Time-fractional Equation Research Articles

Extended Separation Method of Semi‐Fixed Variables Together With Analytical Method for Solving Time Fractional Equation

ABSTRACTIt is well known that investigation of exact solutions on nonlinear fractional partial differential equations (PDEs) is a very difficult work. In this paper, the separation method of semi‐fixed variables is extended. Based on the original separation method, two kinds of new structures of the solutions in a hypothetical way are proposed. With the extended separation method of semi‐fixed variables and the mapping method of Riccati equation combined, a new approach for searching exact solutions on time‐fractional PDEs is introduced. To demonstrate the effect of the extended method, the time‐fractional porous medium equation, time‐fractional Hunter‐Saxton equation, and time‐fractional Fornberg‐Whitham equation are solved under the Riemann‐Liouville fractional differential operator. Different kinds of new exact solutions of the above three equations are obtained. In order to intuitively show the dynamic property of these exact solutions, the 3D‐graphs of some solutions are illustrated as examples. Compared to the previous method, more abundant results can be obtained on solving some complex nonlinear time‐fractional PDEs by use of this extended method.

A finite difference method on uniform meshes for solving the time-space fractional advection-diffusion equation

In order to investigate linear time and spatial fractional advection equations, we present a finite difference scheme (FDS) in this paper. The fractional Taylor series method for u at tj+1 and xi+1 is used to approximate the fractional derivatives. First, we construct our numerical scheme (NS) for the mathematical model. In the second part, we study the stability and convergence of our numerical scheme. Finally, the numerical simulations of the fractional advection equation, using the FDM, is plotted for several values of fractional parameters α and ν. It will be shown that the convergence is achieved properly which confirms the effectiveness of the proposed algorithm.

Asymptotic analysis for time fractional FitzHugh-Nagumo equations

Asymptotic analysis for time fractional FitzHugh-Nagumo equations

Explicit travelling wave solutions to the time fractional Phi-four equation and their applications in mathematical physics

In applied research, fractional calculus plays an important role for comprehending a wide range of intricate physical phenomena. One of the Klein-Gordon model’s peculiar case yields the Phi-four equation. Additionally, throughout the past few decades it has been utilized to explain the kink and anti-kink solitary waveform contacts that occur in biological systems and in the field of nuclear mechanics. In this current work, the key objective is to analyze the consequences of fractional variables on the soliton wave dynamic behavior in a nonlinear time-fractional Phi-four equation. Using the formulation of the conformable fractional derivative it illustrates some of the recovered solutions and analyze their dynamic behavior. The analytical solutions are drawn by using the extended direct algebraic and the Bernoulli Sub-ODE scheme. Various types of soliton solutions are proficiently expressed. Adjusting the specific values of fractional parameters allows to produce the periodic, kink, bell shape, anti-bell shape and W-shaped solitons. The impact of the conformable derivative on the precise solutions of the fractional Phi-four equation is demonstrated with a series of 2D, 3D and contour graphical representations.

Sub-pico-second chirped optical solitons in birefringent fibers for space–time fractional Kaup-Newell equation

The present work is devoted to investigate the chirped bright and dark optical solitons of fractional Kaup-Newell equation (KNE) in birefringent fibers. The study is carried out analytically by the traveling wave hypothesis with the conformable derivative which reduces the governing model to an ordinary differential equation (ODE). The obtained equation is handled with the aid of an exotic integration scheme that utilizes the Jacobi elliptic equation in the form of a first-order nonlinear ODE with three-degree terms. Taking the modulus of Jacobi elliptic function to unity, distinct types of bright and dark optical solitons are derived with their corresponding chirping. The fractional order derivative is noted to have a significant influence on the pulse propagation. Additionally, the nonlinearity amount causes also marked variations in the amplitude and width of solitons. The modulation instability of the KNE is reported by implementing the linear stability analysis which confirms that all solutions are stable. The revealed results can be capitalized in improving the relevant physical and engineering applications in the field of birefringent fiber.

Group Classification, Exact Solutions and Conservation Laws of (2+1)-dimensional Time Fractional Konopelchenko-Dubrovsky Equations

Group Classification, Exact Solutions and Conservation Laws of (2+1)-dimensional Time Fractional Konopelchenko-Dubrovsky Equations

A Note on Exact Results for Burgers-like Equations Involving Laguerre Derivatives

In this work, we consider some Burgers-like equations involving Laguerre derivatives and demonstrate that it is possible to construct specific exact solutions using separation of variables. We prove that a general scheme exists for constructing exact solutions for these Burgers-like equations and extending to more general cases, including nonlinear time-fractional equations. Exact solutions can also be obtained for KDV-like equations involving Laguerre derivatives. We finally consider a particular class of Burgers equations with variable coefficients whose solutions can be obtained similarly.

