A Novel Approach to Solving the Fractional Bogoyavlensky Equation via Modified Extended Direct Algebraic Method
Bilal M
Published on: 2024-12-30
Abstract
An advanced modified extended direct algebraic method for solving nonlinear fractional differential equations (FDEs) precisely is presented in this ground-breaking paper. Through the use of a sophisticated fractional complex transformation, we translate nonlinear FDEs with Jumarie modified Riemann-Liouville derivatives into their corresponding ordinary differential equations. Applying the approach to two nonlinear FDEs, including the time-fractional Bogoyavlensky problem, we show its remarkable strength and adaptability. We indisputably demonstrate the effectiveness of the method in solving a wide range of nonlinear FDEs, opening up new avenues for progress in this ever-evolving subject. The significance of this research lies in its capacity to tackle difficult problems in a range of domains, such as physics, engineering, and finance. Our method to nonlinear FDEs is reliable and efficient, which opens up new possibilities for modelling and analysing real-world processes. The subject of fractional calculus, which has garnered a lot of attention recently due to its enhanced ability to characterise complex systems and processes, is also advanced by this study.
Keywords
Nonlinear Fractional Differential Equations; Modified Extended Direct Algebraic Method; Fractional Complex Transformation; Jumarie Modified Riemann-Liouville Derivatives; Time-Fractional Bogoyavlensky Equation; Fractional CalculusIntroduction
Fractional calculus, which applies derivatives and integrals of any order, is one of the generalisations of classical ordinary calculus in the subject of mathematical analysis. Fractional calculus has a long history dating back several centuries. Spanier and Oldham were the first to introduce the FDEs [18]. Visco elasticity, signal processing, biology, control theory, electrochemistry, and other scientific domains all use nonlinear FDEs [1,17,19,20]. Numerous physical events that these equations can represent depend on the precise solutions of FDEs. Numerous investigations into the analytical and numerical solutions of the nonlinear FDEs have been conducted in recent years. The several definitions of fractional derivations have been illustrated in numerous works. The most popular derivatives are the Caputo and Riemann-Liouville derivatives. Recently, a variety of methods have been developed to solve nonlinear FDEs. Important techniques for creating the analytical and numerical solutions of FDES include the fractional Lie group method [6], the exp-function method [5], the first integral method [15], the simplest equation method [13], the fractional sub-equation method [7,8,11], the Adamian decomposition method [3], and others. The goal of this work is to obtain then on linear FDE, sometimes referred to as the time fractional Bogoyavlensky equation, using the MEDAM technique. We analyse the time fractional Bogoyavlensky equation as follows: