Time Fractional Differential Equations Research Articles

Galerkin approximation for multi-term time-fractional differential equations

Fractional differential equations (FDEs) are utilized as a precise model for describing a wide range of biological and physical processes, benefiting from the inherent symmetry feature in natural phenomena. The introduction of multi-term time-fractional order equations serves to tackle intricate phenomena spanning various physical domains. This study solves multi-term time-fractional DEs using the quadratic B-spline Galerkin method. In Galerkin approach, the quadratic B-spline is utilized as both a test and a basis function. The fractional portion of time is evaluated using the Caputo definition. In addition, the Gauss quadrature formula is used to evaluate the integration of complex function which enhance the accuracy. Lax-Richtmyer's stability criterion is employed to evaluate the stability of the proposed scheme. The E2 and E∞ norms are computed and displayed in the tables to illustrate the robustness and efficiency of the method. We compute simulation for various nodal points as well as various mesh sizes to analyze the solution behavior. The obtained results are compared with the exact and available results, indicating that the proposed scheme is more efficient. The graphical solution of each example is also demonstrated which verifies that exact and approximate solutions graphs closed with each other.

Numerical method for second order singularly perturbed delay differential equations with fractional order in time via fitted computational method

The numerical solution of time fractional parabolic differential equations with singular perturbations and delay is the subject of this article. An arbitrarily small perturbation parameter ɛ(0<ɛ<<1) is multiplied the highest order derivative term. The problem’s solution shows an exponential boundary layer on the right side of the spatial domain for small values of ɛ. The solution and its derivatives are discussed together with their properties and bounds. The Crank–Nicolson approach for time discretization and the fitted operator non-standard finite difference method for space discretization are used to solve the time fractional singularly perturbed time delay problem under consideration. The comparison principle and solution bound are used to examine and analyze the scheme’s stability. The scheme’s uniform convergence is examined and demonstrated. The proposed scheme has a uniform convergence order of Mx−1+(Δt)2−γ. Two numerical illustrations for various values of ɛ are taken into consideration to validate the theoretical analysis of the scheme.The constructed scheme provides a better order of convergence and more accurate results than methods found in the literature.

Analyzing the impact of time-fractional models on chemotherapy's effect on cancer cells

In this study, we employ the Laplace Variational Iterational Method (LVIM) as a sophisticated mathematical tool to investigate into the complex dynamics of cancer cells under the influence of chemotherapy. The LVIM, a method combining Laplace transformations and variational iteration techniques, is specifically adapted to address a system of time fractional differential equations (FDEs) that characterizes the temporal behavior of cancer cells. To enhance the efficacy of our approach, we introduce a semi-analytic version of LVIM, which proves to be a powerful and versatile tool for solving mathematical problems involving fractional derivatives. The focus of our analysis centers on elucidating the impact of chemotherapy, with a particular emphasis on drug diffusion within cancer cells and the fractality of DNA walks. Through numerical exploration encompassing varying fractional order derivatives, our study exposes nonlinear behaviors that remain secret in systems featuring only integer order derivatives. Notably, the methodology we propose is not only applicable to the specific cases examined in this research but also exhibits broad versatility, making it suitable for exploring the effects of different drugs and types of cancers. This research contributes valuable insights into the dynamics of cancer cells, aiding in the understanding of the implications for therapeutic strategies in the context of cancer treatment.

Monotone iterative technique for multi-term time fractional measure differential equations

Monotone iterative technique for multi-term time fractional measure differential equations

Wave solutions in (3 + 1)-dimensional generalized fractional mKdV-ZK equation utilizing jacobi elliptic functions

In this research, we investigate the effects of fractional order on the (3 + 1)-dimensional generalized space-time fractional modified KdV-Zakharov-Kuznetsov (mKdV-ZK) equation. We approach the problem by utilizing the conformable fractional derivative. By reducing the mKdV-ZK equation to an integer order nonlinear ordinary differential equation, we apply the Jacobi elliptic function method to find exact solutions. These solutions are specifically tailored for the fractional order of the (3 + 1)-dimensional generalized mKdV-ZK equation, encompassing solitary waves, shock waves, and periodic waves. We also compare these exact solutions with fractional solutions to gain further insights. Notably, our approach demonstrates the feasibility of solving nonlinear time-fractional differential equations with conformable derivatives. Several diagrams have been included to visually depict the behavior of the solutions under fractional order when certain special parameter values are employed.

