Time-fractional Equation Research Articles

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

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

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.

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.

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

The diffusive-viscous wave (DVW) equation is a widely used model to describe frequency dependent attenuation of seismic wave in fluid-saturated porous medium. In this paper taking power law frequency dependent attenuation into account, we first introduce a modified DVW equation (time fractional DVW equation) in which the first order temporal derivative of viscous term is replaced with a fractional order temporal derivative. In consideration of that most of the existing numerical simulations for seismic wave equations are based on time domain methods and truncation with some specific boundary conditions, we incorporate the absorbing boundary condition as complex-frequency-shifted (CFS) perfectly matched layer (PML) into the time fractional DVW equation, and then develop an adaptive-coefficient (AC) finite difference frequency domain (FDFD) method for numerical simulation. The corresponding analytical solution for homogeneous time fractional DVW equation is provided for model validation, and the effectiveness of the developed AC FDFD method is verified by some numerical examples including the homogeneous model and the layered model. Numerical results show that AC FDFD method is more accurate than the traditional 2nd-order FDFD method for numerical modeling of time fractional DVW equation with CFS PML absorbing boundary condition, while requiring similar computational costs.

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

Diverse solitary wave structures to the conformable time fractional equations through two powerful methods

Diverse solitary wave structures to the conformable time fractional equations through two powerful methods

Lie symmetry analysis of (2+1)-dimensional time fractional Kadomtsev-Petviashvili equation

In this paper, Lie symmetry analysis method is applied to the (2+1)-dimensional time fractional Kadomtsev-Petviashvili (KP) equation with the mixed derivative of Riemann-Liouville time-fractional derivative and integer-order $x$-derivative. We obtained all the Lie symmetries admitted by the KP equation and used them to reduce the (2+1)-dimensional fractional partial differential equation with Riemann-Liouville fractional derivative to some (1+1)-dimensional fractional partial differential equations with Erd\'{e}lyi-Kober fractional derivative or Riemann-Liouville fractional derivative, thereby getting some exact solutions of the reduced equations. In addition, the new conservation theorem and the generalization of Noether operators are developed to construct the conservation laws for the equation studied.

Asymptotic expansion method with respect to a small parameter for fractional differential equations with Riemann–Liouville derivate

In this paper, we proposed an asymptotic approach with respect to a small parameter for fractional differential equations, with the small parameter linked to the fractional order derivative. This approach allows the splitting of the field variable and consequently the Riemann–Liouville Integral as the sum of two contributions. One describes the unperturbed state and the other describes the behavior of the model in the perturbed state due to the presence of the fractional derivative. This approach allows mapping time-fractional differential equations into a system of two coupled partial differential equations with integer order. By solving these partial differential equations we are able to obtain solutions of the assigned fractional differential equations. To obtain solutions we determine the approximate Lie symmetries through which the system of two coupled partial differential equations is reduced to a system of two coupled ordinary differential equations. As an application, we consider the time fractional diffusion–reaction equation with a linear source term; we test the proposed procedure by comparing the profiles of the obtained approximate solutions with solutions found in the fractional context for values of the fractional parameter tending to the integer value.

An inverse problem of reconstructing the unknown coefficient in a third order time fractional pseudoparabolic equation

In this paper, we have considered the problem of reconstructing the time dependent potential term for the third order time fractional pseudoparabolic equation from an additional data at the left boundary of the space interval. This is very challenging and interesting inverse problem with many important applications in various fields of engineering, mechanics and physics. The existence of unique solution to the problem has been discussed by means of the contraction principle on a small time interval and the unique solvability theorem is proved. The stability results for the inverse problem have also been presented. However, since the governing equation is yet ill-posed (very slight errors in the additional input may cause relatively significant errors in the output potential), the regularization of the solution is needed. Therefore, to get a stable solution, a regularized objective function is to be minimized for retrieval of the unknown coefficient of the potential term. The proposed problem is discretized using the cubic B-spline (CB-spline) collocation technique and has been reshaped as a non-linear least-squares optimization of the Tikhonov regularization function. The stability analysis of the direct numerical scheme has also been presented. The MATLAB subroutine $lsqnonlin$ tool has been used to expedite the numerical computations. Both perturbed data and analytical are inverted and the numerical outcomes for two benchmark test examples are reported and discussed.

