Mathematica 7. Mathematica DSolve DSolve[,{ }] u + u = DSolve[D[u[x,y],x]+D[u[x,y],y],u[x,y],{x,y}] {{u[x,y] C[][-x+y]}} C[] uxy (, ) = C( y x) u + u = /( xy) x y DSolve[D[u[x,y],x]+D[u[x,y],y] /(x y),u[x,y],{x,y}] x y ::u@x, yd Log@xD + Log@yD + xc@d@ x + yd yc@d@ x + yd >> x y lnx+ ln y+ x C( y x) y C( y x) ln x ln y uxy (, ) = = C( y x) x y x y u + u = u DSolve[D[u[x,y],x]+D[u[x,y],y] u[x,y]^,u[x,y],{x,y}] ::u@x, yd x C@D@ x + yd >> uxy (, ) = x C ( y x) x y 7
u = a u DSolve[D[u[t,x],t,t] a^ D[u[t,x],x,x],u[t,x],{t,x}] ::u@t, xd C@DB "##### a t + xf + C@DB "##### a t+ xf>> C[] C[] Mathematica a a a> uxt (, ) = C( x at) + C( x+ at) DSolve u + u = DSolve[D[u[x,y],x,x]+D[u[x,y],y,y],u[x,y],{x,y}] {{u[x,y] C[][ x+y]+c[][- x+y]}} C[] C[] xx yy DSolve tt xx u t ( x s) 4at = a uxx uxt (, ) = ϕ( se ) ds a t DSolve DSolve[D[u[t,x],t] a^ D[u[t,x],x,x],u[t,x],{t,x}] DSolve@u H,L @t, xd a u H,L @t, xd, u@t, xd, 8t, x<d Mathematica DSolve utt a uxx x = DSolve [ D[u[x,t],x,x]+D[u[x,t],t,t] x, u[x,t], {x,t} ] DSolve@u H,L @x, td + u H,L @x, td x, u@x, td, 8x, t<d 3 73
DSolve utt uxx =, < x< u e, u xe x x t ( x t) uxt (, ) = e DSolve DSolve[{D[u[t,x],t,t] D[u[t,x],x,x],u[,x] Exp[-x^], Derivative[,][u][,x] x Exp[-x^]},u[t,x],{t,x}] DSolveB:u H,L @t, xd u H,L @t, xd,u@, xd x,u H,L @, xd x x>,u@t, xd, 8t, x<f Mathematica uxx + uyy = u y= = x DSolve DSolve[{D[u[x,y],x]+D[u[x,y],y],u[x,] x^},u[x,y],{x,y}] DSolve@8u H,L @x, yd + u H,L @x, yd, u@x, D x <,u@x, yd, 8x, y<d 7. Mathematica DSolve NDSolve t [,.3] ut = uxx, < x< u, u = u = x( x) NDSolve[{D[u[x,t],t]==D[u[x,t],x,x],u[x,] x(-x),u[,t],u[,t]==},u,{x,,},{t,,.3}] 74
{{u InterpolatingFunction[{{.,.},{.,.3}},<>]}} Mathematica x [,], t [,.3] u(x, t) Plot3D Plot3D[Evaluate[u[x,t]/.First[%]],{x,,},{t,,.3}]...5.5.75.. u. Table[ Plot [Evaluate[u[x,t]/.First[%]],{x,,}],{t,,.3,.}].5..5..5..4.6.8.8.6.4...4.6.8 75
.35.3.5..5..5..4.6.8...8.6.4...4.6.8 Plot [,.5] PlotRange->{,.5} Table[ Plot [Evaluate[u[x,t]/.First[%]],{x,,},PlotRange->{,.5}],{t,,.3,.}].5..5..5..4.6.8.5..5..5..4.6.8 76
.5..5..5..4.6.8.5..5..5..4.6.8 {.,.5,.9} Table[ Plot [Evaluate[u[x,t]/.First[%]],{t,,.3},PlotRange->{,.5}],{x,.,.9,.4}].5..5..5.5..5..5.3.5..5..5.5..5..5.3 77
.5..5..5.5..5..5.3 % Mathematica Solution = NDSolve [ { D[u[x,t],t]==D[u[x,t],x,x], u[x,] x(-x), u[,t], u[,t]== }, u, {x,,}, {t,,.3} ] Table [ Plot [Evaluate[u[x,t]/.First[Solution]],{t,,.3},PlotRange->{,.5}], { x,.,.9,.4} ] t [,] ut = uxx, < x< ux =, ux = u = sin x solution = NDSolveA9 t u@x, td == x,x u@x, td,u@x, D Sin@xD,u H,L @, td ==, u H,L @,td ==,u,8x,, <, 8t,, <E; Plot3D@Evaluate@u@x, tdê. First@solutionDD, 8x,, <, 8t,, <D.75.5.5.8.6.4. 3 78
t [,] ut = uxx + sin t, < x< u =, u = u = sin x solution = NDSolveA9 t u@x, td == x,x u@x, td+ Sin@ tdê, u@x, D Sin@xD,u@, td ==, u@, td ==, u, 8x,, <, 8t,, <E; Plot3D@Evaluate@u@x, tdê. First@solutionDD, 8x,, <, 8t,, <D.75.5.5.8.6.4. 3 3 Mathematica DSolve t [,] utt = uxx, < x< u =, u = u sin x, ut solution = NDSolveA9 t,t u@x, td == x,x u@x, td,u@x, D Sin@ xd, u H,L @x, D, u@, td, u@,td ==,u,8x,, <, 8t,, <E; Plot3D@Evaluate@u@x, tdê. First@solutionDD, 8x,, <, 8t,, <D; 79
