Lecture 16 Solving the Laplace equation in 2-D Remember Phils Problems and your notes = everything Only 6 lectures left Come to see me before the end of term I’ve put more sample questions and answers in Phils Problems I’ve also added all the answers to the tutorial questions in your notes Have a look at homework 2 (due in on 12/12/08)

Poisson’s equation Introduction to PDEs In many physical situations we encounter quantities which depend on two or more variables, for example the displacement of a string varies with space and time: y(x, t). Handing such functions mathematically involves partial differentiation and partial differential equations (PDEs). As (4) in regions containing mass, charge, sources of heat, etc. Electromagnetism, gravitation, hydrodynamics, heat flow. Laplace’s equation Heat flow, chemical diffusion, etc. Diffusion equation Quantum mechanics Schrödinger’s equation Elastic waves, sound waves, electromagnetic waves, etc. Wave equation

SUMMARY of the procedure used to solve PDEs 9. The Fourier series can be used to find the particular solution at all times. 1. We have an equation with supplied boundary conditions 2. We look for a solution of the form 3. We find that the variables ‘separate’ 4. We use the boundary conditions to deduce the polarity of N. e.g. 5. We use the boundary conditions further to find allowed values of k and hence X(x). 6. We find the corresponding solution of the equation for T(t). 7. We hence write down the special solutions. 8. By the principle of superposition, the general solution is the sum of all special solutions.. so

The Laplace equation in 2D can be applied to any system in which the value u does not change with distance. In ‘steady state’ problems where nothing is changing with time, the equation simplifies to which is the Laplace equation. Solving the Laplace equation in 2 dimensions Heat flow is governed by the diffusion equation, We will look at this equation in 2D by considering the following exercise from the notes Exercise Consider a rectangular metal plate 10 cm wide and very long. The two long sides and the far end are held at 0ºC and the base at 100ºC. Find the steady state temperature distribution in x and y inside the plate using the boundary conditions.

Solving the Laplace equation in 2 dimensions Exercise Consider a rectangular metal plate 10 cm wide and very long. The two long sides and the far end are held at 0ºC and the base at 100ºC. Find the steady state temperature distribution in x and y inside the plate using the boundary conditions. Step 1: Separation of the Variables Substitute this into the Laplace equation: Separating variables: Step 2: Rearrange the equation Laplace equation in 2-D is Our boundary conditions are true at special values of x and y, so we look for solutions of the form T (x, y) = X(x)Y(y).

Solving the Laplace equation in 2 dimensions Exercise Consider a rectangular metal plate 10 cm wide and very long. The two long sides and the far end are held at 0ºC and the base at 100ºC. Step 3: Equate to a constant Now we have separated the variables. The above equation can only be true for all x, y if both sides are equal to a constant. So we have which rearranges to(i) which rearranges to (ii) We know that X(0) = X(L) = 0 and we know that in the Y direction up the page we expect an exponential drop or something similar from T(x,0) = 100 to T(x,∞) = 0. It is clear therefore that for a solution in x such that X(x) = 0 more than once, the constant must be negative (like a LHO). For convenience we choose the constant as -k 2 so…. Step 4: Decide based on situation if N is positive or negative

Also since then Solution to (i) is We know that X(0) = X(L) = 0 so A = 0. Step 4 continued: Decide based on situation if N is positive or negative Solving the Laplace equation in 2 dimensions Exercise Consider a rectangular metal plate 10 cm wide and very long. The two long sides and the far end are held at 0ºC and the base at 100ºC. Find the steady state temperature distribution in x and y inside the plate using the boundary conditions. So we have (i) (ii)and Step 5: Solve for the boundary conditions for X(x) We know that X(0) = X(L) = 0 and we know that for Y we expect an exponential drop or something similar from T(x,0) = 100 to T(x,∞) = 0. so we can say

But and L = 10 so this can be written ….. Solution to (ii) is since we know that T (x, y) → 0 as y → ∞ then we can state that D = 0 and Solving the Laplace equation in 2 dimensions Exercise Consider a rectangular metal plate 10 cm wide and very long. The two long sides and the far end are held at 0ºC and the base at 100ºC. Step 6: Solve for the boundary conditions for Y(y) We know that for Y we expect an exponential drop or something similar from T(x,0) = 100 to T(x,∞) = 0. Step 7: Write down the special solution for  (x, y) i.e. T(x,y) =X(x)Y(y) where P = CB

