Presentation on theme: "Chapter 13 Partial differential equations"— Presentation transcript:
1Chapter 13 Partial differential equations Mathematical methods in the physical sciences 3nd edition Mary L. BoasChapter 13 Partial differential equationsLecture 13 Laplace, diffusion, and wave equations
21. Introduction (partial differential equation) ex 1) Laplace equation: gravitational potential, electrostatic potential, steady-state temperature with no sourceex 2) Poisson’s equation:: with sources (=f(x,y,z))ex 3) Diffusion or heat flow equation
3ex 4) Wave equationex 5) Helmholtz equation: space part of the solution of either the diffusion or the wave equation
42. Laplace’s equation: steady-state temperature in a rectangular plate (2D) In case of no heat source
82) 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 cmIn this case, e^ky can not be discarded.
9- To be considered IThis is correct, but makes the problem more complicated.(Please check the boundary condition.)- To be considered IIIn 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 solutions5) Boundary condition (boundary value problem)
113. Diffusion or heat flow equation; heat flow in a bar or slab cf. “Why do we need to choose –k^2, not +k^2?”
12Let’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:
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.