Optimal recovery of solutions of the generalized heat equation in the unit ball from inaccurate data

We consider the problem of optimal recovery of solutions of the generalized heat equation in the unit ball. Information is given at two time instances, but inaccurate. The solution is to be constructed at some intermediate time. We provide the optimal error and present an algorithm which achieves this error level. The application of optimal recovery theory to problems of partial differential equations was started by J. F. Traub and H. Woźniakowski in [1]. In particular, this monograph considered optimal recovery of solutions of the heat equation from finitely many Fourier coefficients of the initial function. Several recovery problems for partial differential equation from noisy information were recently studied in [2]–[7]. The results considered in these papers were based on a general method for optimal recovery of linear operators developed in [8] and [9] (see also [10]). This method extended previous research from [11]. Various problems of optimal recovery from noisy information may be found in [12] (see also [13] where the complexity of differential and integral equations is discussed). Here we consider the optimal recovery problem for solutions of the generalized heat equation in the unit d-ball at the time τ from inaccurate solutions at the times t1 and t2. Set B d = { x = (x1, . . . , xd) : |x| 2 = d ∑