# Preface

This chapter gives an introduction to the wide area, called partial differential equations (PDEs for short). Although our main tool is the separation of variables, we also touch some other useful methods.

Introduction to Linear Algebra with Mathematica

# Partial Differential Equations (PDEs)

In contrast to ordinary differential equations, a partial differential equation (PDE for short) contains partial derivatives of the dependent variable, which is an unknown function in more than one variable. Many important practical problems involve more than one independent variable, so the corresponding mathematical models involve partial, rather than ordinary differential equations. Although one can study PDEs with as many independent variables as one wishes, we will be primarily concerned with PDEs in two independent variables. A solution to the PDE is a function of two or more variables that satisfies the given PDE for all values of the independent variables. Upon introducing shortcut ux for partial derivative ∂u/ ∂x, we can write partial equations in more simple way. Some examples of PDEs (of physical significance) with two independent variable are:

In order for a PDE to have a unique solution, we must impose additional conditions, sometimes called side conditions, on the solution. Since time, usually denoted by t, is a specific variable having nothing in common with spacial variables, conditions imposed on t are called the initial conditions. On the other hand, the conditions on spacial variables are referred to as the boundary conditions. Since time is irreversible, boundary conditions are not appropriate for time variable. Therefore, partial differential equations are accompanied with appropriate initial and boundary conditions that lead naturally to initial boundary value problems (IBVPs) or boundary value problems (BVPs) when time is not involved. The values assigned to the side conditions are called the data. A PDE with side conditions is well posed if it admits a unique solution for any values assigned to the data.

This chapter gives an introduction to this subject and cannot be considered as complete analysis of partial differential equations. We mostly discuss linear partial differential equations of second order because their theory is simpler and for certain modeling purposes, a linear equation may suffice. But in other situations the nonlinear character of the problem is important and even essential. For example, the semilinear equation

$\mbox{div} \left( \frac{\nabla u}{\left( 1 + |\nabla u |^2 \right)^{1/2}} \right) = 0$
for the graph z = u(x,y) defines the smallest surface area for a given boundary curve such as soap films. Another quasilinear PDE arising in physics is the porous medium equation
$u_t = k\,\mbox{div} \left( u^{\gamma} \nabla u \right) ,$
where k > 0 and γ > 1 are constants. An example of a nonlinear PDE that is not quasilinear is the Monge–Ampère equation that arises in differential geometry
$A \left( u_{xx} u_{yy} - u_{xy}^2 \right) + B\,u_{xx} + 2C\,u_{xy} + D\,u_{yy} + E =0$
We present one method of separation of variables that reduces a given initial boundary value problem to a set of ordinary differential equations, which must be solved subject to given boundary conditions. The method of separation of variables is the oldest systematic method, having been used by d'Alambert, Daniel Bernoulli, Euler, and Fourier is their investigations of physical phenomena. The desired solution of an initial boundary value problem is expressed as an infinite series that is usually truncated to finite number of terms in practical applications. This provides a reasonable approximation to the actual solution.

We illustrate the use of separation of variables in solving some practical problems from three basic classes of partial differential equations. Correspondingly, we treat heat and diffusion equations as typical representatives of parabolic PDEs. Our next topic covers wave equations from a wide class of hyperbolic PDEs. Finally, we discuss Laplace's, Helmholtz, and Poisson equations and corresponding boundary value problems from elliptic PDEs and potential theory.

Basic definitions

Let ℤ denote the set of integers, and ℕ denote the natural numbers, i.e., the nonnegative integers. We shall denote n-dimensional Euclidean vector space by ℝn, and points by x = (x1, … , xn). However, we often use the more conventional notation (x,y) and (x,y,z) for points in ℝ² and ℝ³, respectively.

A domain Ω in ℝn is a connected open subset: points x,y ∈ Ω can be connected by a curve in Ω, and for every point in Ω there exists an open ball Br(x) = {y ∈ ℝn : |x - y| < r} from Ω. The boundary ∂Ω of a domain Ω is the set of limit points of Ω, which are not in Ω. If Ω = ℝn, then ∂Ω = ∅, the empty set. The boundary ∂Ω is assumed to be a smooth surface except possibly finite number of points having not equal derivatives from both sides. So domains as, for instance, ♥ are excluded. If Ω ⊆ ℝn, then its boundary is of dimension n-1.

We shall denote the continuous functions on Ω by C(Ω), and those whose first order derivatives are also all continuous by C¹(Ω). Vector functions (as a rule, column functions) are denoted by bold fonts: f. Partial derivatives will be denoted mostly by subscripts: ux ≝ ∂u/∂x. A function f defined on a domain Ω is integrable if ∫|f(x)|dx is defined and finite; we denote all such functions by L¹(Ω) or 𝔏¹(Ω).

The divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem,

$\int_{\Omega} \mbox{div}{\bf F} \,{\text d}V \equiv \int_{\Omega} \left( \nabla \cdot {\bf F} \right) {\text d}V = \oint_{\partial \Omega} \left( {\bf F} \cdot {\bf n} \right) {\text d}S$
holds for bounded domains Ω with piecewise smooth boundary of C¹(∂Ω), and F is a continuously differentiable vector field defined on a neighborhood of Ω. Here n is the outward pointing unit normal field of the boundary ∂Ω.

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