––––––––––––––––––––––––––––––––––––Problem #3Consider the following boundary-value, initial-value problem in the regionR={(x, t)|0≤x≤1,0≤t}, having partial differential equation∂2u(x, t)∂x2=∂u(x, t)∂t,boundary conditions:u(0, t) = 0and∂u(1, t)∂x+∂u(1, t)∂t= 0,and initial condition:u(x,0) =f(x).a.)(10 points)Show that the separation of variables method applied to thisproblem does not lead to a regular Sturm-Liouville problem.b.)(10 points)Continue with solving this problem using the method of sepa-ration of variables anyway, and explain why the hard part of this problemisfitting the initial condition.
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Problem #4Solve foru(x, t)in the regionR={(x, t)|0≤x≤1,0≤t}given the partial differential equation,x∂2u(x, t)∂x2−∂u(x, t)∂x=x3∂u(x, t)∂t,the boundary conditions:u(0, t) = 1,u(1, t) = 0and the initial condition,u(x,0) = 0.––––––––––––––––––––––––––––––––––––2