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Given the partial differential equation:

$$\tau\partial_t\varPhi(x,t)=-\partial_x[(-x+A)\varPhi(x,t)]+D\partial_{xx}\varPhi(x,t)$$ where $\tau$ , $A$ and $D$ are constant parameters.

with the initial conditions:

$$\varPhi(x,0)=\lim\limits_{n\to0^+}\frac{1}{2\sqrt{\pi n}}e^\frac{(x-x_0)^2}{4n}$$

and boundary conditions of the kind:

$$\varPhi(a,t)=0$$ and $$\varPhi(-\infty,t)=0$$

I get this: $$\varPhi(x,t)=[a_1\sin({\lambda\over {\sqrt D}}(x-A)e^{t\over\tau})+a_2\cos({\lambda\over {\sqrt D}}(x-A)e^{t\over\tau})][a_3exp({t\over\tau}-{\lambda^2\over2}e^{2t\over\tau})]$$ How to apply initial and boundary conditions?

Thanks in advance.

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