# Chapter 13 Partial differential equations

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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

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

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

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

1) In the current problem, boundary conditions are

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.

- 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.

-. 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)

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?”

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:

- Using Boundary condition

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.

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

1) case 1

2) case 2

3) Eigenfunctions first harmonic, fundamental second harmonic third fourth

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

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

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

- 2nd step - 3rd step

- 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.

Bessel’s equation 2 - Graph

Bessel’s equation 3 2) Recursion relations

Bessel’s equation 4 3) Orthogonality cf. Comparison

Bessel’s equation 5

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

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.

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

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

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

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

Legendre’s equation 2

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

Legendre’s equation 4 2) Orthogonality

8. Poisson’s equation

Example 1 grounded sphere

‘Method of the images’

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

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