Mariusz Iwaniuk

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dsolve convert second order ODE to first order ODE(to Abel).

Without a reproducible example, we can't say much.

@reza gugheri 

You should copy and paste all your code and definition of U to this thread.

Please paste copy&pastable code instead of screenshots. It is not fair to the people who are trying to assist to make them need to retype all your content based on some images.


Solution for Numeric Fourier transform for sigmoid function.



My answer works for erf(x) function and if Maple can finds fourier transform symbolically, then works for general.

Give an example of a function for which a derivative can not be calculated symbolically ?


You can decrease steps in seq command to make plots more accucurate,but the calculation time will increase.



Digits := 20



F := proc (s, x) options operator, arrow; exp(-(1/20)*sqrt(2)*sqrt(s)*sqrt(49*s^2+280*s+800)/sqrt(s+4))*exp((1/20)*sqrt(2)*sqrt(s)*sqrt(49*s^2+280*s+800)*x/sqrt(s+4))/((-exp((1/20)*sqrt(2)*sqrt(s)*sqrt(49*s^2+280*s+800)/sqrt(s+4))+exp(-(1/20)*sqrt(2)*sqrt(s)*sqrt(49*s^2+280*s+800)/sqrt(s+4)))*s)-exp((1/20)*sqrt(2)*sqrt(s)*sqrt(49*s^2+280*s+800)/sqrt(s+4))*exp(-(1/20)*sqrt(2)*sqrt(s)*sqrt(49*s^2+280*s+800)*x/sqrt(s+4))/((-exp((1/20)*sqrt(2)*sqrt(s)*sqrt(49*s^2+280*s+800)/sqrt(s+4))+exp(-(1/20)*sqrt(2)*sqrt(s)*sqrt(49*s^2+280*s+800)/sqrt(s+4)))*s) end proc

INVLAP := proc (F, tau, x) local oldDigits, n, r, k, t, result, Y, B, W, G; n := 80; oldDigits := Digits; Digits := oldDigits+max(80, 2*n); t := evalf(tau); Y := Vector(1 .. 2*n, proc (k) options operator, arrow; F(k*ln(2.0)/t, x) end proc); B := Matrix(1 .. n, 1 .. 1+n, proc (i, j) options operator, arrow; binomial(i, j-1) end proc); W := Vector(1 .. n, proc (k) options operator, arrow; (-1)^(n-k)*k^n*factorial(2*k)/(factorial(k)*factorial(n-k)*factorial(k)*factorial(k-1)) end proc); G := Vector(1 .. n, proc (k) options operator, arrow; add((-1)^r*B[k, r+1]*Y[k+r], r = 0 .. k) end proc); result := ln(2.0)*add(evalf(W[k]*G[k]), k = 1 .. n)/t; return Re(evalf[oldDigits](result)) end proc

INVLAP(F, 0.84e-1, 1/2)



plots:-pointplot({seq([x, INVLAP(F, 0.84e-1, x)], x = 0 .. 1, 0.1e-1)}, connect = true)


data := [seq([seq([i, j, evalf(INVLAP(F, i, j))], i = 10^(-9) .. 1, 0.5e-1)], j = 0 .. 1, 0.5e-1)]

plots:-surfdata(data, axes = frame, labels = [t, x, z])





@Melvin Brown 

For solving  PDEs use extreme powerfull pdsolve command ,will do the job for you.


Sorry Maple 2019 can't find a solution, give me an errors.


My solution  probably works  only in Maple 2019.


General solution for a=A,a=B:


The short answer is that generally no, for your matrix B:=Matrix([[a, -b, e], [c, l, d], [-e, -k, m]], simplify command can't  simplify it. Mathematica also can't.


@Preben Alsholm 

Thanks for clarification.

Intial condition: D[1](T)(x, 0) = 0 it's seems to be wrong ?

If I change to D[2](T)(x, 10) = 0(I assume),pdsolve(numeric) give me: Error, (in pdsolve/numeric) unable to handle elliptic PDEs.

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