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

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

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

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

6. 1) In the current problem, boundary conditions are

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

9. - To be considered I
This is correct, but makes the problem more complicated.
(Please check the boundary condition.)
- 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.

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)

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?”

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:

13. - Using Boundary condition

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

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

17. 1) case 1

18. 2) case 2

19. 3) Eigenfunctions
first harmonic, fundamental
second harmonic
third
fourth

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

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

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

23. - 2nd step
- 3rd step

27. - Bessel’s equation 1) Equation and solution Bessel’s equation 1
- 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.

28. Bessel’s equation 2
- Graph

29. Bessel’s equation 3
2) Recursion relations

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

31. Bessel’s equation 5

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

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.

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

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

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.

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

40. Legendre’s equation 2

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

42. Legendre’s equation 4
2) Orthogonality

43. 8. Poisson’s equation

44. Example 1
grounded sphere

45. ‘Method of the images’

46. cf. Electric multipoles
monopole
dipole
quadrupole
octopole
2) Expansion for the potential of an arbitrary localized charge distribution

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