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In this work, we suggest a new method for solving linear multi-term time-fractional wave-diffusion equations, which is named the modified fractional reduced differential transform method (m-FRDTM). The importance of this technique is that it suggests a solution for a multi-term time-fractional equation. Very few techniques have been proposed to solve this type of equation, as will be shown in this paper. To show the effectiveness and efficiency of this proposed method, we introduce two different applications in two-term fractional differential equations. The three-dimensional and two-dimensional plots for different values of the fractional derivative are depicted to compare our results with the exact solutions. [ABSTRACT FROM AUTHOR]

Various scholars have lately employed a wide range of strategies to resolve specific types of symmetrical fractional differential equations. This paper introduces a new implicit finite difference method with variable-order time-fractional Caputo derivative to solve semi-linear initial boundary value problems. Despite its extensive use in other areas, fractional calculus has only recently been applied to physics. This paper aims to find a solution for the fractional diffusion equation using an implicit finite difference scheme, and the results are displayed graphically using MATLAB and the Fourier technique to assess stability. The findings show the unconditional stability of the implicit time-fractional finite difference method. This method employs a variable-order fractional derivative of time, enabling greater flexibility and the ability to tackle more complicated problems. [ABSTRACT FROM AUTHOR]

The axisymmetric time-fractional diffusion equation with mass absorption is studied in a circle under the time-harmonic Dirichlet boundary condition. The Caputo derivative of the order 0 < α ≤ 2 is used. The investigated equation can be considered as the time-fractional generalization of the bioheat equation and the Klein–Gordon equation. Different formulations of the problem for integer values of the time-derivatives α = 1 and α = 2 are also discussed. The integral transform technique is employed. The outcomes of numerical calculations are illustrated graphically for different values of the parameters. [ABSTRACT FROM AUTHOR]

This paper is concerned with numerical approach for solving space fractional diffusion equation using shifted Gegenbauer polynomials, where the fractional derivatives are expressed in Caputo sense. The properties of Gegenbauer polynomials are exploited to reduce space fractional diffusion equation to a system of ordinary differential equations, that are then solved using finite difference method. Some selected numerical simulations of space fractional diffusion equations are presented and the results are compared with the exact solution, also with the results obtained via other methods in the literature. The comparison reveals that the proposed method is reliable, effective and accurate. All the computations were carried out using Matlab package. [ABSTRACT FROM AUTHOR]

In this article, we propose a method to solve the time‐fractional hyperbolic heat equation. We first formulate a boundary value problem for the standard hyperbolic heat equation in a finite domain and provide an analytical solution by means of separation of variables and Fourier series. Then, we consider the same boundary value problem for the fractional hyperbolic heat equation. The fractional problem is solved using three different definitions of the fractional derivative: the Caputo fractional derivative and two recently defined alternative versions of this derivative, the Caputo–Fabrizio and the Atangana–Baleanu. A closed form of the solution is provided for each case. Finally, we compare the solutions of the fractional and the standard problem and show numerically that the solution of the standard hyperbolic heat equation can be retrieved from the solution of the fractional equation in the limit γ→2, where γ represents the exponent of the fractional derivative. [ABSTRACT FROM AUTHOR]

The linear non-polynomial spline is used here to solve the fractional partial differential equation (FPDE). The fractional derivatives are described in the Caputo sense. The tensor products are given for extending the one-dimensional linear nonpolynomial spline S1 to a two-dimensional spline S1 ⊗ S2 to solve the heat equation. In this paper, the convergence theorem of the method used to the exact solution is proved and the numerical examples show the validity of the method. All computations are implemented by Mathcad15. [ABSTRACT FROM AUTHOR]

We examine the transition region between two solids which state differs from the state of contacting media. Small thickness of the intermediate region allows us to reduce a three-dimensional problem to a two-dimensional one for a median surface endowed with equivalent physical properties. In the present paper, we consider the generalized boundary conditions of nonperfect thermal contact for the time-fractional heat conduction equation with the Caputo derivative and solve the problem for a composite medium consisting of two semi-infinite regions. Numerical results are illustrated graphically. [ABSTRACT FROM AUTHOR]

