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Question: Integration function variable pde depends on number of variables?

Question:Integration function variable pde depends on number of variables?

janhardo 745 Maple
ode pde differential-equations incomplete-question
4 hours ago
1

The concept of an ode that has a function + integration variable as its solution is similar to that of a pde.

A pde (with two variables as example here) that has a function as its solution + function integration variableIn ode, the shape of the solution curve is determined by the integration variable, and in a pde, the shape of the solution surface is determined by the function integration variable.

"maple.ini in user"

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restart; with(plots); PlotPDEsurface := proc (Ct::procedure) local u_expr, FunsT; u_expr := rhs(pdsolve(diff(u(x, t), x) = 2*x+3*t)); FunsT := select(proc (u) options operator, arrow; type(u, 'function') and has(u, t) and not has(u, x) end proc, indets(u_expr, 'function')); if 0 < nops(FunsT) then u_expr := subs(FunsT[1] = Ct(t), u_expr) end if; return plot3d(u_expr, x = -2 .. 2, t = -2 .. 2, axes = boxed, grid = [60, 60], color = gold, transparency = .2, labels = ["x", "t", "u(x,t)"], title = "Solution surface u(x,t)") end proc

"maple.ini in user"

(1.1)
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PlotPDEsurface(proc (t) options operator, arrow; 0 end proc)

 
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PlotPDEsurface(proc (t) options operator, arrow; sin(t) end proc)

 
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restart; with(PDEtools); eq := diff(u(x, t), x) = 2*x+3*t; bc := u(0, t) = sin(t); sol := pdsolve({bc, eq}); Output; u(x, t) = 3*tx+x^2+sin(t)

"maple.ini in user"

 

u(x, t) = 3*t*x+x^2+sin(t)

 

u(x, t) = 3*tx+x^2+sin(t)

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restart; with(plots); u_expr := rhs(pdsolve(diff(u(x, t), x) = 2*x+3*t)); Cforms := [proc (t) options operator, arrow; 0 end proc, proc (t) options operator, arrow; t^2 end proc, proc (t) options operator, arrow; exp(-t^2) end proc, proc (t) options operator, arrow; sin(t)+3*cos(t) end proc]; Colors := [gold, blue, green, red]; Titles := ["C(t)=0", "C(t)=t^2", "C(t)=exp(-t^2)", "C(t)=sin(t)+3cos(t)"]; plots_list := []; for i to nops(Cforms) do expr_i := subs(f__1(t) = Cforms[i](t), u_expr); surf := plot3d(expr_i, x = -2 .. 2, t = -2 .. 2, color = Colors[i], transparency = .25, grid = [50, 50], axes = boxed, labels = ["x", "t", "u(x,t)"], title = Titles[i]); plots_list := [op(plots_list), surf] end do; display(plots_list, title = "Comparison of different C(t) forms", orientation = [120, 45])

"maple.ini in user"

 

 
 

NULL

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