An efficient meshless method to approximate semi-linear stochastic evolution equations

• Published in: Engineering with computers Vol. 40; no. 1; pp. 61 - 90
• Main Authors: Jalili, Mahdi, Salehi, Rezvan, Dehghan, Mehdi
• Format: Journal Article
• Language: English
• Published: London Springer London 01.02.2024; Springer Nature B.V
• Subjects: Accuracy; Applied mathematics; Approximation; CAE) and Design; Calculus of Variations and Optimal Control; Optimization; Classical Mechanics; Computer Science; Computer-Aided Engineering (CAD; Computers; Control; Finite element method; Linear evolution equations; Math. Applications in Chemistry; Mathematical analysis; Mathematical and Computational Engineering; Meshless methods; Methods; Noise; Numerical analysis; Original Article; Partial differential equations; Physics; Signal processing; Stability analysis; Systems Theory; Radial basis functions; 65C30; 65N35; Kansa method; 65M70; Semi-linear stochastic evolution; Collocation method; Karhunen–Loéve expansion; Gaussian random fields; 65M12; 60G15
• ISSN: 0177-0667; 1435-5663
• DOI: 10.1007/s00366-022-01770-y

In this article, we are concerned with meshless methods to approximate and simulate the solution of semi-linear stochastic evolution equations. We first study the asymmetric Kansa method and then consider its regularized form. Kansa method is an efficient approach that is easy to implement and adapt and has sufficient accuracy and approximation power. We employ Karhunen–Loéve expansion for having faster and better simulations for the stochastic part. The absolute error, standard deviation, root mean square error, and CPU times for showing the accuracy and speed of our methodology are calculated. From the numerical analysis view, the stability of this methodology for time-dependent problems is investigated by numerical factors in the computational part. Experimentally, the performance of both presented methods is more significant, and proportionally they have better results to previous work in this subject.