Boundary-Value Problems in Rectangular Coordinates Chapter 13 Boundary-Value Problems in Rectangular Coordinates

Outline 13.1 Separable Partial Differential Equations 13.2 Classical PDEs and Boundary-Value Problems 13.3 Heat Equation 13.4 Wave Equation 13.5 Laplace’s Equation 13.6 Nonhomogenous BVPs 13.7 Orthogonal Series Expansions 13.8 Fourier Series in Two Variables

Separable Partial Differential Equations The general form of a linear second-order PDE is It is homogeneous if G = 0, and nonhomogeneous otherwise A solution of a linear PDE is a function of two independent variables possessing all partial derivates occurring in the equation and satisfying the equation in some region of the xy-plane

Separable Partial Differential Equations (cont’d.) We are especially interested in particular solutions of linear PDEs since they appear in many applications The method of separation of variables can be used to find a particular solution of the form of a product of functions of x and y With this assumption, it is sometimes possible to reduce a linear PDE in two variables to two ODEs

Separable Partial Differential Equations (cont’d.) We are especially interested in particular solutions of linear PDEs The method of separation of variables can be used to find a particular solution of the form of a product of functions of x and y With this assumption, it is sometimes possible to reduce a linear PDE in two variables to two ODEs

Separable Partial Differential Equations (cont’d.) According to the superposition principle, if are solutions of a homogenous linear PDE, the linear combination where the are constants is also a solution This additional solution may be written as where are constants

Separable Partial Differential Equations (cont’d.) Linear second-order PDEs can be classified according to their coefficients Hyperbolic if Parabolic if Elliptic if The boundary or initial conditions appropriate for solving a PDE depend on the classification

Classical PDEs and Boundary-Value Problems Classical second-order PDEs have important applications in mathematical physics The one-dimensional heat equation The one-dimensional wave equations Laplace’s equation in two dimensions

Classical PDEs and Boundary-Value Problems (cont’d.) “One-dimensional” refers to the fact that x denotes a spatial dimension whereas t represents time “Two-dimensional” means that x and y are both spatial dimensions Laplace’s equation is abbreviated In two dimensions In three dimensions

Classical PDEs and Boundary-Value Problems (cont’d.) Since the heat and wave equations depend on t, we can prescribe what happens at t = 0 (initial conditions, or ICs) One of the following boundary conditions (BCs) can be used to define behavior at boundaries Dirichlet condition Neumann condition Robin condition , h a constant

Heat Equation Consider a thin rod of length L and initial temperature f(x) with ends held at temperature zero The temperature in the rod is determined from

Heat Equation (cont’d.) Use of separation of variables, application of boundary conditions, and superposition lead to the solution to the BVP

Wave Equation The vertical displacement u(x,t) of a string of length L that is freely vibrating in the vertical plane is determined from

Wave Equation (cont’d.) Use of separation of variables leads to the Sturm-Liouville problem, and solutions that satisfy the wave equation and boundary conditions can be written as Note that when string is released from rest, on so

Wave Equation (cont’d.) For the special case of a plucked string where we can see the motion of the string by plotting the displacement for increasing values of t

Wave Equation (cont’d.) Recall that the constant a in the solution of the BVP is given by where  is mass per unit length and T is the magnitude of the tension in the string When T is large enough, the vibrating string produces a musical sound The sound is a result of standing waves The solution to the wave BVP is a superposition of standing waves

Wave Equation (cont’d.) At a fixed value of x each product function represents simple harmonic motion Each point on a standing wave vibrates with a different amplitude but with the same frequency The first standing wave is the fundamental mode of vibration and its frequency is the fundamental frequency or first harmonic

Laplace’s Equation Suppose we wish to find the steady-state temperature in a rectangular plate whose vertical edges are insulated and whose lower and upper edges are maintained at temperatures 0 and f(x)

Laplace’s Equation (cont’d.) When no heat escapes from the lateral faces, we solve the following BVP

Laplace’s Equation (cont’d.) Use of separation of variables, application of boundary conditions, and superposition lead to the solution to the BVP