### Journals Information

Mathematics and Statistics Vol. 11(3), pp. 490 - 508
DOI: 10.13189/ms.2023.110305
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## Solution Analysis of Riccati's Fractional Differential Equations Using the ADM-Laplace Transformation and the ADM-Kashuri-Fundo Transformation

Muhamad Deni Johansyah 1,*, Asep Kuswandi Supriatna 1, Endang Rusyaman 1, Salma Az-Zahra 1, Eddy Djauhari 1, Aceng Sambas 2,3
1 Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Padjadjaran, Sumedang 45363, Indonesia
2 Faculty of Informatics and Computing, Universiti Sultan Zainal Abidin, Kampung Gong Badak, 21300, Terengganu, Malaysia
3 Department of Mechanical Engineering, Universitas Muhammadiyah Tasikmalaya, Tasikmalaya, Jawa Barat 46196, Indonesia

ABSTRACT

Fractional differential equations (FDEs) are differential equations that involve fractional derivatives. Unlike ordinary derivatives, fractional derivatives are defined by fractional powers of the differentiation operator. FDEs can arise in a variety of contexts, including physics, engineering, biology, and finance. They are typically more complex than ordinary differential equations, and their solutions may exhibit unusual properties such as long-range memory, non-locality, and power-law behavior. The solution of the Riccati Fractional Differential Equation (RFDE) is generally challenging due to its nonlinearity and the presence of the fractional power term. The fractional derivative operators in the RFDE are non-local and involve an integral over a certain range of the independent variable. The non-local nature of the fractional derivatives can make the RFDE harder to handle compared to ordinary differential equations. In this paper, we have examined the Riccati Fractional Differential Equation (RFDE) using the combined theorem of the Adomian Decomposition Method and Laplace Transform (ADM-LT). Furthermore, we have compared with Adomian Decomposition Method and Kashuri-Fundo Transformation (ADM-KFT). It is shown that the ADM-LT is equivalent to the ADM-KFT algorithm for solving the Riccati equation. In addition, we have added new theorem of the relationship between the Kashuri Fundo inverse and the Laplace Transform inverse. The main finding of our study shows that the Adomian Decomposition Method and Laplace Transform (ADM-LT) have a good agreement between numerical simulation and exact solution.

KEYWORDS
Fractional Calculus, Laplace Transform, Kashuri-Fundo Transformation, Adomian Decomposition Method

Cite This Paper in IEEE or APA Citation Styles
(a). IEEE Format:
[1] Muhamad Deni Johansyah , Asep Kuswandi Supriatna , Endang Rusyaman , Salma Az-Zahra , Eddy Djauhari , Aceng Sambas , "Solution Analysis of Riccati's Fractional Differential Equations Using the ADM-Laplace Transformation and the ADM-Kashuri-Fundo Transformation," Mathematics and Statistics, Vol. 11, No. 3, pp. 490 - 508, 2023. DOI: 10.13189/ms.2023.110305.

(b). APA Format:
Muhamad Deni Johansyah , Asep Kuswandi Supriatna , Endang Rusyaman , Salma Az-Zahra , Eddy Djauhari , Aceng Sambas (2023). Solution Analysis of Riccati's Fractional Differential Equations Using the ADM-Laplace Transformation and the ADM-Kashuri-Fundo Transformation. Mathematics and Statistics, 11(3), 490 - 508. DOI: 10.13189/ms.2023.110305.