6.3.1 Spherical coordinates

6.3.1.1 [416] No I.C. no B.C.

6.3.1.1 [416] No I.C. no B.C.

problem number 416

Added Jan 10, 2019.

Solve for u(r,θ,ϕ,t) the wave PDE in 3D utt=c22u Using the Physics convention for Spherical coordinates system.

Mathematica

ClearAll["Global`*"]; 
lap = Laplacian[u[r, theta, phi, t], {r, theta, phi}, "Spherical"]; 
pde =  D[u[r, theta, phi, t], {t, 2}] == c^2*lap; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, u[r, theta, phi, t], {r, theta, phi, t}, Assumptions -> {0 < theta < Pi}], 60*10]];
 

{{u(r,θ,ϕ,t){2e12c10(2ϕ+π)tc11(J124c9c2+1(rc11c2)c1+Y124c9c2+1(rc11c2)c2)(e2ϕc10c5+c6)(e2tc11c7+c8)(c42F1(14(c2+4c9c2+2c10+1),14(c2+4c9c2+2c10+1);12;cos2(θ))+c3cos(θ)2F1(14(c2+4c9c2+2c10+3),14(c2+4c9c2+2c10+3);32;cos2(θ)))sinic10(θ)rc0IndeterminateTrue}}

Maple

restart; 
lap:=VectorCalculus:-Laplacian( u(r,theta,phi,t), 'spherical'[r,theta,phi] ); 
pde := diff(u(r,theta,phi,t),t$2)= c^2* lap; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,u(r,theta,phi,t),'build') assuming 0<theta,theta<Pi),output='realtime')); 
sol := simplify(sol);
 

u(r,θ,ϕ,t)=(c7e2t_c4+c8)(c5e2ϕ_c3+c6)(c1BesselJ(c24_c1c22,_c4rc)+c2BesselY(c24_c1c22,_c4rc))(c42hypergeom([2_c3c3c+c24_c14c,2_c3c+3c+c24_c14c],[32],cos(2θ)2+12)|cos(θ)|+c3hypergeom([2_c3c+c+c24_c14c,2_c3cc+c24_c14c],[12],cos(2θ)2+12))(1)i_c32(sini_c3(θ))eϕ_c3t_c4r

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