The approximate solution of one dimensional stochastic evolution equations by meshless methods

Document Type : Research Article


Department of Applied Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, P.O. Box 14115-134, Tehran, Iran


In this article, we develop an iterative scheme based on the meshless methods to simulate the solution of one dimensional stochastic evolution equations using radial basis function (RBF) interpolation under the concept of Gaussian random field simulation. We use regularized Kansa collocation to approximate the mean solution at space and the time component is discretized by the global $ \theta $-weighted method. Karhunen-lo\`{e}ve expansion is employed for simulating the Gaussian random field. Statistical tools for numerical analysis are standard deviation, absolute error, and root mean square. In this work, we solve two major problems for showing the convergence, and stability of the presented method on two problems. The first problem is the semilinear stochastic evolution problem, and the second one is stochastic advection-diffusion model with different control values.