The Existence and Uniqueness of Riccati Fractional Differential Equation Solution and Its Approximation Applied to an Economic Growth Model

Johansyah, Muhamad Deni; Supriatna, Asep Kuswandi; Rusyaman, Endang; Saputra, Jumadil

doi:10.3390/math10173029

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The Existence and Uniqueness of Riccati Fractional Differential Equation Solution and Its Approximation Applied to an Economic Growth Model

by

Muhamad Deni Johansyah

^1,*,

Asep Kuswandi Supriatna

¹,

Endang Rusyaman

¹ and

Jumadil Saputra

^2,*

¹

Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Padjadjaran, Sumedang 45363, Jawa Barat, Indonesia

²

Faculty of Business, Economic and Social Development, Universiti Malaysia Terengganu, Kuala Nerus 21030, Terengganu, Malaysia

^*

Authors to whom correspondence should be addressed.

Mathematics 2022, 10(17), 3029; https://doi.org/10.3390/math10173029

Submission received: 25 July 2022 / Revised: 15 August 2022 / Accepted: 18 August 2022 / Published: 23 August 2022

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Abstract

This work proposes and investigates the existence and uniqueness of solutions of Riccati Fractional Differential Equation (RFDE) with constant coefficients using Banach’s fixed point theorem. This theorem is the uniqueness theorem of a fixed point on a contraction mapping of a norm space. Furthermore, the combined theorem of the Adomian Decomposition Method (ADM) and Kamal’s Integral Transform (KIT) is used to convert the solution of the Fractional Differential Equation (FDE) into an infinite polynomial series. In addition, the terms of an infinite polynomial series can be decomposed using ADM, which assumes that a function can be decomposed into an infinite polynomial series and nonlinear operators can be decomposed into an Adomian polynomial series. The final result of this study is to find a solution of the RFDE approach to the economic growth model with a quadratic cost function using the combined ADM and KIT. The results showed that the RFDE solution on the economic growth model using the combined ADM and KIT showed a very good performance. Furthermore, the numerical solution of RFDE on the economic growth model is presented at the end of this work.

Keywords: Riccati Fractional Differential Equation (RFDE); Kamal’s Integral Transform (KIT); Adomian decomposition method

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MDPI and ACS Style

Johansyah, M.D.; Supriatna, A.K.; Rusyaman, E.; Saputra, J. The Existence and Uniqueness of Riccati Fractional Differential Equation Solution and Its Approximation Applied to an Economic Growth Model. Mathematics 2022, 10, 3029. https://doi.org/10.3390/math10173029

AMA Style

Johansyah MD, Supriatna AK, Rusyaman E, Saputra J. The Existence and Uniqueness of Riccati Fractional Differential Equation Solution and Its Approximation Applied to an Economic Growth Model. Mathematics. 2022; 10(17):3029. https://doi.org/10.3390/math10173029

Chicago/Turabian Style

Johansyah, Muhamad Deni, Asep Kuswandi Supriatna, Endang Rusyaman, and Jumadil Saputra. 2022. "The Existence and Uniqueness of Riccati Fractional Differential Equation Solution and Its Approximation Applied to an Economic Growth Model" Mathematics 10, no. 17: 3029. https://doi.org/10.3390/math10173029

APA Style

Johansyah, M. D., Supriatna, A. K., Rusyaman, E., & Saputra, J. (2022). The Existence and Uniqueness of Riccati Fractional Differential Equation Solution and Its Approximation Applied to an Economic Growth Model. Mathematics, 10(17), 3029. https://doi.org/10.3390/math10173029

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The Existence and Uniqueness of Riccati Fractional Differential Equation Solution and Its Approximation Applied to an Economic Growth Model

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