The time-fractional heat conduction equation with the Caputo derivative is solved for an infinite plane with an external half-infinite crack which surfaces are exposed to the heat flux loading. The solution is obtained using the integral transform technique and is expressed in terms of the Mittag-Leffler function. The stress intensity factor is calculated for different values of the order of fractional derivative. Numerical results are illustrated graphically. [ABSTRACT FROM AUTHOR]

We compare the model of heat transfer proposed by Cattaneo, Maxwell, and Vernotte with another one, formulated in terms of fractional differential equations, in one and two dimensions. These are only some of the numerous models that have been proposed in the literature over many decades to model heat transport and possibly heat waves, in place of the classical heat equation due to Fourier. These models are characterized by sound as well as by critical properties. In particular, we found that the Cattaneo model does not exhibit necessarily oscillations or negative values of the (absolute) temperature when the relaxation parameter, $\tau$, drops below some value. On the other hand, the fractional derivative model may be affected by oscillations, depending on the specific initial profile. We also estimate the error made when the Cattaneo equation is adopted in place of the heat equation, and show that the approximation error is of order $\tau$. Moreover, the solution of the Cattaneo equation converges uniformly to that of the heat equation as $\tau \to 0+$ in the full closed time interval $[0,T]$ (for any given $T > 0$), while this does not occur for the time derivative, and the higher-order time derivatives blow up. [ABSTRACT FROM AUTHOR]

We consider the initial value problem of the time fractional diffusion equation on the whole line and the fractional derivative is described in Caputo sense. A fully discrete Hermite pseudospectral approximation scheme is structured basing Hermite-Gauss points in space and finite difference in time. Unconditionally stability and convergence are proved. Numerical experiments are presented and the results conform to our theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, a new approach for solving space fractional order diffusion equations is proposed. The fractional derivative in this problem is in the Caputo sense. This approach is based on shifted Chebyshev polynomials of the fourth kind with the collocation method. The finite difference method is used to reduce the equations obtained by our approach for a system of algebraic equations that can be efficiently solved. Numerical results obtained with our approach are presented and compared with the results obtained by other numerical methods. The numerical results show the efficiency of the proposed approach. [ABSTRACT FROM AUTHOR]

In this paper, we present an analysis and modeling of the electrical diffusion equation using the fractional calculus approach. This alternative representation for the current density is expressed in terms of the Caputo derivatives, the order for the space domain is and for the time domain is . We present solutions for the full fractional equation involving space and time fractional derivatives using numerical methods based on Fourier variable separation. The case with spatial fractional derivatives leads to Levy flight type phenomena, while the time fractional equation is related to sub- or super diffusion. We show that the mathematical concept of fractional derivatives can be useful to understand the behavior of semiconductors, the design of solar panels, electrochemical phenomena and the description of anomalous complex processes. [ABSTRACT FROM AUTHOR]

• The time-fractional telegraph equation with moving time-harmonic source is considered on a real line. • Two characteristic versions of this equation: the "wave-type" with the second and Caputo fractional time-derivatives as well as the "heat-type" with the first and Caputo fractional time-derivatives are investigated. • The solution to the "wave-type" equation contains wave fronts and describes the Doppler effect contrary to the solution for the "heat-type" equation. • For the time-fractional telegraph equation it is impossible to consider the quasi-steady-state corresponding to the solution being a product of a function of the spatial coordinate and the time-harmonic term. • The derived solutions can be successfully used when the source term can be expanded into a Fourier series. The time-fractional telegraph equation with moving time-harmonic source is considered on a real line. We investigate two characteristic versions of this equation: the "wave-type" with the second and Caputo fractional time-derivatives as well as the "heat-type" with the first and Caputo fractional time-derivatives. In both cases the order of fractional derivative 1 < α < 2. For the time-fractional telegraph equation it is impossible to consider the quasi-steady-state corresponding to the solution being a product of a function of the spatial coordinate and the time-harmonic term. The considered problem is solved using the integral transforms technique. The solution to the "wave-type" equation contains wave fronts and describes the Doppler effect contrary to the solution for the "heat-type" equation. Numerical results are illustrated graphically for different values of nondimensional parameters. [ABSTRACT FROM AUTHOR]