Determination of a time dependent source in semilinear pseudoparabolic equations with Caputo fractional derivative

Determination of a time dependent source in semilinear pseudoparabolic equations with Caputo fractional derivative

Application of the Atangana–Baleanu operator in Caputo sense for numerical solutions of the time-fractional Burgers–Fisher equation using finite difference approaches

Application of the Atangana–Baleanu operator in Caputo sense for numerical solutions of the time-fractional Burgers–Fisher equation using finite difference approaches

Modulation instability, and dynamical behavior of solitary wave solution of time M- fractional clannish random Walker's Parabolic equation via two analytic techniques

Modulation instability, and dynamical behavior of solitary wave solution of time M- fractional clannish random Walker's Parabolic equation via two analytic techniques

A new approach of generalized shifted Vieta-Fibonacci polynomials to solve nonlinear variable order time fractional Burgers-Huxley equations

Abstract In recent years, advancements in optimization techniques and the widespread availability of high-performance computing have made it increasingly feasible to study and develop approximation strategies for nonlinear models. This progress has empowered researchers to address more intricate and realistic challenges associated with these models. This paper introduces the application of a novel polynomial, the generalized shifted Vieta-Fibonacci polynomials (GSVFPs), in solving nonlinear variable order time fractional Burgers-Huxley equations. To mitigate storage and computational costs, new operational matrices (OMs) are devised. The proposed algorithm integrates GSVFPs, OMs, and Lagrange multipliers to achieve optimal approximations. Through convergence analysis and numerical examples, the effectiveness and accuracy of the proposed algorithm in solving these equations are demonstrated. The provided numerical illustrations further bolster this assertion.

The hyperbolic multi-term time fractional integro-differential equation with generalized Caputo derivative and error estimate in $$L_{p,\gamma ,\upsilon }$$ space

The hyperbolic multi-term time fractional integro-differential equation with generalized Caputo derivative and error estimate in $$L_{p,\gamma ,\upsilon }$$ space

Symmetry analysis of the time fractional potential-KdV equation

Symmetry analysis of the time fractional potential-KdV equation

A Robust and higher order numerical technique for a time-fractional equation with nonlocal condition

A Robust and higher order numerical technique for a time-fractional equation with nonlocal condition

The Influence of the Caputo Fractional Derivative on Time-Fractional Maxwell’s Equations of an Electromagnetic Infinite Body with a Cylindrical Cavity Under Four Different Thermoelastic Theorems

This paper introduces a new mathematical modeling of a thermoelastic and electromagnetic infinite body with a cylindrical cavity in the context of four different thermoelastic theorems; Green–Naghdi type-I, type-III, Lord–Shulman, and Moore–Gibson–Thompson. Due to the convergence of the four theories under study and the simplicity of putting them in a unified equation that includes these theories, the theories were studied together. The bunding plane of the cavity surface is subjected to ramp-type heat and is connected to a rigid foundation to stop the displacement. The novelty of this work is considering Maxwell’s time-fractional equations under the Caputo fractional derivative definition. Laplace transform techniques were utilized to obtain solutions by using a direct approach. The Laplace transform’s inversions were calculated using Tzou’s iteration method. The temperature increment, strain, displacement, stress, induced electric field, and induced magnetic field distributions were obtained numerically and represented in figures. The time-fractional parameter of Maxwell’s equations has a significant impact on all the mechanical studied functions and does not affect the thermal function. The time-fractional parameter of Maxwell’s equations works as a resistance to deformation, displacement, stress, and induced magnetic field distributions, while it acts as a catalyst to the induced electric field through the material.

Adaptive-coefficient finite difference frequency domain method for time fractional diffusive-viscous wave equation arising in geophysics

Adaptive-coefficient finite difference frequency domain method for time fractional diffusive-viscous wave equation arising in geophysics

Anomalous Random Flights and Time-Fractional Run-and-Tumble Equations

Random flights (also called run-and-tumble walks or transport processes) represent finite velocity random motions changing direction at any Poissonian time. These models in d-dimension, can be studied giving a general formulation of the problem valid at any spatial dimension. The aim of this paper is to extend this general analysis to time-fractional processes arising from a non-local generalization of the kinetic equations. The probabilistic interpretation of the solution of the time-fractional equations leads to a time-changed version of the original transport processes. The obtained results provide a clear picture of the role played by the time-fractional derivatives in this kind of random motions. They display an anomalous behavior and are useful to describe several complex systems arising in statistical physics and biology. In particular, we focus on the one-dimensional random flight, called telegraph process, studying the time-fractional version of the classical telegraph equation and providing a suitable interpretation of its stochastic solutions.

State-Space Approach to the Time-Fractional Maxwell’s Equations under Caputo Fractional Derivative of an Electromagnetic Half-Space under Four Different Thermoelastic Theorems

This paper introduces a new mathematical modelling method of a thermoelastic and electromagnetic half-space in the context of four different thermoelastic theorems: Green–Naghdi type-I, and type-III; Lord–Shulman; and Moore–Gibson–Thompson. The bunding plane of the half-space surface is subjected to ramp-type heat and traction-free. We consider that Maxwell’s time-fractional equations have been under Caputo’s fractional derivative definition, which is the novelty of this work. Laplace transform techniques are utilized to obtain solutions using the state-space approach. Laplace transform’s inversions were calculated using Tzou’s iteration method. The temperature increment, strain, displacement, stress, induced electric field, and induced magnetic field distributions were obtained numerically and are illustrated in figures. The time-fraction parameter of Maxwell’s equations had a major impact on all the studied functions. The time-fractional parameter of Maxwell’s equations worked as resistant to the changing of temperature, particle movement, and induced magnetic field, while it acted as a catalyst to the induced electric field through the material. Moreover, all the studied functions have different values in the context of the four studied theorems.

Some Grönwall Inequalities for a Class of Discretizations of Time Fractional Equations on Nonuniform Meshes

Some Grönwall Inequalities for a Class of Discretizations of Time Fractional Equations on Nonuniform Meshes

Group classification of time fractional Black-Scholes equation with time-dependent coefficients

Group classification of time fractional Black-Scholes equation with time-dependent coefficients