Nonlinear fractional-order differential equations: New closed-form traveling-wave solutions

Abstract The fractional-order differential equations (FO-DEs) faithfully capture both physical and biological phenomena making them useful for describing nature. This work presents the stable and more effective closed-form traveling-wave solutions for the well-known nonlinear space–time fractional-order Burgers equation and Lonngren-wave equation with additional terms using the exp ( − Φ ( ξ ) ) (-\Phi (\xi )) expansion method. The main advantage of this method over other methods is that it provides more accuracy of the FO-DEs with less computational work. The fractional-order derivative operator is the Caputo sense. The transformation is used to reduce the space–time fractional differential equations (FDEs) into a standard ordinary differential equation. By putting the suggested strategy into practice, the new closed-form traveling-wave solutions for various values of parameters were obtained. The generated 3D graphical soliton wave solutions demonstrate the superiority and simplicity of the suggested method for the nonlinear space–time FDEs.

Sharp analysis of [formula omitted] method on graded mesh for time fractional parabolic differential equation

In this paper, we study the L1−2 method for solving numerically a class of time fractional parabolic differential equations on a graded mesh. Based on the nonsmooth regularity assumptions, the stability and convergence of the proposed numerical method are investigated. Using rigorous and sharp theoretical analysis, we obtain the optimal graded parameter r=(3−α)/(1+α) of the graded mesh. Numerical experiments are given to demonstrate that even if the α is small, the graded mesh based on the optimal graded parameter is still able to recover the optimal convergence order of the L1−2 method.

The kink solitary wave and numerical solutions for conformable non-linear space–time fractional differential equations

This paper is about the conformable non-linear space–time fractional Telegraph and the non-linear space–time fractional Kolmogorov–Petrovskii–Piskunov (KPP) equations. To construct some real and analytical wave solutions of both equations the simplest equation method is applied. As a consequence, a series of analytical wave solutions are obtained as well as verify these solutions by using Mathematica software. These solutions denote that this method is useful and straightforward. We show some graphs in 2-D and 3-D of our solution to illustrate the behaviour of waves. Finally, we introduce the numerical solutions for the conformable non-linear space–time fractional Telegraph and the non-linear space–time fractional KPP equations using the finite difference method and make comparisons with analytical solutions.

On the $S$-asymptotically $\omega$-periodic mild solutions for multi-term time fractional measure differential equations

In this paper, based on regulated functions and fixed point theorem, a class of nonlocal problem of multi-term time-fractional measure differential equations involving nonlocal conditions in Banach spaces. Firstly, we introduce the concept of $S$-asymptotically $\omega$-periodic mild solution, on the premise of by utilizing $(\beta,\gamma_k)$-resolvent family and measure functional (Henstock-Lebesgue-Stieltjes integral), the existence of $S$ -asymptotically $\omega$ periodic mild solutions for the mentioned system are obtained. Finally, as the application of abstract results, the existence $S$-asymptotically $\omega$-periodic mild solution for a classes of measure driven differential equation are discussed.

Numerical Solution of Time-fractional BBMB Equation via LRPSM

The time-fractional Benjamin-Bona-Mahony-Burger (BBMB) differential equation plays an important role in explaining the unidirectional propagation of long waves in definite nonlinear differential systems. This work presents an analytical numerical solution of considered equation by Laplace transform with residual power series method (LRPSM) which is a generalized Taylor series together with Laplace transform and the residual error function. Using the proposed approach, the series solution of this equation is obtained. The analytical numerical solution shows that the LRPSM is a reliable and powerful method for solving the time fractional differential equations (FDEs) with less number of terms. The obtained results of numerical solutions are also compared with the exact solution and presented as absolute errors at different time levels.

