Mat-F February 21, 2005 Separation of variables Åke Nordlund Niels Obers, Sigfus Johnsen / Anders Svensson Kristoffer Hauskov Andersen Peter Browne Rønne

Overview Follow-up Computer problems Maple TA problems Maple TA changes New web pages Chapter 19 Separation of variables Exercises today

Follow-up: Computer problems Client-server problems Over-loaded servers Over-loaded network?? Browser problems Mozilla / Maple / JAVA Netscape plugin missing Don’t panic! “… it said in warm, red letters …”

Follow-up: Computer problems Some problems will be fixed Browser: Netscape with plugin Some hardware constraints need work- arounds Network limitations?? Possibly use D315 & A122 CPU limitations Make sure logins and windows are on localhost

Follow-up: Maple TA Maple TA improvements No longer accepts “0” or “whatever” as solutions Remaining limitations Still accepts less than general PDE solutions  Use human judgment! I will !

Follow-up: Maple TA Maple TA changes For those who could not come Wednesday For continued exercises (computer problems) For additional training Maple TA exercises will be (are) available in an open version from the day (Thursday) after the exercises.

New web pages Maple Examples Examples / templates for turn-in assignments Exercise Groups Time & room schedule added Rooms could possibly change next week But not this week!

19. Partial Differential Equations: Separation of variables … Main principles Why? Techniques How?

Main principles Why? Because many systems in physics are separable In particular systems with symmetries Such as spherically symmetric problems Example: Spherical harmonics u(r,Θ,φ,t) = Y lm n (r,Θ,φ) e iωt Example: Spherical harmonics u(r,Θ,φ,t) = Y lm n (r,Θ,φ) e iωt

19. Partial Differential Equations: Separation of Variables … Main principles Why? Common in physics! Greatly simplifies things! Techniques How?

19.1 Separation of variables: the general method Suppose we seek a PDE solution u(x,y,z,t) Ansatz: u(x,y,z,t) = X(x) Y(y) Z(z) T(t) Ansatz: u(x,y,z,t) = X(x) Y(y) Z(z) T(t) This ansatz may be correct or incorrect! If correct we have This ansatz may be correct or incorrect! If correct we have  u/  x = X’(x) Y(y) Z(z) T(t)  2 u/  x 2 = X”(x) Y(y) Z(z) T(t)  u/  x = X’(x) Y(y) Z(z) T(t)  2 u/  x 2 = X”(x) Y(y) Z(z) T(t)

19.1 Separation of variables: the general method Suppose we seek a PDE solution u(x,y,z,t): Ansatz: u(x,y,z,t) = X(x) Y(y) Z(z) T(t) Ansatz: u(x,y,z,t) = X(x) Y(y) Z(z) T(t) This ansatz may be correct or incorrect! If correct we have This ansatz may be correct or incorrect! If correct we have  u/  y = X(x) Y’(y) Z(z) T(t)  2 u/  y 2 = X(x) Y”(y) Z(z) T(t)  u/  y = X(x) Y’(y) Z(z) T(t)  2 u/  y 2 = X(x) Y”(y) Z(z) T(t)  u/  y = X(x) Y’(y) Z(z) T(t)  2 u/  y 2 = X(x) Y”(y) Z(z) T(t)  u/  y = X(x) Y’(y) Z(z) T(t)  2 u/  y 2 = X(x) Y”(y) Z(z) T(t)

19.1 Separation of variables: the general method Suppose we seek a PDE solution u(x,y,z,t): Ansatz: u(x,y,z,t) = X(x) Y(y) Z(z) T(t) Ansatz: u(x,y,z,t) = X(x) Y(y) Z(z) T(t) This ansatz may be correct or incorrect! If correct we have This ansatz may be correct or incorrect! If correct we have  u/  y = X(x) Y’(y) Z(z) T(t)  2 u/  y 2 = X(x) Y”(y) Z(z) T(t)  u/  y = X(x) Y’(y) Z(z) T(t)  2 u/  y 2 = X(x) Y”(y) Z(z) T(t)  u/  z = X(x) Y(y) Z’(z) T(t)  2 u/  z 2 = X(x) Y(y) Z”(z) T(t)  u/  z = X(x) Y(y) Z’(z) T(t)  2 u/  z 2 = X(x) Y(y) Z”(z) T(t)