Solving the Laplace equation in 2 dimensions Exercise Consider a rectangular metal plate 10 cm wide and very long. The two long sides and the far end are held at 0ºC and the base at 100ºC. Find the steady state temperature distribution in x and y inside the plate using the boundary conditions. Step 8: Constructing the general solution for  (x, y) The general solution of our equation is the sum of all special solutions: This already satisfies the boundary conditions for x, namely that T(0,y) = T(L,y) = 0. All that remains is calculate the required values of P n such that the T(x,0) =100 is satisfied Step 9: Use Fourier series to find values of P n Since the temperature at y = 0 is 100 then

Solving the Laplace equation in 2 dimensions Exercise Consider a rectangular metal plate 10 cm wide and very long. The two long sides and the far end are held at 0ºC and the base at 100ºC. Find the steady state temperature distribution in x and y inside the plate using the boundary conditions. Step 9 continued: Use Fourier series to find values of P n So if temperature at y = 0 is 100 so general solution here is Here is a lateral jump that isn’t obvious!!!! As you know the Half-range sine series is given by: where Look how similar this is to the expression for T(x,0) if we set f(x)=100. So all we have to do now is calculate the half-range sine series in the usual way in order to find the values of P n.

Solving the Laplace equation in 2 dimensions Exercise Consider a rectangular metal plate 10 cm wide and very long. The two long sides and the far end are held at 0ºC and the base at 100ºC. Find the steady state temperature distribution in x and y inside the plate using the boundary conditions. Step 9 continued: Use Fourier series to find values of P n In effect we’re finding the half range sine series for a pulse as defined below. Half-range sine series coefficient PnPn n So in order for the boundary conditions for T(x,0) = 100 to be satisfied, we must take the following values of P n in the sum.

Solving the Laplace equation in 2 dimensions Exercise Consider a rectangular metal plate 10 cm wide and very long. The two long sides and the far end are held at 0ºC and the base at 100ºC. Find the steady state temperature distribution in x and y inside the plate using the boundary conditions. Step 10: Finding the full solution for all x and y So in order for the boundary conditions for T(x,0) = 100 to be satisfied: Remember from step 8 the general solution of our equation is the sum of all special solutions: So the full solution can be written as:

Let’s check that this fulfils all boundary conditions Solving the Laplace equation in 2 dimensions Exercise Consider a rectangular metal plate 10 cm wide and very long. The two long sides and the far end are held at 0ºC and the base at 100ºC. Find the steady state temperature distribution in x and y inside the plate using the boundary conditions. T(x, 0) = 100 : T(x, ∞) = 0 : T(0, y) = 0 : T(L, y) = 0 : This is Fourier series for f(x) = 100 we just found Just think about how T(x, y) would look if you were to plot it on a graph

Revision for the exam Above is a sample exam paper for this course There are 5 questions. You have to answer Q1 but then choose any 2 others I’ll put previous years maths question papers up on Phils Problems very soon Q1: Basic questions to test elementary concepts. Looking at previous years you can expect complex number manipulation, basic integration, solving ODEs, applying boundary conditions, plotting functions. Easy stuff. Q2-5: More detailed questions usually centred about specific topics: InhomoODE, damped SHM equation, Fourier series, Half range Fourier series, Fourier transforms, convolution, partial differential equation solving (including applying an initial condition to general solution for a specific case), Cartesian 3D systems, Spherical polar 3D systems, Spherical harmonics The notes are the source of examinable material – NOT the lecture presentations I wont be asking specific questions about Quantum mechanics outside of the notes

Revision for the exam The notes are the source of examinable material – NOT the lecture presentations Things to do now Read through the notes using the lecture presentations to help where required. At the end of each section in the notes try Phils problem questions, then try the tutorial questions, then look at your problem and homework questions. If you can do these questions (they’re fun) then you’re in excellent shape for getting over 80% in the exam. Any problems – see me in my office or me Same applies over holidays. I’ll be in the department most days but a question or tell me you want to meet up and I’ll make sure I’m in. Look at the past exam papers for the style of questions and the depth to which you need to know stuff. You’ll have the standard maths formulae and physical constants sheets (I’ll put a copy of it up on Phils Problems so you are sure what’s on it). At present you don’t need to know any equations e.g. Fourier series or transforms, wave equation, polars.