A Novel Approach for Solving Nonlinear Time Fractional Fisher Partial Differential Equations

This study focuses on solving non-linear time fractional Fisher partial differential equations using analytical series solutions. The authors consider the Caputo fractional derivative in their formulas, which adds accuracy to the results. They introduce a novel method called LRPS which proves to be a powerful tool for obtaining precise analytical and numerical solutions for these equations. The LRPS method emphasizes precision, effectiveness, and practical application, making it suitable for various fields such as physics, engineering, and finance. Due to the importance of accuracy, effectiveness and method of application in this approach, it is highlighted that accurate solutions can be obtained when there is a pattern in the parts of the series, while approximate estimates are provided otherwise. The LRPS method is presented as a powerful technique specifically designed for solving nonlinear fractional Fisher partial differential equations.

On the completely positive kernels for nonuniform meshes

The complete positivity for convolutional kernels is an important property for the positivity property and asymptotic behaviors of Volterra equations. We investigate the discrete analogue of the complete positivity properties, especially for convolutional kernels on nonuniform meshes. Through an operation which we call pseudo-convolution, we introduce the complete positivity property for discrete kernels on nonuniform meshes and establish the criterion for the complete positivity. We then apply our theory to the L1 discretization of time fractional differential equations on nonuniform meshes.

PMNN: Physical model-driven neural network for solving time-fractional differential equations

In this paper, an innovative Physical Model-driven Neural Network (PMNN) method is proposed to solve time-fractional differential equations. It establishes a temporal iteration scheme based on physical model-driven neural networks which effectively combines deep neural networks (DNNs) with interpolation approximation of fractional derivatives. Specifically, once the fractional differential operator is discretized, DNNs are employed as a bridge to integrate interpolation approximation techniques with differential equations. On the basis of this integration, we construct a neural-based iteration scheme. Subsequently, by training DNNs to learn this temporal iteration scheme, approximate solutions to the differential equations can be obtained. The proposed method aims to preserve the intrinsic physical information within the equations as far as possible. It fully utilizes the powerful fitting capability of neural networks while maintaining the efficiency of the difference schemes for fractional differential equations. The experimental results show that the PMNN maintains precision comparable to traditional methods while exhibiting superior computational efficiency. This implies the potential of PMNN in addressing large-scale problems. Moreover, when considering both error and convergence rate, PMNN consistently outperforms fPINN. Additionally, the performance of PMNN on L2−1σ surpasses that on L1 in an overall comparison. The data and code can be found at https://github.com/DouMiao1226/PMNN.

A novel method based on fractional order Gegenbauer wavelet operational matrix for the solutions of the multi-term time-fractional telegraph equation of distributed order

In this article, we propose an effective scheme based on a combination of the Tau method and fractional-order Gegenbauer wavelets for solving the multi-term time-fractional differential equations of distributed order. First, we define fractional order Gegenbauer wavelets and then obtain operational matrices of these orthogonal functions. Applying the Legendre–Gauss quadrature for the integral term, we use function approximations obtained by the presented wavelets and the Tau method for the solution of the distributed-order multi-term time-fractional telegraph equation. The proposed method reduces the numerical solution of multi order time-fractional equations to a system of algebraic equations. Then, the convergence analysis and error bounds of the proposed scheme are studied. Three illustrative examples are solved to justify the effectiveness of the proposed method compared with some previously published results.

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.

A Hybrid Non-Polynomial Spline Method and Conformable Fractional Continuity Equation

This paper presents a groundbreaking numerical technique for solving nonlinear time fractional differential equations, combining the conformable continuity equation (CCE) with the Non-Polynomial Spline (NPS) interpolation to address complex mathematical challenges. By employing conformable descriptions of fractional derivatives within the CCE framework, our method ensures enhanced accuracy and robustness when dealing with fractional order equations. To validate our approach’s applicability and effectiveness, we conduct a comprehensive set of numerical examples and assess stability using the Fourier method. The proposed technique demonstrates unconditional stability within specific parameter ranges, ensuring reliable performance across diverse scenarios. The convergence order analysis reveals its efficiency in handling complex mathematical models. Graphical comparisons with analytical solutions substantiate the accuracy and efficacy of our approach, establishing it as a powerful tool for solving nonlinear time-fractional differential equations. We further demonstrate its broad applicability by testing it on the Burgers–Fisher equations and comparing it with existing approaches, highlighting its superiority in biology, ecology, physics, and other fields. Moreover, meticulous evaluations of accuracy and efficiency using (L2 and L∞) norm errors reinforce its robustness and suitability for real-world applications. In conclusion, this paper presents a novel numerical technique for nonlinear time fractional differential equations, with the CCE and NPS methods’ unique combination driving its effectiveness and broad applicability in computational mathematics, scientific research, and engineering endeavors.

