On the series solution of the stochastic Newell Whitehead Segel equation

Javed Hussain; Javed Hussain

doi:10.3934/math.20231100

AIMS Mathematics

2023, Volume 8, Issue 9: 21591-21605. doi: 10.3934/math.20231100

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On the series solution of the stochastic Newell Whitehead Segel equation

Javed Hussain ^,

Department of Mathematics, Sukkur IBA University, Sukkur, 65200, Pakistan

Received: 13 February 2023 Revised: 21 April 2023 Accepted: 26 April 2023 Published: 07 July 2023
MSC : 58J35, 60H15, 60H35

The purpose of this paper is to present a two-step approach for finding the series solution of the stochastic Newell-Whitehead-Segel (NWS) equation. The proposed two-step approach starts with the use of the Wiener-Hermite expansion (WHE) technique, which allows the conversion of the stochastic problem into a set of coupled deterministic partial differential equations (PDEs) by components. The deterministic kernels of the WHE serve as the solution to the stochastic NWS equation by decomposing the stochastic process. The second step involves solving these PDEs using the reduced differential transform (RDT) algorithm, which enables the determination of the deterministic kernels. The final step involves plugging these kernels back into the WHE to derive the series solution of the stochastic NWS equation. The expectation and variance of the solution are calculated and graphically displayed to provide a clear visual representation of the results. We believe that this two-step technique for computing the series solution process can be used to a great extent for stochastic PDEs arising in a variety of sciences.
- amplitude equations,
- Wiener-Hermite expansion,
- differential transform,
- series solution
Citation: Javed Hussain. On the series solution of the stochastic Newell Whitehead Segel equation[J]. AIMS Mathematics, 2023, 8(9): 21591-21605. doi: 10.3934/math.20231100

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Abstract

The purpose of this paper is to present a two-step approach for finding the series solution of the stochastic Newell-Whitehead-Segel (NWS) equation. The proposed two-step approach starts with the use of the Wiener-Hermite expansion (WHE) technique, which allows the conversion of the stochastic problem into a set of coupled deterministic partial differential equations (PDEs) by components. The deterministic kernels of the WHE serve as the solution to the stochastic NWS equation by decomposing the stochastic process. The second step involves solving these PDEs using the reduced differential transform (RDT) algorithm, which enables the determination of the deterministic kernels. The final step involves plugging these kernels back into the WHE to derive the series solution of the stochastic NWS equation. The expectation and variance of the solution are calculated and graphically displayed to provide a clear visual representation of the results. We believe that this two-step technique for computing the series solution process can be used to a great extent for stochastic PDEs arising in a variety of sciences.

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