19.1 Separation of variables: the general method Suppose we seek a PDE solution u(x,y,z,t): Ansatz: u(x,y,z,t) = X(x) Y(y) Z(z) T(t) Ansatz: u(x,y,z,t) = X(x) Y(y) Z(z) T(t) This ansatz may be correct or incorrect! If correct we have This ansatz may be correct or incorrect! If correct we have  u/  y = X(x) Y’(y) Z(z) T(t)  2 u/  y 2 = X(x) Y”(y) Z(z) T(t)  u/  y = X(x) Y’(y) Z(z) T(t)  2 u/  y 2 = X(x) Y”(y) Z(z) T(t)  u/  t = X(x) Y(y) Z(z) T’(t)  2 u/  t 2 = X(x) Y(y) Z(z) T”(t)  u/  t = X(x) Y(y) Z(z) T’(t)  2 u/  t 2 = X(x) Y(y) Z(z) T”(t)

Examples u(x,y,z,t) = xyz 2 sin(bt) u(x,y,z,t) = xy + zt u(x,y,z,t) = (x 2 +y 2 ) z cos(ωt)

Examples u(x,y,z,t) = xyz 2 sin(bt) u(x,y,z,t) = xy + zt u(x,y,z,t) = (x 2 +y 2 ) z cos(ωt) Separable? No! Yes! Hm… ?

Examples u(x,y,z,t) = xyz 2 sin(bt) u(x,y,z,t) = xy + zt u(x,y,z,t) = (x 2 +y 2 ) z cos(ωt) Separable? No! Yes! Hm… ? (x 2 +y 2 )  p 2

Examples u(x,y,z,t) = xyz 2 sin(bt) u(x,y,z,t) = xy + zt u(p,z,t) = p 2 z cos(ωt) Separable? No! Yes! Yes!

Example PDEs The wave equation where c 2  2 u =  2 u/  t 2  2   2 /  x 2 +  2 /  y 2 +  2 /  z 2 Ansatz: u(x,y,z,t) = X(x) Y(y) Z(z) T(t) Ansatz: u(x,y,z,t) = X(x) Y(y) Z(z) T(t)

Separation of variables in the wave equation For the ansatz to work we must have (lets set c=1 for now to get rid of a triviality!) Divide though with XYZT, for: X”YZT + XY”ZT + XYZ”T = XYZT” X”/X + Y”/Y + Z”/Z = T”/T This can only work if all of these are constants!

Separation of variables in the wave equation Let’s take all the constants to be negative Negative  bounded behavior at large x,y,z,t X”/X + Y”/Y + Z”/Z = T”/T  = - µ 2 This is typical: If / when a PDE allows separation of variables, the partial derivatives are replaced with ordinary derivatives, and all that remains of the PDE is an algebraic equation and a set of ODEs – much easier to solve! These are called separation constants

Separation of variables in the wave equation Solutions of the ODEs: Example: x-direction X”/X = - 2 Example: x-direction X”/X = - 2 General solution: X(x) = A exp(i x) + B exp(-i x) or X(x) = A’ cos( x) + B’ sin( x) General solution: X(x) = A exp(i x) + B exp(-i x) or X(x) = A’ cos( x) + B’ sin( x)

Separation of variables in the wave equation Combining all directions (and time) Example: u = exp(i x) exp(i  y) exp(i x) exp(-i cµt) or u = exp(i ( x +  y + x - ω t)) Example: u = exp(i x) exp(i  y) exp(i x) exp(-i cµt) or u = exp(i ( x +  y + x - ω t)) This is a traveling wave, with wave vector {, , } and frequency ω. A general solution of the wave equation is a super-position of such waves.

Example PDEs The diffusion equation where   2 u =  u/  t  2   2 /  x 2 +  2 /  y 2 +  2 /  z 2 Ansatz: u(x,y,z,t) = X(x) Y(y) Z(z) T(t) Ansatz: u(x,y,z,t) = X(x) Y(y) Z(z) T(t)

Separation of variables in the diffusion equation For the ansatz to work we must have (lets set  =1 for now to again get rid of that triviality) Divide though with XYZT, for: X”YZT + XY”ZT + XYZ”T = XYZT’ X”/X + Y”/Y + Z”/Z = T’/T Again, this can only work if all of these are constants!

Separation of variables in the diffusion equation Again, we take all the constants to be negative Negative  bounded behavior at large x,y,z,t X”/X + Y”/Y + Z”/Z = T’/T  = - µ We see the typical behavior again: All that remains of the PDE is an algebraic equation and a set of ODEs – much easier to solve!

