Question: differential equation, recurrent sequence

Dear all;

I hope an idea to solve the following problem.

P(n,t): probability at time t

Lambda and mu : two given parameters ( can be fixed ), and we assure mu greater then lambda

 t : time parameter ( we can say that 0<= t <=T , and T fixed ) 

 for n an integrer  greater then 1 we have the following equation

eq := diff(P(n, t), t) = -(lambda+mu)*P(n, t)+lambda*P(n-1, t)+mu*P(n+1, t);

for n=0, we have

ics := diff(P(0, t), t) = -lambda*P(0, t)+mu*P(1, t);

How can we compute P(n,t) ,

is there an exact solution or a numerical method to get the value of P(n,t)

Then we plot( P(n,t),t=0..100) ;


Many thinks






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