spherical polar coordinates. The solution of the three-dimensional Schrödinger equation in a central field with applications to hydrogen and helium atoms is discussed. We also consider the forced vibration of a rectangular membrane described by the three-dimensional, nonhomogeneous wave equation with moving boundaries.