Question, solve the given PDE :
$$ \frac{\partial C}{\partial t} = a \frac{\partial^2 C}{\partial x^2} -kC $$ where a, k are constants.
boundary conditions : $C= C_0$ at $x=0$, $C=0 $ at $x =$ infinitum
initial condition : $C=0$ at $t=0$
My attempt :
$$C(x,t) = X(x)~T(t)$$ $$ C = X~T$$ $$ \tag{1}\boxed{ \frac{\partial C}{\partial t} = T^{\prime} X,~ \space \frac{\partial C}{\partial x}= X^{\prime}T, ~ \frac{\partial^2 C}{\partial x^2} = X^{\prime \prime } T}$$ replacing the partial derivatives into the Original equation : $$ T^{\prime}X = a ~X^{\prime \prime}T-k(X~T) $$ $$ X (T^{\prime} +KT)=a ~X^{\prime \prime}T $$ $$\frac{T^{\prime}}{T} = a ~\frac{ X^{\prime \prime}}{X} -k$$ $$ \tag{2}\boxed{\frac{T^{\prime}}{T} = J, ~\frac{ X^{\prime \prime}}{X} -k = J}$$ equating each side to a constant $J = -\lambda^2$ $$ \tag{3}\boxed{\frac{d T}{d t} = -\lambda^2 T ,~ \frac{d^2 X}{d x^2} = \left (\frac{-\lambda^2 + k}{a} \right) X}$$ solving each differential equation : $$T(t) = Ae^{- \lambda t} , ~ ~~~X(x) = B ~e^{\left (x \sqrt{\frac{ k - \lambda^2}{a}} \right)} + C ~e^{\left (-x \sqrt{\frac{ k - \lambda^2}{a}} \right) } $$ Note : D = AB and E = AC
Corrected equation : $$\tag{4} \boxed{{C(x,t) = \left(D \ e^ {x \sqrt{\frac{k-\lambda^2}{a}}} + E e^ {-x \sqrt{\frac{k-\lambda^2}{a}}} \right) e^{-\lambda t}}}$$
confused with Boundary conditions and initial condition, could you guys help please