## Design and analysis for Bayesian approximation of solutions to linear differential equations

##### Abstract

The problem we are addressing is that of designing and analyzing experiments for numerically solving linear differential equations of the form: Σ-(i=0)^n [g_i (t) y^((i) ) = h(t)] for all t in a specified domain. In general, for a linear differential equation of the order n, we are given n pieces of information to use as boundary conditions. This information may consist of a boundary condition for y and each derivative of y up to y(n-1); or, we may be given multiple boundary conditions for y and/or any of the (n-1) derivatives, as long as there are n pieces of information in all. We use a Bayesian approach in which y is regarded as a realization of a stationary Gaussian stochastic process Y, over the domain T of t. Thus, E(Y(t )) = µ, VAR(Y(t )) = σ2 , and Corr(Y(t ),Y(s )) = R(s-t), for t, s ϵ T. Next we select m enforcement sites, {tn+1 , tn+2,…,tn+m} in T, where the approximation of y, ŷ , must satisfy the differential equation. Finally, we apply Bayes’ Theorem and replace the prior stationary process with a posterior (generally non-stationary) process. Every realization of this posterior process will satisfy each boundary condition and the differential equation at each enforcement site; we use the mean of the posterior process, ŷ, as our approximation for y(t). We wrote a computer program which, given a first-order linear differential equation, a boundary condition, enforcement sites, and a correlation function, R(s-t), will generate ŷ , the approximate solution to the differential equation. To demonstrate the use of this method, we have included some examples of first-order linear differential equations for which the solution is known so that the approximate solution generated by the computer program can be examined in relation to the true solution. We also included an example of the approximate solution to a second-order linear differential equation generated by a similar, expanded program.