Question

Partial Differential Equation

Solve the following boundary value problems

\[\Large \frac{\partial^{2} u}{\partial x \partial y} (x,y) =3x^{2} , u(x,0) = x^n (n > 0) , u(0,y) = 0 \]

Answer 1

I've tried solve it

\[ \Large \frac{\partial}{\partial x} \left(\frac{\partial u}{\partial y} (x,y) \right) = 3x^{2} \]

Integrate respect to \(x\)

\[ \Large \frac{\partial u}{\partial y} (x,y) = x^3 + f(y) \]

Integrate respect to \(y\)

\[ \Large u(x,y) = x^{3} y + F(y) + g(x) ; \frac{\partial}{\partial y} F(y) = f(y) \]


\[\Large u(x,0) = x^n \implies x^{3} \cdot 0 + F(0) + g(x) = x^{n} \implies F(0) + g(x) = x^n\]

\[\Large u(0,y) = 0 \implies 0 \cdot y + F(y) + g(0) = 0 \implies F(y) + g(0) = 0\]

\[ \Large u(x,y) = x^3y - g(0) + x^n - F(0) \]

\[ \Large u(0,y) = 0 \implies 0 - g(0) + 0 -F(0) = 0 \implies g(0) = - F(0) \]

so..,
\[ \Large u(x,y) = x^3y + x^n \]

is this correct?

Answer 2

While I am not learning this currently, the work seems to be correct, and the answer seems to agree with your initial problem (checking by taking partial derivative).

Answer 3

thank you...