Time-fractional Problem Research Articles

Higher-order fractional equations and related time-changed pseudo-processes

We study Cauchy problems of fractional differential equations in both space and time variables by expressing the solution in terms of “stochastic composition” of the solutions to two simpler problems. These Cauchy sub-problems respectively concern the space and the time differential operator involved in the main equation. We provide some probabilistic and pseudo-probabilistic applications, where the solution can be interpreted as the pseudo-transition density of a time-changed pseudo-process. To extend our results to higher order time-fractional problems, we introduce stable pseudo-subordinators as well as their pseudo-inverses. Finally, we present our results in the case of more general differential operators and we interpret the results by means of a linear combination of pseudo-subordinators and their inverse processes.

A virtual element scheme for the time-fractional parabolic PDEs over distorted polygonal meshes

We extend the virtual element method to the two-dimensional time-fractional parabolic PDE, characterized by a fractional derivative of order α∈(0,1) in time. To illustrate the working of this fractional virtual element scheme, a numerical investigation of the following time-fractional problem over distorted polygonal meshes is conducted. cDtαu(z,t)−Δu=f(z,t)inz∈Ω,t∈(0,T],where Ω is a spatial domain, α is fractional order, and t is time variable. Our methodology is based on the fundamental technical component, fractional version of the Grunwald–Letnikov approximation. We prove the method’s well-posedness, that is the approximate solution’s existence and uniqueness. The fully discrete scheme inherently maintains stability and consistency by leveraging the discrete maximal regularity and the energy projection operator. The convergence in the L2-norm and H1-norm over distorted mesh configuration is validated by numerical results, underlining the practical effectiveness of the proposed method.

Analytical Scheme for Time Fractional Kawahara and Modified Kawahara Problems in Shallow Water Waves

The Kawahara equation exhibits signal dispersion across lines of transmission and the production of unstable waves from the water in the broad wavelength area. This article explores the computational analysis for the approximate series of time fractional Kawahara (TFK) and modified Kawahara (TFMK) problems. We utilize the Shehu homotopy transform method (SHTM), which combines the Shehu transform (ST) with the homotopy perturbation method (HPM). He’s polynomials using HPM effectively handle the nonlinear terms. The derivatives of fractional order are examined in the Caputo sense. The suggested methodology remains unaffected by any assumptions, restrictions, or hypotheses on variables that could potentially pervert the fractional problem. We present numerical findings via visual representations to indicate the usability and performance of fractional order derivatives for depicting water waves in long-wavelength regions. The significance of our proposed scheme is demonstrated by the consistency of analytical results that align with the exact solutions. These derived results demonstrate that SHTM is an effective and powerful scheme for examining the results in the representation of series for time-fractional problems.

Roundoff error problems in interpolation methods for time-fractional problems

Roundoff errors often disrupt interpolation methods for time-fractional equations, potentially causing suboptimal convergence or even failure. These issues primarily result from catastrophic cancellations. To address this, we introduce a novel framework for computing coefficients in standard and fast interpolation methods on nonuniform meshes. We propose δ-cancellation and associated threshold conditions to prevent such cancellations. If the thresholds aren't met, a Taylor expansion technique can be applied. Numerical experiments demonstrate our method's accuracy, on par with the Gauss–Kronrod quadrature, but significantly more efficient, allowing for extensive simulations with hundreds of thousands of time steps.

Existence and uniqueness solution analysis of time-fractional unstable nonlinear Schrödinger equation

The time-fractional unstable nonlinear Schrödinger (NLS) equations capture the time evolution of disturbances within media, tailored for describing phenomena in unstable media to help model and understand the intricate dynamics of systems prone to instability, the behavior of disturbances in complex and unstable environments and many more. In this study, we investigate the exact and analytical solutions of time-fractional unstable nonlinear Schrödinger (NLS) equations with the time-fractional β-time derivative, a powerful technique known as the sine-Gordon expansion (SGE) method is used. Non-locality behaviors that cannot be modeled using classical calculus are included in this equation. It discusses the relationships between waves of different frequencies or the same frequency but with distinct polarizations having a diverse set of key applications in nonlinear optics. We begin by using a fixed-point argument to prove the existence and uniqueness of solutions to the time-fractional problem under consideration. The partial differential equation of fractional order is transformed into an ordinary differential equation using the complex transformation. Numerous exact solutions are achieved with all the arbitrary parameters. The specific values of the obtained solutions' associated parameters are analyzed graphically with 3D plots as multiple soliton types of different wave solutions with the help of symbolic software. Understanding that the features of the solutions are far from intuitive because it is based on the parameter selection from the figures is difficult. The main objective of this paper is to investigate some fresh and further general solitary wave solutions to the suggested equation that properly explain the above-stated phenomena.

