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1 Chapter 13 Partial differential equations Mathematical methods in the physical sciences 3nd edition Mary L. Boas Lecture 13 Laplace, diffusion, and wave equations

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2 1. Introduction (partial differential equation) ex 1) Laplace equation ex 2) Poisson’s equation: ex 3) Diffusion or heat flow equation : gravitational potential, electrostatic potential, steady-state temperature with no source : with sources (=f(x,y,z))

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3 ex 4) Wave equation ex 5) Helmholtz equation : space part of the solution of either the diffusion or the wave equation

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4 2. Laplace’s equation: steady-state temperature in a rectangular plate (2D) In case of no heat source

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6 1) In the current problem, boundary conditions are

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8 2) How about changing the boundary condition? Let us consider a finite plate of height 30 cm with the top edge at T=0. In this case, e^ky can not be discarded. T=0 at 30 cm

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9 - To be considered I - To be considered II In case that the two adjacent sides are held at 100 (ex. C=D=100 ), the solution can be the combination of C=100 solutions (A, B, D: 0 ) and D=100 (A, B, C: 0 ) solutions. This is correct, but makes the problem more complicated. (Please check the boundary condition.)

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10 -. Summary of separation of variables. 1) A solution is a product of functions of the independent variables. 2) Separate partial equation into several independent ordinary equation. 3) Solve the ordinary differential eq. 4) Linear combination of these basic solutions 5) Boundary condition (boundary value problem)

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11 3. Diffusion or heat flow equation; heat flow in a bar or slab cf. “Why do we need to choose –k^2, not +k^2?”

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12 Let’s take a look at one example. At t=0, T=0 for x=0 and T=100 x=l. From t=0 on, T=0 for x=l. For T(x=0)=0 and T(x=l)=100 at t=0, the initial steady-state temperature distribution:

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13 - Using Boundary condition

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14 For some variation, when T 0, we need to consider u f.as the final state, maybe a linear function. In this case, we can write down the solution simply like this.

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15 4. Wave equation; vibrating string Under the assumption that the string is not stretched, x=0x=l node

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17 1) case 1

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18 2) case 2

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19 3) Eigenfunctions first harmonic, fundamentalsecond harmonic third fourth

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20 Chapter 13 Partial differential equations Mathematical methods in the physical sciences 2nd edition Mary L. Boas Lecture 14 Using Bessel equation

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21 5. Steady-state temperature in a cylinder For this problem, cylindrical coordinate (r, , z) is more useful.

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22 In order to say that a term is constant, 1) function of only one variable 2) variable does not elsewhere in the equation. - 1 st step

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nd step - 3 rd step

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27 - Bessel’s equation 1) Equation and solution - named equation which have been studied extensively. - “Bessel function”: solution of a special differential equation. - being something like damped sines and cosines. - many applications. ex) problems involving cylindrical symmetry (cf. cylinder function); motion of pendulum whose length increases steadily; small oscillations of a flexible chain; railway transition curves; stability of a vertical wire or beam; Fresnel integral in optics; current distribution in a conductor; Fourier series for the arc of a circle. Bessel’s equation 1

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28 - Graph Bessel’s equation 2

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29 2) Recursion relations Bessel’s equation 3

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30 3) Orthogonality cf. Comparison Bessel’s equation 4

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31 Bessel’s equation 5

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32 6. Vibration of a circular membrane (just like drum)

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33 cf. They are not integral multiples of the fundamental as is true for the string (characteristics of the bessel function). This is why a drum is less musical than a violin.

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35 American Journal of Physics, 35, 1029 (1967) (m,n)=(1,0)(2,0) (1,1)

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36 Chapter 13 Partial differential equations Mathematical methods in the physical sciences 2nd edition Mary L. Boas Lecture 15 Using Legendre equation

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37 7. Steady-state temperature in a sphere - Sphere of radius 1 where the surface of upper half is 100, the other is 0 degree.

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39 - Legendre’s equation 1) Equation and solution Legendre’s equation 1

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40 Legendre’s equation 2

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41 - Legendre polynomials- Associated Legendre polynomials Legendre’s equation 3

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42 2) Orthogonality Legendre’s equation 4

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43 8. Poisson’s equation

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44 Example 1 grounded sphere

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45 ‘Method of the images’

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46 monopoledipole quadrupoleoctopole 2) Expansion for the potential of an arbitrary localized charge distribution cf. Electric multipoles

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47 - n = 0 : monopole contribution - n = 1 : dipole - n = 2 : quadrupole - n = 3 : octopole

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