How to solve this equation?

Questions
,
1

Would you like to explain whether this equation can be solved and how?

In[1308]:= eq1 = 
\!\(\*SuperscriptBox[\(C2\), 
TagBox[
RowBox[{"(", 
RowBox[{"0", ",", "1"}], ")"}],
Derivative],
MultilineFunction->None]\)[R1, T] == 1 - C2[R1, T] + 
\!\(\*SuperscriptBox[\(C2\), 
TagBox[
RowBox[{"(", 
RowBox[{"1", ",", "0"}], ")"}],
Derivative],
MultilineFunction->None]\)[R1, T]/R1  + (Rv 
\!\(\*SuperscriptBox[\(C2\), 
TagBox[
RowBox[{"(", 
RowBox[{"1", ",", "0"}], ")"}],
Derivative],
MultilineFunction->None]\)[R1, T])/R1 + 
\!\(\*SuperscriptBox[\(C2\), 
TagBox[
RowBox[{"(", 
RowBox[{"2", ",", "0"}], ")"}],
Derivative],
MultilineFunction->None]\)[R1, T]

Out[1308]= 
\!\(\*SuperscriptBox[\(C2\), 
TagBox[
RowBox[{"(", 
RowBox[{"0", ",", "1"}], ")"}],
Derivative],
MultilineFunction->None]\)[R1, T] == 1 - C2[R1, T] + 
\!\(\*SuperscriptBox[\(C2\), 
TagBox[
RowBox[{"(", 
RowBox[{"1", ",", "0"}], ")"}],
Derivative],
MultilineFunction->None]\)[R1, T]/R1 + (Rv 
\!\(\*SuperscriptBox[\(C2\), 
TagBox[
RowBox[{"(", 
RowBox[{"1", ",", "0"}], ")"}],
Derivative],
MultilineFunction->None]\)[R1, T])/R1 + 
\!\(\*SuperscriptBox[\(C2\), 
TagBox[
RowBox[{"(", 
RowBox[{"2", ",", "0"}], ")"}],
Derivative],
MultilineFunction->None]\)[R1, T]

In[1309]:= eq2 = Rv C2[1, T] + 
\!\(\*SuperscriptBox[\(C2\), 
TagBox[
RowBox[{"(", 
RowBox[{"1", ",", "0"}], ")"}],
Derivative],
MultilineFunction->None]\)[1, T] == 0

Out[1309]= Rv C2[1, T] + 
\!\(\*SuperscriptBox[\(C2\), 
TagBox[
RowBox[{"(", 
RowBox[{"1", ",", "0"}], ")"}],
Derivative],
MultilineFunction->None]\)[1, T] == 0
2

Equation can be solved.I used Maple 2019.2

LateX:

\text{C2}(\text{R1},T)=1-\frac{\text{Rv} \text{R1}^{-\frac{\text{Rv}}{2}} I_{\frac{\text{Rv}}{2}}(\text{R1})}{\text{Rv} I_{\frac{\text{Rv}}{2}}(1)+I_{\frac{\text{Rv}}{2}+1}(1)}

Wolfram language:

C2[R1, T] == 1 - (Rv R1^(-(Rv/2))*BesselI[Rv/2, R1])/(Rv*BesselI[Rv/2, 1] + BesselI[Rv/2 + 1, 1])

Check:

Derivative[0, 1][C2][R1, T] == 
1 - C2[R1, T] + 
 Derivative[1, 0][C2][R1, T]/R1 + (Rv*Derivative[1, 0][C2][R1, T])/
 R1 + Derivative[2, 0][C2][R1, T] /. 
 C2 -> Function[{R1, T}, 
1 - (Rv R1^(-(Rv/2))*BesselI[Rv/2, R1])/(
 Rv*BesselI[Rv/2, 1] + BesselI[Rv/2 + 1, 1])] // FullSimplify
 (*True*)

 Rv*C2[1, T] + Derivative[1, 0][C2][1, T] == 0 /. 
  C2 -> Function[{R1, T}, 
  1 - (Rv R1^(-(Rv/2))*BesselI[Rv/2, R1])/(
  Rv*BesselI[Rv/2, 1] + BesselI[Rv/2 + 1, 1])] // FullSimplify
 (*True*)

Regards M.I.