Supervised learning and meshless methods for two-dimensional fractional PDEs on irregular domains

Recently, several numerical methods have been developed for solving time-fractional differential equations not only on rectangular computational domains but also on convex and non-convex non-rectangular computational geometries. On the other hand, due to the existence of integrals in the definition of space-fractional operators, there are few numerical schemes for solving space-fractional differential equations on irregular regions. In this paper, we develop a novel numerical solution based on the machine learning technique and a generalized moving least squares approximation for two-dimensional fractional PDEs on irregular domains. The scheme is constructed on the monomials, and this is the strength of this technique. Moreover, it will be used to approximate the space derivatives on convex and non-convex non-rectangular computational domains. The numerical results are extended to solve the fractional Bloch–Torrey equation, fractional Gray–Scott equation, and fractional Fitzhugh–Nagumo equation.

Conformable non-polynomial spline method: A robust and accurate numerical technique

The article introduces a novel and developed a numerical technique that solves nonlinear time fractional differential equations. It combines finite difference and non-polynomial spline methods while using conformable descriptions of fractional derivatives. The strategy has proven to be useful to analyzing intricate datasets across various domains, including finance, science, and engineering. The applicability and validity of the approach are demonstrated through numerical examples. The study assessed the approach's stability using the Fourier method and found it to be unconditionally stable for a specific parameter range. The proposed method has been analyzed for convergence, and the analysis reveals that it exhibits a convergence order of six. The paper also includes graphs comparing the present solution to an analytical one. The method is tested with some examples of the model that have a wide range of applications in fields such as biology, ecology, and physics called the Burgers-Fisher equations and compared to previous approaches to demonstrate its effectiveness and applicability. The research evaluated the approach's accuracy and efficiency using the (L2 and L∞) norm errors. Also, the effects of time and fractional derivatives have been studied, and to the best of our knowledge, it has not been previously published in any academic or industry publication.

A study on mild solutions for multi-term time fractional measure differential equations

In this paper, we investigate the existence and uniqueness of the S-asymptotically ω-periodic mild solutions to a class of multi-term time-fractional measure differential equations with initial conditions in Banach spaces. Firstly, we look for a suitable concept of S-asymptotically ω-periodic mild solution to our concerned problem, by means of the Laplace transform and -resolvent family . Secondly, the existence of S-asymptotically ω-periodic mild solutions for the mentioned system is obtained by utilizing regulated functions and fixed point theorem. Finally, as the application of abstract results, an example is given to illustrate our main results.

A numerical solution of fractional reaction–convection–diffusion for modeling PEM fuel cells based on a meshless approach

The purpose of this contribution is to present or implement generalized finite difference method (GFDM) for the first time in order to solve the reaction convection Diffusion equation (FRCDE) model. The FRCDE equations are nonlinear time fractional differential equations that can be used to describe mathematical models in physics and engineering, such as fuel cells and transport in inhomogeneous media. This model is a generalization of the classical RCDE, with time terms considered by Caputo derivative sense for 0<α(x,t)<1. The meshless GFDM, in conjunction with the ϑ-weighted finite difference method, is developed to approximate processes in the spatial direction. GFDM is based on the Taylor series expansion and the moving least squares (MLS) method to evaluate the derivatives of unknown variables. The proposed method, as a truly meshless approach, is very promising in the numerical approximation of engineering problems within convex, non-convex, irregular, and regular domains (complex domains). The reliability and accuracy of the proposed method are shown by considering a variety of computational domains. The sensitivity of the selection of local collocation points and ϑ is also reported.