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Solve analytically the following partial differential equations (PDE's)?

GROUPS:

I have three PDE's and I want to solve it analytically. But I could not find any method to solve it. Can anyone suggest me which method is suitable for these type of PDE's. Details are given in the attached file. Anyone can help me it would be highly appreciated.

Attachments:
POSTED BY: Mirza Farrukh Baig
Answer
14 days ago

Do you have the equations in a safer format? I don't open Microsoft formatted files. Sometimes they contain viruses.

POSTED BY: John Hendrickson
Answer
13 days ago

Yes, I have. Please see the attached file.

Attachment

Attachments:
POSTED BY: Mirza Farrukh Baig
Answer
13 days ago

A Mathematica notebook works much better than a jpg image, for the purpose at hand. Just posting in Mathematica InputForm would be better still.

POSTED BY: Daniel Lichtblau
Answer
13 days ago

Kindly see the attached file.

Attachments:
POSTED BY: Mirza Farrukh Baig
Answer
13 days ago

OK, I translated into the Wolfram Language and made a code for numerical integration. It is not possible to solve the problem analytically.

A1 = 1; A2 = 1; A3 = 1; A4 = 1; \[Epsilon] = 1; \[Kappa] = 1;
Bi = 1; Subscript[U, p] = 1; eq = {D[Subscript[\[Theta], c][x, y], 
     x] - A4/(\[Epsilon]*(y^2 A1 + y*A2 + A3))*
     D[Subscript[\[Theta], c][x, y], y, y] == 0  , 
  D[Subscript[\[Theta], s][x, y], y, y] - 
    Bi*(Subscript[\[Theta], s][x, y] - 
       Subscript[\[Theta], f][x, y]) == 
   0  , \[Kappa]*D[Subscript[\[Theta], f][x, y], y, y] + 
    Bi*(Subscript[\[Theta], s][x, y] - 
       Subscript[\[Theta], f][x, y]) - \[Kappa]*Subscript[U, p]*
     D[Subscript[\[Theta], f][x, y], x] == 0   }; 
ic = {Subscript[\[Theta], s][0, y] == 0, 
   Subscript[\[Theta], f][0, y] == 0, 
   Subscript[\[Theta], c][0, y] == 0};
bc = {DirichletCondition[ {Subscript[\[Theta], s][x, y] == 1, 
     Subscript[\[Theta], f][x, y] == 0, 
     Subscript[\[Theta], c][x, y] == 0} , y == 1], 
   DirichletCondition[ {Subscript[\[Theta], s][x, y] == 0, 
     Subscript[\[Theta], f][x, y] == 0, 
     Subscript[\[Theta], c][x, y] == 1} , y == 0] };   
sol = NDSolve[{eq, ic, bc}, {Subscript[\[Theta], c], 
   Subscript[\[Theta], s], Subscript[\[Theta], f]}, {x, 0, 1}, {y, 0, 
   1}]

{Plot3D[Evaluate[Subscript[\[Theta], c][x, y] /. sol], {x, 0, 1}, {y, 
   0, 1}, Mesh -> None, ColorFunction -> Hue, 
  AxesLabel -> {"x", "y", ""}], 
 Plot3D[Evaluate[Subscript[\[Theta], s][x, y] /. sol], {x, 0, 1}, {y, 
   0, 1}, Mesh -> None, ColorFunction -> Hue, 
  AxesLabel -> {"x", "y", ""}], 
 Plot3D[Evaluate[Subscript[\[Theta], f][x, y] /. sol], {x, 0, 1}, {y, 
   0, 1}, Mesh -> None, ColorFunction -> Hue, 
  AxesLabel -> {"x", "y", ""}]}

fig1

POSTED BY: Alexander Trounev
Answer
11 days ago

Thanks for your answer and suggestion. I tried to solve it according to your method it does not give me the answer.

Attachments:
POSTED BY: Mirza Farrukh Baig
Answer
11 days ago

Ok, can you call $Version? I checked the code on 11.01 and 11.3. It works. But on 10.3 and earlier versions it does not work.

POSTED BY: Alexander Trounev
Answer
11 days ago

Group Abstract Group Abstract