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Fibonacci wavelets-based numerical method for solving fractional order (1 + 1)-dimensional dispersive partial differential equation

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Abstract

In this study, third-order fractional (1 + 1)-dimensional dispersive partial differential equations are numerically solved using the generalized fractional-order Fibonacci wavelet functions. Fibonacci wavelet functions and functional integration matrices are intended to be used. Firstly, we constructed the functional matrix of the Fibonacci wavelets. Secondly, we developed a new technique called the Fibonacci wavelet collocation scheme (FWCS) to solve the fractional partial differential equations (FPDEs). Here, we transform the considered FPDEs into a system of algebraic equations using properties of the Fibonacci wavelets and their matrices. The Newton–Raphson method/fsolve command in MATLAB is used to solve the given system of algebraic equations to determine the unknown coefficients. The projected Fibonacci wavelet collocation method (FWCM) applicability, validity, and accuracy are illustrated with numerical examples. We also discuss the convergence analysis through the theorems and the error bound of the current technique. We execute the vital role of the wavelets, and the proposed method is far better than other methods in the literature through tables.

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Data availability

The data supporting this study's findings are available within the article.

Code availability

Mathematica code for finding the functional matrix of integration is available in the manuscript.

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Acknowledgements

The authors are grateful to the referees and the editor for carefully checking the details and for helpful comments that improved this paper. The author thanks the University Grants Commission (UGC), Govt. of India, for financial support under the UGC-BSR Research Start-Up Grant for 2021-2024:F.30-580/2021(BSR) Dated: 23rd Nov. 2021.

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Authors and Affiliations

  1. Department of Mathematics, Bangalore University, Bengaluru, India

    S. Kumbinarasaiah & Mallanagoud Mulimani

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  1. S. Kumbinarasaiah
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Contributions

KS proposed the main idea of this paper. KS and MM prepared the manuscript and performed all the steps of the proofs in this research. Both authors contributed equally and significantly to writing this paper. Both authors read and approved the final manuscript.

Corresponding author

Correspondence to S. Kumbinarasaiah.

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Appendices

Algorithm to obtain Fibonacci wavelet coefficients and solution

Step 1 Compute Fibonacci wavelet basis using Eq. (2.2).

Step 2 Evaluate the functional matrix of integration for the desired order.

Step 3 Consider Eq. (4.3) depending upon the order of the selected PDE (4.1).

Step 4 Integrate Eq. (4.3) and obtain the required partial derivatives \(V,{V}_{t},{V}_{x},{V}_{xx}\) and \({V}_{xxx}\).

Step 5 Fit the obtained derivatives \(V,{V}_{t},{V}_{x},{V}_{xx}\) and \({V}_{xxx}\) in Eq. (4.1).

Step 6 Now, collocate Eq. (4.1) from the collocation points given in Eq. (4.12) and obtain the algebraic equations.

Step 7 Solve a set of algebraic equations using the fsolve command in MATLAB or Newton Raphson Method to obtain the unknown coefficients.

Step 8 Substitute the unknown coefficients in the Eq. (4.10) to find the FWCS solution.

Mathematica code for finding the functional matrix of integration up to the fourth integration

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Kumbinarasaiah, S., Mulimani, M. Fibonacci wavelets-based numerical method for solving fractional order (1 + 1)-dimensional dispersive partial differential equation. Int. J. Dynam. Control 11, 2232–2255 (2023). https://doi.org/10.1007/s40435-023-01129-1

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