.5 -.5 -.8.6.4. 3 t [,] utt = uxx, < x< ux =, ux = u cos x, ut solution = NDSolveA9 t,t u@x, td == x,x u@x, td,u@x, D Cos@ xd,u H,L @x, D, u H,L @, td ==, u H,L @,td ==,u,8x,, <, 8t,, <E; Plot3D@Evaluate@u@x, tdê. First@solutionDD, 8x,, <, 8t,, <D;.5 -.5 -.5.5 3 t [,6] 8
utt = uxx, 6< x< 6 u 6= u 6 x u e, ut solution=ndsolve[{d[u[t, x], t, t] == D[u[t, x], x, x], u[, x] == Exp[-x^], Derivative[,][u][, x] ==, u[t, -6] == u[t, 6]}, u, {t,, 6}, {x, -6, 6}]; Plot3D[Evaluate[u[t,x]/.First[solution]],{t,,6},{x,-6,6}].75.5.5 4 6-5 -.5.5 5 PlotPoints 5 NDSolve[{D[u[t, x], t, t] == D[u[t, x], x, x], u[, x] == Exp[-x^], Derivative[,][u][, x] ==, u[t, -6] == u[t, 6]}, u, {t,, 6}, {x, -6, 6}]; Plot3D[Evaluate[u[t,x]/.First[%]],{t,,6},{x,-6,6}, PlotPoints 5].75.5.5 4 6-5 -.5.5 5 8
Mathematica DSolve t [,] utt = uxx, < x< u =, u = u x /, ut solution = NDSolveA9 t,t u@x, td == x,x u@x, td,u@x, D x ê,u H,L @x, D, u@, td, u@,td ==,u,8x,, <, 8t,, <E; Plot3D@Evaluate@u@x, tdê. First@solutionDD, 8x,, <, 8t,, <D.75.5.5 3.75.5.5 4 NDSolve y [,] uxx + uyy =, < x<, y > u, u = u y= cos x, uy y= solution = uê. FirstANDSolveA9 x,x u@x, yd+ y,y u@x, yd ==, u@x, D == Cos@ xd,u H,L @x, D ==, u@, yd ==, u@, yd == =,u,8x,, <, 8y,, <EE; Plot3D@solution@x, yd, 8x,, <, 8y,, <D; 8
.5 -.5-5..4.6.8.8.6.4. Mathematica Plot3D[solution[x,y],{x,,},{y,,},PlotPoints->8] 9-9.75.5.5.75.5.5 NDSolve ut = uux, < x< u u u = sin x solution = uê. First@NDSolve@8 t u@x, td == u@x, td x u@x, td,u@x, D == Sin@ xd,u@, td == u@, td<, 8u<, 8x,, <, 8t,,.5<DD; Plot3D@solution@x, td, 8x,, <, 8t,,.5<D; 83
.5.4 -.3...4.6..8 utt = uxx + u + u < x< u = u x u e, ut ( )( ), solution=ndsolve[{d[u[t,x],t,t] D[u[t,x],x,x]+(-u[t,x]^)(+u[t,x]),u[,x] Exp[-x^],Derivative[,][u][,x],u[t,-] u[t,]},u,{t,,},{x,-,}] {{u InterpolatingFunction[{{.,.},{...,-.,.,...}},<>]}} Plot3D[Evaluate[u[t,x]/.First[solution]],{t,,},{x,-,},PlotPoints 8] 5-4 -5 6 8 - DensityPlot[Evaluate[u[-t,x]/.First[solution]],{x,-,},{t,,},PlotPoints,Mesh False] 84
8 6 4 - -5 5 7.3 ut = uxx, < x< 4 u, u 4= u = 6sin( x/ ) + 3sin x t su ( x) 6sin( x / ) + 3sin x = U ''( x), < x < 4 U() =, U(4) = ( U x) ( uxt, ) U( x) uxt (, ) Mathematica eq = t u@x, td == x,x u@x, td;e= LaplaceTransform@eq, t, sd e = e ê. 9u@x, D 6 Sin@ xêd + 3 Sin@ xd, LaplaceTransform@u@x, td,t,sd U@xD, LaplaceTransformAu H,L @x, td,t,se D@U@xD, 8x, <D= ans = DSolve@8e, U@D, U@4D <, U@xD,xD@@,, DD sol = InverseLaplaceTransform@ans, s, td 85
s LaplaceTransform@u@x, td,t,sd u@x, D LaplaceTransform@u H,L @x, td,t,sd 6 SinB x F 3Sin@ xd + su@xd U @xd 6 I4 SinA x E + 4sSinA x E SinA x E + CosA x 4 + 5 s+ 4s 6 t i j 3 t 4 + CosB x k Fy z SinB x { F E + 4 s CosA x E SinA x EM Plot3D[sol,{x,,},{t,,}] 4 3.5.5.75.5.5 utt = a uxx, < x< 4 u = u, ut b Mathematica eq = t,t u@x, td a x,x u@x, td;e= LaplaceTransform@eq,t,sD; e = e ê. 9u@x, D, t u@x, D b, LaplaceTransform@u@x, td, t,sd U@xD, LaplaceTransformAu H,L @x, td,t,se D@U@xD, 8x, <D=; ans = DSolve@8e, U@D <, U@xD,xD@@,, DD; sol = InverseLaplaceTransform@ans, s, td a DiracDelta@tD U @xd 86