Partial differential equations

 

Partial differential equations

Brent J. Lewis, ... Andrew A. Prudil, in Advanced Mathematics for Engineering Students, 2022

Definitions

A partial differential equation is an equation containing an unknown function of two or more variables and its partial derivatives with respect to these variables.

The order of a partial differential equations is that of the highest-order derivatives. For example, $\frac{{\partial }^2�}{\partial �\partial �}=2�-�$ is a partial differential equation of order 2.

A solution of a partial differential equation is any function that satisfies the equation identically.

A general solution is a solution that contains a number of arbitrary independent functions equal to the order of the equation. A particular solution is one that is obtained from the general solution by a particular choice of arbitrary functions. For example, the general solution of the partial differential equation $\frac{{\partial }^2�}{\partial �\partial �}=2�-�$ is $�={�}^2�-\frac{1}{2}�{�}^2+�(�)+�(�)$. With $�(�)=2\sin �$ and $�(�)=3{�}^4-5$, the particular solution is $�={�}^2�-\frac{1}{2}�{�}^2+2\sin �+3{�}^4-5$.

A singular solution is one that cannot be obtained from the general solution by a particular choice of arbitrary functions.

A boundary value problem involving a partial differential equation seeks all solutions of a partial differential equation which satisfy conditions called boundary conditions. When time is one of the variables, the solution u (or $\frac{\partial �}{\partial �}$ or both) must satisfy initial conditions at $�=0$.

Partial Differential Equations

Martha L. Abell, James P. Braselton, in Differential Equations with Mathematica (Fourth Edition), 2016

10.1.1 Introduction

We begin our study of partial differential equations with an introduction of some of the terminology associated with the topic. A linear second-order partial differential equation (PDE) in the two independent variables x and y has the form

(10.1)$\begin{array}{l}\hfill �(�,�)\frac{{�}^2�}{�{�}^2}+�(�,�)\frac{{�}^2�}{����}+�(�,�)\frac{{�}^2�}{�{�}^2}+�(�,�)\frac{��}{��}+�(�,�)\frac{��}{��}+�(�,�)�=�(�,�),\end{array}$

where the solution is u(x, y). If G(x, y) = 0 for all x and y, we say that the equation is homogeneous. Otherwise, the equation is nonhomogeneous.

Example 10.1.1

Classify the following partial differential equations: (a) uxx + uyy = u; (b) uux = x.

Solution

(a) This equation satisfies the form of the linear second-order partial differential equation (10.1) with A = C = 1, F = −1, and B = D = E = 0. Because G(x, y) = 0, the equation is homogeneous. (b) This equation is nonlinear, because the coefficient of ux is a function of u. It is also nonhomogeneous because G(x, y) = x.

Definition 40 Solution of a Partial Differential Equation

A solution of a partial differential equation in some region R of the space of the independent variables is a function that possesses all of the partial derivatives that are present in the PDE in some region containing R and satisfies the PDE everywhere in R.

Example 10.1.2

Show that u(x, y) = y2 − x2 and $�(�,�)={�}^{�}sin�$ are solutions to Laplace’s equation uxx + uyy = 0.

Solution

For u(x, y) = y2 − x2, ux(x, y) = −2x, uy(x, y) = 2y, uxx(x, y) = −2, and uyy(x, y) = 2, so we have that uxx + uyy = (−2) + 2 = 0, which we verify with Mathematica.

Clear[u]u[x_, y_]=y ∧2−x ∧2;D[u[x, y], {x, 2}]+D[u[x, y], {y, 2}]

0

Similarly, for $�(�,�)={�}^{�}sin�$, we have ${�}_{�}={�}^{�}cos�$, ${�}_{�}={�}^{�}cos�$, ${�}_{��}=-{�}^{�}sin�$, and ${�}_{��}={�}^{�}sin�$. Therefore, ${�}_{��}+{�}_{��}=\left(-{�}^{�}sin�\right)+{�}^{�}sin�=0$, so the equation is satisfied for both functions.

Clear[u]u[x_, y_]=Exp[y]Sin[x];D[u[x, y], {x, 2}]+D[u[x, y], {y, 2}]

0

We notice that the solutions to Laplace’s equation differ in form. This is unlike solutions to homogeneous linear ordinary differential equations. There, we found that solutions were similar in form. (Recall, all solutions could be generated from a general solution.)

Some of the techniques used in constructing solutions of homogeneous linear ordinary differential equations can be extended to the study of partial differential equations as we see with the following theorem.

Theorem 38 Principle if Superposition

If u1, u2, $\dots$, um are solutions to a linear homogeneous partial differential equation in a region R, then

$\begin{array}{l}\hfill {�}_1{�}_1+{�}_2{�}_2+\cdots +{�}_{�}{�}_{�}=\sum _{�=1}^{�}{�}_{�}{�}_{�},\end{array}$

where c1, c2, $\dots$, cm are constants is also a solution in R.

The Principle of Superposition will be used in solving partial differential equations throughout the rest of the chapter. In fact, we will find that equations can have an infinite set of solutions so that we construct another solution in the form of an infinite series.