Exploring Stochastic Heat Equations: A Numerical Analysis with Fast Discrete Fourier Transform Techniques

This paper presents an innovative numerical technique for specific classes of stochastic heat equations. Our approach uniquely combines a sixth-order compact finite difference algorithm with fast discrete Fourier transforms. While traditional discrete sine transforms are effective for approximating...

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Bibliographic Details
Published in:Axioms Vol. 13; no. 12; p. 886
Main Authors: Khattab, Ahmed G., Semary, Mourad S., Hammad, Doaa A., Fareed, Aisha F.
Format: Journal Article
Language:English
Published: Basel MDPI AG 01.12.2024
Subjects:
Algorithms
Approximation
Brownian motion
compact finite difference
fast discrete Fourier transforms
Fourier transforms
Heat treatment
Mathematical analysis
Mathematics
Methods
Numerical analysis
Partial differential equations
Polynomials
stochastic heat equations
stochastic partial differential equations
Thermodynamics
white noise
ISSN:2075-1680, 2075-1680
Online Access:Get full text
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Summary:This paper presents an innovative numerical technique for specific classes of stochastic heat equations. Our approach uniquely combines a sixth-order compact finite difference algorithm with fast discrete Fourier transforms. While traditional discrete sine transforms are effective for approximating second-order derivatives, they are inadequate for first-order derivatives. To address this limitation, we introduce an innovative variant based on exponential transforms. This method is rigorously evaluated on two forms of stochastic heat equations, and the solutions are compared with those obtained using the established stochastic ten non-polynomial cubic-spline method. The results confirm the accuracy and applicability of our proposed method, highlighting its potential to enhance the numerical treatment of stochastic heat equations.
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ISSN:2075-1680
2075-1680
DOI:10.3390/axioms13120886