Separation of variables in the diffusion equation Solutions of the ODEs: Example: x-direction X”/X = - 2 Example: x-direction X”/X = - 2 General solution: X(x) = A exp(i x) + B exp(-i x) or X(x) = A’ cos( x) + B’ sin( x) General solution: X(x) = A exp(i x) + B exp(-i x) or X(x) = A’ cos( x) + B’ sin( x)

Time-direction: T’/T = - µ Time-direction: T’/T = - µ Separation of variables in the diffusion equation General solution: T(t) = A exp(- µ t) + B exp(µ t) General solution: T(t) = A exp(- µ t) + B exp(µ t)

Separation of variables in the diffusion equation Combining all directions (and time) Example: u = exp(i x) exp(i  y) exp(i x) exp(-µ t) or u = exp(i ( x +  y + x )) exp(- µ t) Example: u = exp(i x) exp(i  y) exp(i x) exp(-µ t) or u = exp(i ( x +  y + x )) exp(- µ t) This is a damped wave, with wave vector {, , } and damping constant µ. A general solution of the diffusion equation is a super-position of such waves.

Example PDEs The Laplace equation where   2 u = 0  2   2 /  x 2 +  2 /  y 2 +  2 /  z 2 Ansatz: u(x,y,z,t) = X(x) Y(y) Z(z) Ansatz: u(x,y,z,t) = X(x) Y(y) Z(z)

Separation of variables in the Laplace equation For the ansatz to work we must have Divide though with XYZT, for: X”YZT + XY”ZT + XYZ”T = 0 X”/X + Y”/Y + Z”/Z = 0 Again, this can only work if all of these are constants!

Separation of variables in the Laplace equation This time we cannot take all constants to be negative! X”/X + Y”/Y + Z”/Z =  = 0 At least one of the terms must be positive  imaginary wave number; i.e., exponential rather than sinusoidal behavior! Let’s assume for simplicity (as in the example in the book) that there is only x & y dependence!

Separation of variables in the Laplace equation Solutions of the ODEs: Example: x-direction X”/X = - 2 < 0 assumed Example: x-direction X”/X = - 2 < 0 assumed General solution: X(x) = A exp(i x) + B exp(-i x) or X(x) = A’ cos( x) + B’ sin( x) General solution: X(x) = A exp(i x) + B exp(-i x) or X(x) = A’ cos( x) + B’ sin( x)

Separation of variables in the Laplace equation Solutions of the ODEs: Example: y-direction Y”/Y = + 2 > 0 follows! Example: y-direction Y”/Y = + 2 > 0 follows! General solution: Y(y) = C exp( y) + D exp(- y) or Y(y) = C’ cosh( y) + B’ sinh( y) General solution: Y(y) = C exp( y) + D exp(- y) or Y(y) = C’ cosh( y) + B’ sinh( y)

Separation of variables in the Laplace equation Combining x & y directions Example: u = [A exp(i x) + B exp(-i x)] [C exp( y) + D exp(- y)] or u = [A’ cos( x) + B’ sin( x)] [C’ cosh( y) + D’ sinh(- y)] Example: u = [A exp(i x) + B exp(-i x)] [C exp( y) + D exp(- y)] or u = [A’ cos( x) + B’ sin( x)] [C’ cosh( y) + D’ sinh(- y)] A general solution of the Laplace equation is a superposition of such functions (which everywhere have a ’saddle-like’ behavior – net curvature = 0)

19.2 Superposition of separated solutions If the PDE with separable solutions is linear … Wave equation Diffusion equation Laplace & Poisson eq. Schrødinger eq. … then any linear combination (superposition) is also a solution Boundary conditions will make a limited selection!

Exercises today: Exercise 19.1 Separation of variables Exercise 19.2 Diffusion of heat Exercise 19.3 Wave equation on a stretched membrane

Examples in the text of the book Heat diffusion Stationary (  u/  t = 0) equivalent to Laplace equation problem! Various boundary conditions selects particular solutions NOTE: a diffusion (heat flow) equation may well have a mixed sinusoidal & exponential behavior in space! consider the limit when the time dependence is weak!  2 u = 0   2 u =  u/  t  2 u = (1/  )  u/  t →  = - µ → 0

Enough for today! Good luck with the exercises 10:15-12:00