More on Laplace s Equation February 18 2009 Overview Equations in Spheres Vector Calculus and Complex Variables Review last two classes Solutions of Laplace s equation in cylindrical coordinates Spherical coordinate systems Diffusion equation in a sphere Laplace equation in a sphere Legendre polynomials as orthogonal eigenfunctions Results from complex analysis and vector calculus for Laplace s equation Larry Caretto Mechanical Engineering 501B Seminar in Engineering Analysis February 18 2009 2 Review Cylindrical Solutions Review Cylindrical Solutions II Laplace s Equation in two dimensional cylindrical region 0 z L and 0 r R Laplace s Equation in two dimensional cylindrical region 0 z L and 0 r R u r 0 0 and u 0 z is finite u r z Cm sin m z I 0 m r m 0 Cm 2 I 0 m R L u r 0 0 and u 0 z is finite u z 1 u 2u r 0 u R z u R z u r L 0 r r r z 2 0 z L m 2m 1 2L u r z m 0 L sin m z u R z dz 0 4 10 20 9 18 16 n 0 n 1 n 4 7 Since J x I and K are real 6 5 2 12 I0 0 1 6 4 2 0 0 0 1 2 3 4 x 5 10 8 1 Solution is z AI x BK x n 0 n 1 n 4 14 All Kn become infinite as x approaches 0 4 3 d dy 2 2 z z y 0 dz dz z ME 501B Engineering Analysis 0 8 Kn x sin m z u R z dz Review Modified Bessel Functions I x i J ix K x i Y ix i2 1 Satisfy modified differential equation I 0 m R L 3 d2y dy x x x 2 2 y 0 dx 2 dx Equation above transforms to m L m L 2 Cm Modified Bessel Functions 2 Cm sin m z I 0 m r In x 1 u u 0 u R z u R z r r r r z 2 2 5 6 7 0 1 2 3 4 5 6 7 x I x i J ix K x i Y ix 6 1 More on Laplace s Equation February 18 2009 Review Hollow Cylinder Review Hollow Cylinder II Consider various boundary conditions Nonzero conditions on upper or lower surface only gives Bessel eignefunctions Nonzero conditions on inner or outer surface gives sine or cosine eigenfunctions Laplace s Equation in two dimensional cylindrical region 0 z L and Ri r Ro u r 0 u r L u Ri z 0 1 u 2u r 0 u Ro z u R z r r r z 2 I R m u r z Cm sin m z I 0 m r 0 m i K 0 m r m K 0 m Ri L m 1 L 2 K 0 m Ri Cm sin m z u R z dz I 0 m R0 K 0 m Ri I 0 m Ri K 0 m R0 0 7 8 Review Hollow Cylinder III Review Eigenvalues f Ri R0 Laplace s Equation in two dimensional cylindrical region 0 z L and Ri r Ro 1 u 2u r 0 u r L u N r r r r z 2 Eigenvalues m m R0 and eigenfunction R R J 0 m i Y0 m J 0 m Y0 m i 0 R R 0 0 Eigenfunction P0 mr radius ratio 0 5 9 0 25 radius ratio 0 9 0 35 0 20 40 60 80 100 10 Review Conclusions Eigenfunction expansion in P0 mr Y0 mR0 J0 mr J0 mR0 Y0 mr Approach to solving Laplace equation is similar to that of diffusion equation R0 m J 0 m Ri 2 ru N r P0 m r dr Ri sinh m L 2 J 02 m Ri J 02 m R0 Solution for uN r U a constant sinh m z J 0 m Ri P0 m r m 1 sinh m L J 0 m Ri J 0 m R0 u r z U 11 ME 501B Engineering Analysis 0 05 radius ratio 0 1 Review Hollow Cylinder IV Cm 0 05 0 15 Cm sinh m z Y0 m R0 J 0 mr J 0 m R0 Y0 m r m 1 R R J 0 m i Y0 m J 0 m Y0 m i 0 R0 R0 0 15 u r 0 u Ri z u Ro z 0 u r z 0 25 Main difference is that second dimension y or r in Laplace equation gives closed boundary instead of open boundary in time Use separation of variables Have eigenfunction solution sine cosine Bessel or other in one dimension Use eigenfunction expansion to fit condition at one boundary 12 2 More on Laplace s Equation February 18 2009 Review Conclusions II Spherical Coordinates z Use superposition to solve Laplace equation with more than one nonzero boundary Additional cylindrical geometry considerations z r cos Coordinate system used in Kreyszig r y Complex Bessel functions when radial boundary is not eigenfunction solution Must include both Y0 and J0 when radial coordinate does not start at zero must have zero boundary at inner radius x x r sin cos y r sin sin See p A72 is called the polar angle is same in cylindrical system Some works reverse and 13 14 Spherical Geometry Spherical Geometry II Look at diffusion equation in a sphere 1 u t u u 1 2 u 1 1 sin r r 2 r r r 2 sin2 2 r 2 sin 2 1 u 1 2 u Spherical symmetry r has r derivatives only t r 2 r r Initial condition u r 0 f r Boundary conditions u R t uR and u 0 t is finite Write u r t v r t uR and use separation of variables v r t T t P r 1 T t T t t 1 2 P r 2 r r r 2 P r r dT t 2 T t 0 T t Ae t dt d 2 dP r 2 2 Sturm Liouville r r P r 0 problem with weight dr dr function p r r2 15 Spherical Geometry III Define W r such that for P r W r r and radial equation becomes d 2W r 2W r 0 W A sin r B cos r 2 dx W r sin r cos r P r A B r r r Finite solution at r 0 gives B 0 and v 0 at R r gives n n r R C e n t sin n r u r t v r t u R n uR r n 1 17 ME 501B Engineering Analysis 2 16 Spherical Geometry IV Use eigenfunction expansion for initial conditions to evaluate constants Cn C sin n r u r 0 f r v r 0 u R n uR r n 1 m r sin R dr …