Numerical treatment for time fractional order phytoplankton-toxic phytoplankton-zooplankton system

<abstract><p>The study of time-fractional problems with derivatives in terms of Caputo is a recent area of study in biological models. In this article, fractional differential equations with phytoplankton-toxic phytoplankton-zooplankton (PTPZ) system were solved using the Laplace transform method (LTM), the Adomain decomposition method (ADM), and the differential transform method (DTM). This study demonstrates the good agreement between the results produced by using the specified computational techniques. The numerical results displayed as graphs demonstrate the accuracy of the computational methods. The approaches that have been established are thus quite relevant and suitable for solving nonlinear fractional models. Meanwhile, the impact of changing the fractional order of a time derivative and time $ t $ on populations of phytoplankton, toxic-phytoplankton, and zooplankton has been examined using graphical representations. Furthermore, the stability analysis of the LTM approach has been discussed.</p></abstract>

Study on fuzzy fractional European option pricing model with Mittag-Leffler kernel

This research paper presents an innovative approach for modeling and analyzing complex systems with uncertain data. Our strategy leverages fuzzy calculus and time-fractional differential equations to achieve this goal. Specifically, we propose the utilization of the fuzzy Atangana-Baleanu time-fractional derivative, which incorporates non-singular kernels for fuzzy functions. This derivative type is particularly suitable for qualitative analysis of fractional differential equations in fuzzy space. We establish the existence and uniqueness of solutions for fuzzy linear time-fractional problems based on this differentiability concept. Additionally, we introduce a numerical solution method, namely the fuzzy homotopy perturbation transform method (FHPTM), to solve these problems. To demonstrate the effectiveness and practical applicability of our approach, we provide concrete examples such as the fuzzy time-fractional Advection-Dispersion equation, the fuzzy time-fractional Diffusion equation, and the fuzzy time-fractional Black-Scholes European option pricing problem. These examples not only illustrate the solution steps involved but also showcase the potential of our method in addressing real-world problems. The outcomes of our research underscore the significance of considering fuzzy calculus and time-fractional differential equations when modeling and analyzing intricate systems with uncertain data.

Explicit Chebyshev–Galerkin scheme for the time-fractional diffusion equation

The time-fractional diffusion equation is applied to a wide range of practical applications. We suggest using a potent spectral approach to solve this equation. These techniques’ main objective is to efficiently solve the linear time-fractional problem by transforming it into a system of linear algebraic equations in the expansion coefficients, together with the problem’s initial and boundary conditions. The main advantage of our technique is that the resulting linear systems have special structures which facilitate their computational solution. The numerical methods are supported by a thorough convergence study for the suggested Chebyshev expansion. Some test problems are offered to demonstrate the suggested methods’ broad applicability and a high degree of accuracy.

An efficient computational approach for fractional-order model describing the water transport in unsaturated porous media

This paper focuses on the application of an efficient technique, namely, the fractional natural decomposition method (FNDM). The numerical solutions of the model containing the water transport in unsaturated porous media, called Richards equation, are extracted. This model is used to describe the non-locality behaviors which cannot be modeled under the framework of classical calculus. To demonstrate the effectiveness and efficiency of the scheme used, two cases with time-fractional problems are considered in detail. The numerical stimulation is presented with results accessible in the literature, and corresponding consequences are captured with different values of parameters of fractional order. The attained consequences confirm that the projected algorithm is easy to implement and very effective to examine the behavior of nonlinear models. The reliable algorithm applied in this paper can be used to generate easily computable solutions for the considered problems in the form of rapidly convergent series.

A Survey of the L1 Scheme in the Discretisation of Time-Fractional Problems

A Survey of the L1 Scheme in the Discretisation of Time-Fractional Problems

A spectral order method for solving the nonlinear fourth-order time-fractional problem

A spectral order method for solving the nonlinear fourth-order time-fractional problem

A fast numerical scheme for a variably distributed-order time-fractional diffusion equation and its analysis

We develop a fast numerical method for a variably distributed-order time-fractional diffusion equation modeling the anomalous diffusion with uncertainties in inhomogeneous medium. Different from the analysis techniques of the commonly-used schemes for time-fractional problems like L1 methods, error estimates are proved based on a novel discretization coefficient splitting. The proposed method has the same accuracy as traditional schemes, while only O(N3/2log⁡N) computations and O(Nlog⁡N) storage are required for generating and storing temporal discretization coefficients, where N refers to the number of time steps. We further design a corresponding fast divide and conquer algorithm to reduce the computational complexity of solving the linear system from O(MN2) of the time-stepping method to O(MNlog3⁡N) with M being the number of spatial nodes. Numerical experiments are presented to substantiate the theoretical results.

On Effects of a New Method for Fractional Initial Value Problems

The aim of this study is to analyze nonlinear Liouville-Caputo time-fractional problems by a new technique which is a combination of the iterative and ARA transform methods and is denoted by IAM. First, the ARA transform method and its inverse are utilized to get rid of time fractional derivative. Later, the iterative method is applied to establish the solution of the problem in infinite series form. The main advantages of this method are that it converges to analytic solution of the problem rapidly and implementation of method is easy. Finally, outcomes of the illustrative examples prove the efficiency and accuracy of the method.

