In this article, we investigate the fractional-order Fokker-Planck equations with the help of the Yang transform decomposition method (YTDM). The YTDM combines Yang transform, Adomian decomposition method, and Adomian polynomials into one method. In the Caputo sense, fractional derivatives of space and time are studied. The convergent series form solution demonstrates the method’s efficiency in resolving several types of fractional differential equations. Compared to other methods of finding approximate and exact solutions for nonlinear partial differential equations, this technique is more efficient and time-consuming.
We use a new integral transform approach to solve the fractional Harry Dym equation and fractional Rosenau-Hyman equation in this work. The Elzaki transform and the integral transformation are combined in the suggested method (ET). To handle two nonlinear problems, we first construct the Elzaki transforms of the Caputo fractional derivative (CFD) and Atangana-Baleanu fractional derivative (ABFD). The ultimate purpose of this study is to find an error analysis that demonstrates that our final result converges to the exact and approximate result. The convergent series form solution demonstrates the method’s efficiency in resolving several types of fractional differential equations. Furthermore, the solutions obtained in this study agree well with the exact solutions; thus, this strategy is powerful and efficient as an alternate way for obtaining approximate solutions to both linear and nonlinear fractional differential equations.
The current article discusses the new fuzzy iterative transform method, a hybrid methodology based on fuzzy logic and an iterative transformation technique. We demonstrate the consistency of our technique by employing the Caputo derivative under generalized Hukuhara differentiability to construct fractional fuzzy Klein-Gordon equations with the initial fuzzy condition. The series produced result was calculated and compared to the exact result’s recommended equations. Two problems were used to verify our method, with the results approximated in fuzzy form. The upper and lower half of the fuzzy results were approximated in each of the two examples using two distinct fractional orders between zero and one. Because it globalizes the dynamical behavior of the specified equation, it produces all forms of fuzzy results at any fractional order between 0 and 1. Since fuzzy numbers offer their results in a fuzzy form with lower and upper branches, the unknown amount also adds fuzziness. It is crucial to emphasize that the suggested fuzziness method is intended to demonstrate the efficiency and superiority of numerical solutions to nonlinear fractional fuzzy partial differential equations found in complex and physical structures.
The analytical behavior of fractional differential equations is often puzzling and difficult to predict under uncertainty. It is crucial to develop a robust, extensive, and extremely successful theory to address these problems. An application of fuzzy fractional differential equations can be found in applied mathematics and engineering. Using the iterative transform technique, the study determines the analytic solution of fractional fuzzy Emden-Fowler equation in the sense of the Caputo operator, which is applied to evaluate the physical model range in several scientific and engineering disciplines. The derived solutions to the fractional fuzzy Emden-Fowler equations are more generic and applicable to a broader range of problems. Through the translation of fractional fuzzy differential equations into equivalent crisp systems of fractional differential equations, we obtain a parametric description of the solutions. The graphical and numerical representation demonstrates the symmetry among the upper and lower fuzzy solution representations in their simplest form, which may aid in the comprehension of artificial intelligence, control system models, computer science, image processing, quantum optics, medical science, physics, measure theory, stochastic optimization theory, biology, mathematical finance, and other domains, as well as nonfinancial evaluation.
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