Investigation of the wave solutions of two space–time fractional equations in physics

In this study, two fractional differential equations describing the characteristics of three-dimensional nonlinear ion-acoustic waves (IAWs) in plasmas are considered with the conformable fractional derivative (CFD). 3+1-dimensional space–time fractional Jimbo–Miwa (JM) equation and 3+1-dimensional space–time fractional modified KdV–Zakharov–Kuznetsov (mKdV-ZK) equation are reduced into ordinary differential equations and an investigation into the exact solutions is carried out through the modified exponential function method (MEFM). These exact wave solutions are beneficial in interpreting the interior structure and characteristics of physical phenomena modeled by the aforementioned nonlinear evolution equations (NLEEs) arising in magnetized plasma. Various analytical methods were applied to such equations. Therefore, it is vital to determine the proper technique for the present problem. The proposed scheme has demonstrated a different spectrum of wave structures. The portrait of the exact solutions permits the exploration of plasmas, plasma models, and optics. The software system Mathematica is employed and the rational, exponential, trigonometric, and hyperbolic function solutions are acquired. The graphical results are presented for the physical nature of the solutions. The results are compared with the integer-order models.

Two Novel Difference Schemes for the One-Dimensional Multi-Term Time Fractional Oldroyd-B Equation

Two Novel Difference Schemes for the One-Dimensional Multi-Term Time Fractional Oldroyd-B Equation

An Efficient IMEX Compact Scheme for the Coupled Time Fractional Integro-Differential Equations Arising from Option Pricing with Jumps

An Efficient IMEX Compact Scheme for the Coupled Time Fractional Integro-Differential Equations Arising from Option Pricing with Jumps

Laplace-fPINNs: Laplace-Based Fractional Physics-Informed Neural Networks for Solving Forward and Inverse Problems of a Time Fractional Equation

Laplace-fPINNs: Laplace-Based Fractional Physics-Informed Neural Networks for Solving Forward and Inverse Problems of a Time Fractional Equation

On the construction of various soliton solutions of two space-time fractional nonlinear models

Abstract In this article, we investigate a couple of nonlinear fractional models of eminent interests subsequently the conformable derivative sense is used to designate the fractional order derivatives. The given structures are transformed into nonlinear ordinary differential equations of integer order, and the extended simple equation technique is then employed to solve the resulting equations. Initially, the nonlinear space time fractional Klein–Gordon equation is considered emerging from quantum and classical relativistic mechanics, which have application in plasma physics, dispersive wave phenomena, quantum field theory, and optical fibres. Later, the (2 + 1)-dimensional time fractional Zoomeron equation is analysed which is convenient to explore the innovative phenomena related to boomerons and trappons. As a result, various new soliton solutions are successfully established. The reported results offer a key implementation for analysing the soliton solutions of nonlinear fractional models which are extremely encouraging arising in the recent era of science and engineering. The 3D simulations have been carried out to demonstrate dynamics of the various soliton solutions for a given set of parameters.

NEW WAVE SOLUTIONS OF TIME FRACTIONAL CHAFEE-INFANTE EQUATION WITH BETA DERIVATIVE

In this article, we discuss the exact solutions forthe Chafee-Infante equation involving beta fractional derivative. Beta fractional derivative which is a local derivative, is a modification of conformable fractional derivative. Using the Modified Kudryashov Method, we obtain the general solution of the time fractional Chafee-Infante equation with the help of Wolfram Mathematica. We use chain rule and wave transform to convert the equation into integer order nonlinear ordinary differential equation. Hence, we don’t need any discretization, normalization, or reduction. Moreover, 3D graphical representations are given. With the help of these representations, we can have an idea on the physical and geometrical behavior of the solutions.