Solutions for time-fractional coupled nonlinear Schrödinger equations arising in optical solitons

In this work, an efficient novel technique, namely, the q-homotopy analysis transform method (q-HATM) is applied to obtain analytical solutions for a system of time-fractional coupled nonlinear Schrödinger (TF-CNLS) equations with the time-fractional derivative taken in the Caputo sense. This system of equations incorporate nonlocality behaviors which cannot be modeled under the framework of classical calculus. With numerous important applications in nonlinear optics, it describes interactions between waves of different frequencies or the same frequency but belonging to different polarizations. We first establish existence and uniqueness of solutions for the considered time-fractional problem via a fixed point argument. To demonstrate the effectiveness and efficiency of the q−HATM, two cases each of two time-fractional problems are considered. One important feature of the q−HATM is that it provides reliable algorithms which can be used to generate easily computable solutions for the considered problems in the form of rapidly convergent series. Numerical simulation are provided to capture the behavior of the state variables for distinct values of the fractional order parameter. The results demonstrate that the general response expression obtained by the q−HATM contains the fractional order parameter which can be varied to obtain other responses. Particularly, as this parameter approaches unity, the responses obtained for the considered fractional equations approaches that of the corresponding classical equations.

On the fuzzy solutions of time-fractional problems

The main purpose of this paper is to obtain an analytical solution for the time-fractional fuzzy equation. To do this, the time-fractional equation is transformed into an algebraic equation using the fuzzy Laplace and Fourier transforms. The fractional derivatives are described in the Caputo gH-differentiability. In addition, to demonstrate the efficiency of the method some various examples are solved.

Finite difference/generalized Hermite spectral method for the distributed-order time-fractional reaction-diffusion equation on multi-dimensional unbounded domains

Distributed-order fractional differential equations, where the differential order is distributed over a range of values rather than being just a fixed value as it is in the classical differential equations, offer a powerful tool to describe multi-physics phenomena. In this article, we develop and analyze an efficient finite difference/generalized Hermite spectral method for the distributed-order time-fractional reaction-diffusion equation on one-, two-, and three-dimensional unbounded domains. Considering the Gauss-Legendre quadrature rule for the distributed integral term in temporal direction, we first approximate the original distributed-order time-fractional problem by the multi-term time-fractional differential equation. Then, we apply the L2-1σ formula for the discretization of the multi-term Caputo fractional derivatives. Moreover, we employ the generalized Hermite functions with scaling factor for the spectral approximation in space. The detailed implementations of the method are presented for one-, two-, and three-dimensional cases of the fractional problem. The stability and convergence of the method are strictly established, which shows that the proposed method is unconditionally stable and convergent with second-order accuracy in time. In addition, the optimal error estimate is derived for the space approximation. Finally, we perform numerical examples to support the theoretical claims.

(2004-5828) On the fuzzy solutions of time-fractional problems

The main purpose of this paper is to obtain an analytical solution for the time-fractional fuzzy equation. To do this, the time-fractional equation is transformed into an algebraic equation using the fuzzy Laplace and Fourier transforms. The fractional derivatives are described in the Caputo gH-differentiability. In addition, to demonstrate the efficiency of the method some various examples are solved.

A Meshless Local Radial Point Collocation Method for Simulating the Time-Fractional Convection-Diffusion Equations on Surfaces

The time-fractional problem is a class of important models to represent the real world. It is an open problem to study how the fractional operator acts on the surface. In this work, we present and analyze a meshless local radial point collocation method for numerically solving time-fractional convection-diffusion equations on closed surfaces embedded in [Formula: see text]. The second-order shifted Grünwald scheme is applied in time discretization. All computations use only extrinsic coordinates to avoid coordinate distortions and singularities. Moreover, the stability and convergence of the method are proven by the energy estimate. Numerical experiments are provided to support the convergence analysis and numerical performance of the proposed method.

Promoted residual power series technique with Laplace transform to solve some time-fractional problems arising in physics

Physical applications involving time-fractional derivatives are reflecting some memory characteristics. These inherited memories have been identified as a homotopy mapping of the fractional-solution into the integer-solution preserving its physical shapes. The aim of the current work is threefold. First, we present a new technique which is constructed by combining the Laplace transform tool with the residual power series method. Precisely, we provide the details of implementing the proposed method to treat time-fractional nonlinear problems. Second, we test the validity and the efficiency of the method on the temporal-fractional Newell–Whitehead–Segel model. Then, we implement this new methodology to study the temporal-fractional (1+1)-dimensional Burger’s equation and the Drinfeld–Sokolov–Wilson system. Further, for accuracy and reliability purposes, we compare our findings with other methods being used in the literature. Finally, we provide 2-D and 3-D graphical plots to support the impact of the fractional derivative acting on the behavior of the obtained profile solutions to the suggested models.

Closed-form solutions to the conformable space-time fractional simplified MCH equation and time fractional Phi-4 equation

The time-fractional problem is a class of important models to represent the real phenomena. We construct new solitary waves for the space-time fractional simplified modified Camassa-Holm (MCH) and the time fractional Phi-4 equations using the unified solver. The fractional derivatives are defined in the sense of the new conformable fractional derivative. The acquired solutions may be useful for various vital observations in nuclear and particle physics and fluid mechanics. The proposed unified solver is a sturdy mathematical tool for solving various classes of fractional partial differential equations in applied science. Moreover, the simulation of some selected solutions has been demonstrated with the aid of matlab software.