3

Dear Mariusz,
I used the PDEtools of MAPLE and I didn’t get the result.
Would you like share the codes with me?
Thanks a lot.

4

Oh, my mistake PDE is solved in Maple 2020 beta version.

I’m a beta tester. Yes Maple 2019.2 can’t solve.

Regards M.I.

I added a Maple file PDE solution.mw you must only chage a file extension to *.mw

PDE solution.nb (36.2 KB)

5

Dear Mariusz,
I have no Maple 2020 beta version.
Would you like to run the original problem in this version?
Thanks a lot.

eq1 = 
\!\(\*SuperscriptBox[\(C2\), 
TagBox[
RowBox[{"(", 
RowBox[{"0", ",", "1"}], ")"}],
Derivative],
MultilineFunction->None]\)[R1, T] == 1 - C2[R1, T] + 
\!\(\*SuperscriptBox[\(C2\), 
TagBox[
RowBox[{"(", 
RowBox[{"1", ",", "0"}], ")"}],
Derivative],
MultilineFunction->None]\)[R1, T]/(Rd (R0 + R1 Rd)) + (Rv 
\!\(\*SuperscriptBox[\(C2\), 
TagBox[
RowBox[{"(", 
RowBox[{"1", ",", "0"}], ")"}],
Derivative],
MultilineFunction->None]\)[R1, T])/(Rd (R0 + R1 Rd)) + 
\!\(\*SuperscriptBox[\(C2\), 
TagBox[
RowBox[{"(", 
RowBox[{"2", ",", "0"}], ")"}],
Derivative],
MultilineFunction->None]\)[R1, T]/Rd^2

eq2 = Rv C2[1, T] + (Rx 
\!\(\*SuperscriptBox[\(C2\), 
TagBox[
RowBox[{"(", 
RowBox[{"1", ",", "0"}], ")"}],
Derivative],
MultilineFunction->None]\)[1, T])/Rd == 0
6

Unfortunately Maple 2020 does not solve.

7

I appreciate your help and wish you happy new year 2020.

8

How about this one?

eq1 = 
\!\(\*SuperscriptBox[\(C1\), 
TagBox[
RowBox[{"(", 
RowBox[{"0", ",", "1"}], ")"}],
Derivative],
MultilineFunction->None]\)[R, T] == 1 - C1[R, T] + 
\!\(\*SuperscriptBox[\(C1\), 
TagBox[
RowBox[{"(", 
RowBox[{"1", ",", "0"}], ")"}],
Derivative],
MultilineFunction->None]\)[R, T]/R + (Rv 
\!\(\*SuperscriptBox[\(C1\), 
TagBox[
RowBox[{"(", 
RowBox[{"1", ",", "0"}], ")"}],
Derivative],
MultilineFunction->None]\)[R, T])/R + 
\!\(\*SuperscriptBox[\(C1\), 
TagBox[
RowBox[{"(", 
RowBox[{"2", ",", "0"}], ")"}],
Derivative],
MultilineFunction->None]\)[R, T]

eq2 = Rv C1[Rx, T] + Rx 
\!\(\*SuperscriptBox[\(C1\), 
TagBox[
RowBox[{"(", 
RowBox[{"1", ",", "0"}], ")"}],
Derivative],
MultilineFunction->None]\)[Rx, T] == 0
9

Unfortunately Maple does not solve this system analytically.

Do you have any reason to think there is a closed form? Most PDE’s don’t have one. Maybe the best you can do is numerical methods.

All computer algebra systems, including Mathematica,Maple, are limited in their capabilities.

10

I absolutely agree.
This one is just a little different from that one which can be solved by Maple 2020 beta.
With respect to the mathematical structure, they are same.
The numerical solutions have been done earl