Numerical solution of partial differential equations with help of fuzzy transform

Noise Reduction on Numerical Solutions of Partial Differential Equations using Fuzzy Transform

“…Many valuable properties of this method has been shown and, as consequence many applications have been published, from geology [2] to electrical engineering [6], including image processing [3] and differential equations [7]. As in other approximation techniques, it can be shown that if an original function is replaced by an approximation model, then a certain simplification of complex computations could be achieved, the F-transform methodology has been demonstrated to be a robust approximation method.…”

“…As in other approximation techniques, it can be shown that if an original function is replaced by an approximation model, then a certain simplification of complex computations could be achieved, the F-transform methodology has been demonstrated to be a robust approximation method. It has been proved that, under some conditions, it can remove periodical noise and it can significantly reduce random noise [4,7].…”

“…Moreover, F-transform approach, on FD schemes, works well when noise is presented on the external force factor [7]. In this work our approach is slightly different, we assume the initial condition is perturbed by noise.…”

“…The calculation for the general case are quite tedious, the ideas presented for the wave equation is enough to understand the process. The derivation described below follows the ideas presented in [7]. Let us consider a set D = [a, b] × [0, T ] and let u(x, t) be a continuous solution of the following equation on D,…”

“…[7] The purpose of this work is to show how F-transform impacts finite-difference (FD) solutions of certain classes of partial differential equations (PDEs) when the initial condition is contaminated by noise. Fuzzy transform works well in removing noise.…”