Superposition principle for the homogeneous case. If the functions 1.4.1 Exercises
1 Problem 1.1 Consider the problemClearly, the function is a solution. Is this solution unique? Does the answer depend on ?
2 Problem 1.2 Consider the problemIs the solution unique? ( is a given function).Under what condition on a solution exists?
3 Problem 1.3 Suppose are solutions of the linear differential equation , where . Under what condition on the constant coefficients is the linear combination also a solution of this equation?
4 Problem 1.4 Consider the nonlinear ODE .Show that and both are solutions, but is not a solution.For which value of is a solution? What about ?
5 Problem 1.5 Show that each of the following equations has a solution of the form for a proper choice of constants . Find the constants for each example.
6 Problem 1.6Consider the equation . Write the equation in the coordinates .Find the general solution of the equation.Consider the equation . Write it in the coordinates and .
7 Problem 1.7Show that for satisfiesFind a linear combination of s that satisfies .Solve the Dirichlet problem
1.5 Some Equations of Mathematical Physics
Throughout the book, we focus on the study of some of the basic PDEs of mathematical physics. These equations involve differential operators as the gradient and Laplacian, acting on
(1.5.1)
These equations describe fundamental physical phenomena as follows:
(1.5.2)
Here
1.5.1 The Poisson Equation
The Poisson equation is a BVP of the form
(1.5.3)
Below are some common phenomena modeled by the Poisson's equation.
Electrostatics: To describe the components of the Maxwell's equations, associating the electric‐ and potential fields and to the charge density , roughly speaking, we haveand with a Dirichlet boundary condition on .
Fluid mechanics: The velocity field of a rotation‐free fluid flow satisfies and hence, is a, so‐called, gradient field: , with being a scalar potential field. The rotation‐free incompressible fluid flow satisfies , which yields the Laplace's equation for its potential. At a solid boundary, this problem will be associated with homogeneous Neumann boundary condition, due to the fact that in such a boundary, the normal velocity is zero.
Statistical physics: The random motion of particles inside a container until they hit the boundary is described by the probability of a particle starting at the point winding up to stop at some point on , where means that it is certain and means that it never happens. It turns out that solves the Laplace's equation , with discontinuity at the boundary: on and on where . Poisson's equation is of vital importance in describing, the coupling of nonlinearity aspects, in the system of gas kinetic/dynamic equations.
1.5.2 The Heat Equation
1.5.2.1 A Model Problem for the Stationary Heat Equation in
Below we model the heat conduction in a thin heat‐conducting wire stretched between the two endpoints of an interval
Figure 1.3 A heat‐conducting one‐dimensional wire.
To this end, let
By the Fundamental Theorem of Calculus,
