Fractional Differential Equations Solutions

Efficient numerical techniques for computing the Riesz fractional-order reaction-diffusion models arising in biology

Stable and functional solutions of the Klein-Fock-Gordon equation with nonlinear physical phenomena

Abstract The present article uses a modified <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mfenced close=")" open="("> <mml:mrow> <mml:mstyle displaystyle="false"> <mml:mfrac> <mml:mrow> <mml:mi>G</mml:mi> <mml:mo accent="false">′</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>G</mml:mi> </mml:mrow> </mml:mfrac> </mml:mstyle> </mml:mrow> </mml:mfenced> </mml:math> -expansion method and the generalized Kudryashov method on Klein-Fock-Gordon (KFG) equation and receives some stable and functional solutions. The obtained results are checked by putting backwards into the physical model and are very beneficial over various existing processes. The diverse variety of stable and functional outcomes such as kink-type shape, bright and dark lump shape, bright and dark singular kinky shape, periodic bright and dark lump shape, multiple bright and dark lump shape, the lump with rough wave shape, the rough wave shape and the kinky shape are taken. The above procedure could also be employed to get stable and functional solutions for other integral and fractional nonlinear models in physics, mathematics, and engineering.

Exploring the multistability, sensitivity, and wave profiles to the fractional Sharma–Tasso–Olver equation in the mathematical physics

In this work, we study the solitary wave profiles of the fractional-Sharma–Tasso–Olver equation, which is applicable to particle fission and fusion mechanisms in nuclear physics. In numerical and analytical theories, exact solitary wave solutions are of the uttermost importance for such equations. Improved analytical methods are essential for a deeper understanding of dynamics, despite their widespread implementation. In this study, we use the advanced analytical techniques known as generalized Arnous method, modified generalized Riccati equation mapping technique, and Riccati extended simple equation approach for securing a variety of solutions. This study marks a significant milestone by applying the prescribed techniques to the proposed equation using truncated M-fractional derivatives and providing a significant contribution to the existing literature. This equation is widely regarded as a model that illustrates the propagation of nonlinear dispersive waves in inhomogeneous media. Using the suitable wave transformation with the fractional-derivative, the governing equation is converted into an ordinary differential equation to get the required solutions. Various types of solutions, such as mixed, dark, singular, bright–dark, bright, complex, and combined solitons, are extracted. Moreover, another important aspect of this study is to discuss the multistability and sensitivity analysis of the studied model by the assistance of the Galilean transformation and perturbation term. The utilized methods have strong computing capacity, which helps them effectively handle the exact solutions with high accuracy in these systems. In addition, we depict 3D and 2D phase portrait graphs with appropriate parameters to illustrate the solution’s behavior.

On the exact and numerical solutions to a nonlinear model arising in mathematical biology

This study acquires the exact and numerical approximations of a reaction-convection-diffusion equation arising in mathematical bi- ology namely; Murry equation through its analytical solutions obtained by using a mathematical approach; the modified exp(-Ψ( η ))-expansion function method. We successfully obtained the kink-type and singular soliton solutions with the hyperbolic function structure to this equa- tion. We performed the numerical simulations (3D and 2D) of the obtained analytical solutions under suitable values of parameters. We obtained the approximate numerical and exact solutions to this equa- tion by utilizing the finite forward difference scheme by taking one of the obtained analytical solutions into consideration. We investigate the stability of the finite forward difference method with the equation through the Fourier-Von Neumann analysis. We present the L 2 and L ∞ error norms of the approximations. The numerical and exact approx- imations are compared and the comparison is supported by a graphic plot. All the computations and the graphics plots in this study are car- ried out with help of the Matlab and Wolfram Mathematica softwares. Finally, we submit a comprehensive conclusion to this study.

Investigation of the exact solutions via sub-equation neural network method to the nonlinear systems in fluid and nuclear physics

This paper aims to explore the nonlinear dynamics of the well-known nonlinear partial differential equations, namely, Estevez–Mansfield–Clarkson (EMC) and Sharma–Taso–Olver (STO) equations. The presented models have useful applications in various fields. The EMC equation clarifies the complex dynamics of waves in shallow water and fluid physics. In nuclear physics, the STO model is pertinent to particle fission and fusion processes. This work offers Riccati sub-equation neural networks to provide exact solutions for space–time partial differential equations. The proposed method incorporates the solutions of the Riccati problem into neural networks. Neural networks are multi-layer computer models with activation functions and weight functions that connect neurons across the input, hidden, and output layers. In this approach, each neuron in the first hidden layer is assigned to the solutions of the Riccati equation. Consequently, the new trial functions are established. The proposed method provides exact solutions to the studied models in the forms of bright, dark, singular, combined, and complex solitons. Moreover, generalized hyperbolic function solutions, trigonometric function solutions, and generalized rational solutions are also recovered. This study introduces innovative solutions as the proposed methodology is used in the neural network model. A variety of graphs have been sketched for the physical behavior of the obtained solutions. By establishing the dependability of the method used, this research’s outcomes could advance our grasp of nonlinear behavior in targeted systems.

Analytical Study of Variable‐Order Fractional Differential Equations With Initial and Terminal Antiperiodic Boundary Conditions

This study investigates the existence, uniqueness, and stability of solutions to Riemann–Liouville fractional differential equations with fractional variable‐order and antiperiodic boundary conditions. By employing the Banach fixed point theorem, we establish conditions for the uniqueness of solutions, while Schauder’s fixed point theorem is used to prove their existence in a Banach space. We further demonstrate Ulam–Hyers–Rassias stability, ensuring solution robustness against perturbations. The variable‐order framework enables modeling of complex systems with evolving memory, offering advantages over constant‐order models in applications such as physics and epidemiology. A concrete example illustrates the practical applicability of our results. This work provides a rigorous theoretical foundation, bridging pure mathematics with potential applications in science and engineering, and sets the stage for future numerical and applied studies.

Exploring the optical soliton and solitary wave solutions for the nonlinear Akbota equation via improved expansion approach

In the present research, we explored the various kinds of optical solitons and many other solitary wave solutions for the nonlinear Akbota equation by utilizing the symbolic computational simulation on the basis of the improved F-expansion approach. The nonlinear Akbota equation has applications in physics and engineering. The examined solitary wave and soliton solutions have interesting physical structures, including anti-kink wave solitons, bright solitons, kink wave solitons, dark solitons, periodic wave solitons, peakon bright solitons, peakon dark solitons, mixed bright–dark periodic solitons, mixed solitons in bright–dark form, and solitary wave structures. The newly extracted soliton solutions in this study shed light on the fact that the utilized approach is more efficient, concise, powerful, effective, straightforward, and simple, and we can also utilize it for other higher order nonlinear complex models. The extracted solutions will be helpful to understand the nonlinear phenomena in various areas of nonlinear sciences and engineering, including quantum physics, laser optics, nonlinear optics, optical fibers, ocean engineering, and electronic engineering. The physical interpretation of the extracted solutions is visualized in two-dimensional, three-dimensional, and contour graphics based on numerical simulation by using the computer software Mathematica. The presented research will be helpful for further investigation of analytical solitary wave and soliton solutions to the complex, higher order nonlinear evolution equations.