# How to find the solution of a PDE

I'm trying to find the solution of the following PDE and failing miserably. This is the equation:

DSolve[{\[CurlyEpsilon] fB +
D[y[fB, a], {fB, 1}]/
y[fB, a] (\[CurlyEpsilon] \[Gamma]B - c (fB - fbar)) +
1/2 \[CurlyEpsilon] (\[CurlyEpsilon] - 1) sD^2 +
1/2 D[y[fB, a], {fB, 2}]/y[fB, a] \[Gamma]B^2 (1/sD^2 + 1/sS^2) -
D[y[fB, a], {a, 1}]/y[fB, a] == 0, y[fB, 0] == 1}, y[fB, a], {fB, a}]


The point is that I already know that the solution exists and it is:

ySol[fB_, a_] := Exp[\[CurlyEpsilon] (fbar (a) - 1/c (fB - fbar) (E^(-c a) - 1)) +
1/2 \[CurlyEpsilon]^2 (\[Gamma]B^2/sD^2 + \[Gamma]B^2/
sS^2) (1/c^2 a + 1/c^3 (E^(-c a) - 1) -
1/2 1/c^3 (E^(-c a) - 1)^2) +
1/2 \[CurlyEpsilon] (\[CurlyEpsilon] -
1) sD^2 a + \[CurlyEpsilon]^2 \[Gamma]B/
c (a + 1/c (E^(-c a) - 1))];


So my question is, how could I find it?

This is something I've been trying out. Because I know for sure what the derivatives with respect to fB should be:

D[ySol[fB, a], {fB, 1}]/ ySol[fB, a] == -(((-1 + E^(-a c)) \[CurlyEpsilon])/c)


and

D[ySol[fB, a], {fB, 2}]/ ySol[fB, a] == ((-1 + E^(-a c))^2 \[CurlyEpsilon]^2)/c^2


I thought I could replace them in the original PDE to get an ODE and solve it just by integrating as:

Integrate[\[CurlyEpsilon] fB - ((-1 + E^(-c (a))) \[CurlyEpsilon])/   c (\[CurlyEpsilon] \[Gamma]B - c (fB - fbar)) +   1/2 \[CurlyEpsilon] (\[CurlyEpsilon] - 1) sD^2 +   1/2 ((-1 + E^(-c (a)))^2 \[CurlyEpsilon]^2)/   c^2 \[Gamma]B^2 (1/sD^2 + 1/sS^2), a]


Now my issue is that, although this solution clearly satisfies the original PDE, the derivatives are not what I set them to be initially :(

Temp[a_, fB_] :=  Exp[1/2 \[CurlyEpsilon] (a (2 fbar +
sD^2 (-1 + \[CurlyEpsilon])) + (
E^(-2 a c) (-1 + 4 E^(a c)) (sD^2 +
sS^2) \[Gamma]B^2 \[CurlyEpsilon])/(
2 c^3 sD^2 sS^2) + (\[Gamma]B (2 E^(-a c) +
a (1/sD^2 + 1/sS^2) \[Gamma]B) \[CurlyEpsilon])/c^2 + (
2 E^(-a c) (-fB + fbar + a E^(a c) \[Gamma]B \[CurlyEpsilon]))/
c)];


Meaning:

D[Temp[a, fB], {fB, 1}]/Temp[a, fB] == -((E^(-a c) \[CurlyEpsilon])/c)


And not what I initially replaced for... Does anybody have any other ideas on how to solve this PDE? Or maybe some thoughts on why the usual DSolve[...] is not working in this context?

Thanks thanks!

• See comparsion for solving PDE symbolically Maple vs Mathematica: 12000.org/my_notes/pde_in_CAS/pde_in_cas.htm .Mathematica solved 58.46%. Maple solved 89.23% – Mariusz Iwaniuk Mar 22 '18 at 15:27
• @MariuszIwaniuk are you suggesting I should better try to solve in Maple? – NinjaCowAndForks Mar 22 '18 at 15:29
• I suggest to try Maple+Mathematica together. You can increase your chances to solve. – Mariusz Iwaniuk Mar 22 '18 at 15:31
• I tried with Maple 2018 and give me only general solution with exp,KummerM and KummerU functions. – Mariusz Iwaniuk Mar 22 '18 at 17:18