n := 0
u[0] := x^3+(1/2)*A*x^2
for k from 0 to n do
A[k] := sum((Diff(u[i], x))*(Diff(u[k-i], x)), i = 0 .. k);
A[k] := sum((diff(u[i], x))*(diff(u[k-i], x)), i = 0 .. k)
end do;
 

it gives A[0]:=0 which is incorrect. why? and how it will give correct answer?

pde := [diff(u(x, y), x, x)+diff(u(x, y), y, y) = 2*Pi*(2*Pi*y^2-2*Pi*y-1)*exp(Pi*y*(1-y))*sin(Pi*x), u(0, y) = sin(Pi*y), u(1, y) = exp(Pi)*sin(Pi*y), u(x, 2) = exp(-2*Pi)*sin(Pi*x), u(x, 0) = u(x, 1)]pdsolve(pde)

pdsolve(pde)

it does not return any solution and answer, kindly help.

can i have step by step procedure of the solution of the follwoing problem

sys[1] := [-(diff(u(x, t), t, t))-(diff(u(x, t), x, x))+u(x, t) = 2*exp(-t)*(x-(1/2)*x^2+(1/2)*t-1), u(x, 0) = x^2-2*x, u(x, 1) = u(x, 1/2)+((1/2)*x^2-x)*exp(-1)-((3/4)*x^2-(3/2)*x)*exp(-1/2), u(0, t) = 0, eval(diff(u(x, t), x), x = 1) = 0]

sys[1] := [-(diff(u(x, t), t, t))-(diff(u(x, t), x, x))+u(x, t) = 2*exp(-t)*(x-(1/2)*x^2+(1/2)*t-1), u(x, 0) = x^2-2*x, u(x, 1) = u(x, 1/2)+((1/2)*x^2-x)*exp(-1)-((3/4)*x^2-(3/2)*x)*exp(-1/2), u(0, t) = 0, eval(diff(u(x, t), x), x = 1) = 0]

Error, (in PDEtools:-ToJet) found functions to be rewritten in jet notation, {u(1, t)}, having different dependency than the indicated in [u(x, t)]

what is the meaning of above error and how to resolve this to get the solution from pdsolve command?

how to find the closed form solution from the given expression in maple?

u = 140-(1/239500800)*x*t^12+(1/479001600)*x^2*t^12-119*t+40*x-(1/4435200)*t^11-(1/19958400)*x*t^11+(1/39916800)*x^2*t^11-(5/72576)*t^9-(1/60480)*x*t^9+(1/120960)*x^2*t^9+(1/2016)*t^8+(1/10080)*x*t^8-(1/20160)*x^2*t^8-(23/2520)*t^7-(1/420)*x*t^7+(1/840)*x^2*t^7+(23/360)*t^6-(1/144)*x^2*t^6+(1/72)*x*t^6-(7/12)*t^5+(1/12)*x^2*t^5-(1/6)*x*t^5+(19/6)*t^4-(3/8)*x^2*t^4+(3/4)*x*t^4-(95/6)*t^3+(5/2)*x^2*t^3-5*x*t^3+54*t^2-7*x^2*t^2+14*x*t^2+21*x^2*t-42*x*t-42*exp(-t)*x+21*exp(-t)*x^2-21*exp(-t)*t-(1/39916800)*t^12-140*exp(-t)-20*x^2