Using the collage method to solve one-dimensional two-point boundary value problems at steady-state Original Research Article

We consider an inverse problem for two-point boundary value problems (BVPs) D(u)=f(x)−q(x)u(x),u(A)=uA,u(B)=uBD(u)=f(x)−q(x)u(x),u(A)=uA,u(B)=uB, where the goal is to recover an approximation of the physically meaningful functions of xx and parameters in the differential operator DD. We rewrite the BVP inverse problem as an initial value problem (IVP) inverse problem and recast this inverse problem in terms of approximating a “target” function by the fixed point of a contractive operator on a complete metric space. We formulate different collage coding frameworks to solve such problems. Although the approach has broad applicability, we focus on two model problems: approximating the variable diffusivity in a steady-state heat-flow equation, and approximating the flexural rigidity in a steady-state vibrational beam. We present two examples each treated with three different frameworks.