By the method of separation of variables, solve the boundary value problem
∂u/∂x=4∂u/∂y, u(0,y)=8e-3y
let the solution is of the form u(x,y)=X(x)Y(y).
put u(x,y) in the given pde.
"Y\\frac{\\partial X}{\\partial x}=4X\\frac{\\partial Y}{\\partial y}\\\\\n\\frac{1}{X}\\frac{\\partial X}{\\partial x}=\\frac{4}{Y}\\frac{\\partial Y}{\\partial y}=\\lambda(say)\\\\\n\\text{first solve}\\\\\n\\frac{1}{X}\\frac{\\partial X}{\\partial x}=\\lambda\\\\\n\\implies X=c_1e^{\\lambda x}\\\\\n\\text{Second solve}\\\\\n\\frac{4}{Y}\\frac{\\partial Y}{\\partial y}=\\lambda\\\\\n\\implies Y=c_1e^{ \\frac{\\lambda y}{4}}\\\\\n\\text{Therefore, the solution is given by}\\\\\nu(x,y)=c_1c_2e^{\\lambda(x+\\frac{ y}{4})}\n=ce^{\\lambda(x+\\frac{ y}{4})}\\\\\n\\text{Now, apply the given condition, we get}\\\\\nc=8e^{-y(3+ \\frac{\\lambda}{4})}\n\\text{Therefore, the solution is}\\\\\nu(x,y)=8e^{-y(3+ \\frac{\\lambda}{4})}e^{\\lambda(x+y)}\\\\\n\\text{ Since, it is a boundary value problem. Therefore, when the boundary conditions given then we can calculate the value of lambda and get the solution u(x,y).}"
Comments
Leave a comment