In a real solid there are different types of defects. During sudden cooling, near cracks, there can appear high thermal stresses. In this paper, the time-fractional heat conduction equation is studied in an infinite space with an external circular crack with the interior radius R in the case of axial symmetry. The surfaces of a crack are exposed to the constant heat flux loading in a circular ring R < r < ρ . The stress intensity factor is calculated as a function of the order of time-derivative, time, and the size of a circular ring and is presented graphically. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we study the asymptotics of the solutions to the anomalous diffusion equations. The fractional anomalous diffusion equations are obtained from the existing anomalous diffusion and typical diffusion equations by replacing the first-order time derivative with fractional derivatives of order for the sub-diffusion and for the super-diffusion respectively. In most situations, fractional derivatives mean Riemann–Liouville derivative or Caputo derivative. In this paper, we use these two kinds of fractional derivatives. Using Laplace transform and Fourier transform, we obtain the asymptotics estimates of solutions to the anomalous diffusion equations. [Copyright &y& Elsevier]

The diffusion equations with the local and the nonlocal fractional derivatives have been used to describe the flow through disorder media. Recently, the variational iteration method is successfully developed to find approximate solutions of the two kinds of fractional differential equations. This study reveals the new development of the method and compares the applications in two types of fractional diffusions. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, a note on the finite element method for the space-fractional advection diffusion equation with non-homogeneous initial-boundary condition is given, where the fractional derivative is in the sense of Caputo. The error estimate is derived, and the numerical results presented support the theoretical results. [Copyright &y& Elsevier]

In this study, exact and approximate solutions of higher-dimensional time-fractional diffusion equations were obtained using a relatively new method, the fractional reduced differential transform method (FRDTM). The exact solutions can be found with the benefit of a special function, and we applied Caputo fractional derivatives in this method. The numerical results and graphical representations specified that the proposed method is very effective for solving fractional diffusion equations in higher dimensions. [ABSTRACT FROM AUTHOR]

• Numerical analysis of fractional neutron diffusion equation. • Anomalous diffusion behavior with step, ramp and sinusoidal reactivity effects. • Application of L1 approximation technique for discretization of time derivative. • Application of collocation method based on QBS basis function for discretization of space derivative. • The numerical experiments show the accuracy and efficiency of the method. In this paper, we construct an efficient numerical technique for solving a fractional neutron diffusion equation with step, ramp and sinusoidal reactivity effects combined with one group of delayed neutron precursor concentration equation. This problem describes neutron transport in a nuclear reactor. We use L 1 approximation technique for discretization of time derivative and a collocation method based on quintic B-spline (QBS) basis function for discretization of space derivative. Some numerical experiments are performed to show the accuracy and efficiency of the method. The effects of order of fractional derivative, relaxation time and radioactive decay constant on the neutron flux profile are examined. It is shown that the suggested method is of order O (k 1 + h 4) convergence, where k and h represent the step size for time and space, respectively. The numerical result of fractional model is compared with that of classic one. The CPU time is provided to show the computational efficiency of the method. [ABSTRACT FROM AUTHOR]

The heat conduction equations with Caputo fractional derivative are considered in two joint half-planes under the conditions of perfect thermal contact. The fundamental solution to the Cauchy problem as well as the fundamental solution to the source problem are examined. The Fourier and Laplace transforms are employed. The Fourier transforms are inverted analytically, whereas the Laplace transform is inverted numerically using the Gaver–Stehfest method. We give a graphical representation of the numerical results. [ABSTRACT FROM AUTHOR]

The time-fractional heat conduction equation follows from the law of conservation of energy and the corresponding time-nonlocal extension of the Fourier law with the "long-tail" power kernel. The time-fractional heat conduction equation with the Caputo derivative is solved for an infinite plane with two external half-infinite slits with the prescribed heat flux across their surfaces. The integral transform technique is used. The solution is obtained in the form of integrals with integrand being the Mittag–Leffler function. A graphical representation of numerical results is given. [ABSTRACT FROM AUTHOR]

The time-fractional diffusion equation with mass absorption in a sphere is considered under harmonic impact on the surface of a sphere. The Caputo time-fractional derivative is used. The Laplace transform with respect to time and the finite sin-Fourier transform with respect to the spatial coordinate are employed. A graphical representation of the obtained analytical solution for different sets of the parameters including the order of fractional derivative is given. [ABSTRACT FROM AUTHOR]