Lecture 21: Partial Differential Equations I: Laplace equation

Transcription

1 Lecture 21: Partial Differential Equations I: Laplace equation 1 Key points Separation of variable Boundary conditions: Dirichlet boundary conditions, Neumann boundary condition 2 Examples of PDE in physics Laplace's equation: Poisson's equation: Helmholtz equation: Diffusion equation: Wave equation: Schrödinger equation: 3 Type of boundary conditions boundary conditions in space Dirichlet boundary conditions: Neumann boundary conditions: boundary condition in time Initial condition: boundary condition, too This function must satisfy a spatial 4 Methods to solve PDE 1 Separation of variables 2 Integral transformation (Fourier transform and Laplace transform) 51 Example: Steady state temperature profile Consider a temperature profile in an open room with three walls as shown in Figure Two walls at and have temperature T=0 The temperature at

2 is also 0 Only the wall at has T=100 T=0 (y= ) T=0 T=0 y= x= T= x= Step 1: Find an appropriate differential equation Temperature profile is known to satisfy a Laplace equation Step 2: Find the boundary conditions imposed by the problem Step 3: Choose a method to solve the differential equation We use the method of variable separation, which converts the partial differential equation to multiple ordinary differential equations

3 Substituting this product to the original PDE, The left hand side depends only on and the right hand side only on That is impossible! Hence, the neither side depends on or That means they are constant: Then, we have two ODEs:, Step 4: Find general solutions to the ODEs These equations can be solved analytically However, the type of solution depends on if,, or Although it is automatically determined by boundary conditions, it is convenient to know the type of solutions before applying the boundary conditions When, is oscillatory However, it is unlikely since the temperature is expected to decay to zero as increases When, is linear Since the left and right walls have the same temperature, only possible solution is = constant, which is again unlikely So, we assume that Then, the solutions are Step 5: Determine the unknown constants by applying the boundary conditions There are five unknowns,,,,, and On the other hand, there are only four boundary conditions Don't worry You will see

4 Remark 1: Since there are many 's, we have also many 's Remark 2: and are combined to a single constant Noting that the last boundary condition is a Fourier series, we can determine using the theory of Fourier series right hand side which is simplified to Now, evalute it from to a sufficiently large integer Step 6: Construct the final expression of the solution We plot it for L=1

5 Maple PDE := diff(w(x, y), x, x)+diff(w(x, y), y, y) = 0 (511) ans := pdsolve(pde) Here and are arbitrary functions (512) 52 Example: Scalar potential and electric field Two infinitely long rounded metal plates,at and,are connected at by metal strips maintained at

6 a constant potential, as shown in Figure A thin layer of insulation at each comer prevents them from shorting outfind the potential inside the resulting rectangular pipe Show also the electric field inside the pipe Step 1: Find an appropriate differential equation Since the pipe is infinitely long, the potential does not depend on the coordinate Therefore, the potential satisfies the two-dimensional laplace equation Step 2: Find the boundary conditions imposed by the problem Step 3: Choose a method to solve the differential equation We use the method of variable separation, which converts the partial differential equation to two independent ordinary differential equations Substituting this product to the original PDE, we obtain The left hand side depends only on and the right hand side only on That is impossible! Hence, the both sides must depend neither on or That means they are constant:

7 Then, we have two differential equations:, Step 4: Find general solutions to the ODEs These equations can be solved analytically However, the type of solution depends on if,, or Although it is automatically determined by boundary conditions, it is convenient to know the type of solutions before applying the boundary conditions Since the potential is zero at the boundaries in the direction, it will be convenient to have basis functions that can take zero When, is oscillatory and can take zero So, we assume that Then, the solutions to the ODEs are Step 5: Determine the unknown constants by applying the boundary conditions There are five unknowns,,,,, and On the other hand, there are only four boundary conditions Don't worry The last two bouondary conditions suggest that conditions are simply Hence, the boundary, where Noting that this is a Fourier series, we can determine by

8 right hand side which is simplified to Step 6: Construct the final expression of the solution

9

10 53 Example: Spherical coordinates A spherical shell of radius with an insulating ring in the plane has its upper hemisphere at potential and its lower hemisphere at Find the potential inside the sphere Step 1: Find an appropriate differential equation We use the spherical coordinates Step 2: Find the boundary conditions imposed by the problem

11 where Step 3: Separation of variables Separating variables, the PDE can be written as At this point, is separated from and Introducing a separation constant, the PDE can be split to Now, we can separate and Introducing another separation constant [, the ODE for is given by, and for, Step 4: Find general solutions to the ODEs The ODE for can be solved immediately and its general solution is The ODE for can be also solved easily Using Maple ODE solver, =

12 Solving the ODE for needs additional steps Introducing a new variable, The ODE becomes where This ODE is nothing but a generalized Legendre equation Since the solution must be real, the general solution is associate Legendre polynomials, and must be a non-negative integer Combining general all individual solutions,, Step 5: Determine the unknown constants by applying the boundary conditions Boundary condition at The potential should not diverge at Therefore, Boundary condition at First we note that the potential does not depends on Only satisfies the condition Therefore, At this point, the potential is written as and the boundary condition is Using the orthogonality of Legendre polynomials

13 , and using, we obtain the final expression Step 6: Plotting the results: For plotting purpose, we assume and Constructing the expansion coefficients: Constructing the potential: Draw the boundaries: Contour plotting of the potential:

14 Homework Due: 11/21 11am 211 Find the steady-state temperature distribution in a metal plate of 10 cm square if one side is held at 100 and the other three sides at 0 Find numericallt the temperautre at the center of the plate