Caputo Fractional Derivative Research Articles

(2+1)-dimensional Burgers equation with a Caputo fractional derivative: Lie symmetry analysis, optimal system, exact solutions and conservation laws

In this paper, the Lie group method is applied to analyze the (2+1)-dimensional Burgers equation with Caputo fractional derivative. The infinitesimal generators of the equation are investigated. Then, using the Lie point symmetry of the equation, a one-dimensional optimal system is constructed. Subsequently, within the framework of the optimal system, the equation is symmetrically simplified and the exact solutions of the original equation are determined using the invariant subspace method and the homogeneous balance method. In the process of simplifying the equation to a (1+1)-dimensional fractional differential equation, it is deliberately avoided to use complex Erdélyi-Kober fractional derivative operator. Instead, a simple transformation is utilized to obtain a low-dimensional fractional differential equation. Finally, based on the concept of nonlinear self-adjointness, the conservation laws of the fractional differential equation are obtained.

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

This paper devoted to study unique solvability of inverse source problem for a semilinear time fractional pseudoparabolic equation perturbed by a damping term. Inverse problem consists of recovering a solely time dependent coefficient of right-hand side under a measurement in an integral form. The damped term acts in the equation as a nonlinear source or as an absorption, depending on its sign of coefficient, where it is positive or negative, respectively. In these both cases, we have established sufficient conditions on data, where the inverse problem has a global or local in time unique weak and strong solution.

Modified Gradient Descent Approach Involving Caputo Fractional Derivative with Metaheuristic Optimizer

Modified Gradient Descent Approach Involving Caputo Fractional Derivative with Metaheuristic Optimizer

An efficient computational method for solving the fractional form of the European option price PDE with transaction cost under the fractional Heston model

This paper introduces a new method for precise option pricing analysis by utilizing the fractional form of the option price partial differential equation (PDE) and implementing a collocation technique based on the first Fermat polynomial sequence. The key innovation of this study is the exploration of the fractional formulation of European option pricing within the context of the fractional Heston model, which employs a collocation scheme utilizing Fermat polynomials to generate operational matrices characterized by numerous zeros, thereby enhancing computational efficiency. To achieve this, we first derive the option price PDE and convert it to its fractional form in the Caputo sense. We then solve the fractional PDE using fractional Fermat functions, expressing the solution as a series of multivariate Fermat functions with unknown coefficients. Following this, we compute the operational matrices for the Caputo fractional derivative and related partial derivatives, demonstrating how this computational framework transforms the primary problem into a nonlinear system of equations. Additionally, we conduct a convergence analysis of the collocation method. We conclude by presenting several numerical examples that illustrate the applicability and effectiveness of the proposed method, with its robust theoretical foundations and successful numerical tests indicating significant potential for practical applications in finance.

Effects of Memory on Inventory Control and Pricing Policy for Imperfect Production with Rework Process

Fractional calculus is a pertinent way to study the memory effect in an inventory model for investigating its dynamical behavior. Since inventory management is a memory-dependent process, fractional calculus approach may be employed to discover some fruitful insights and can help to gain more profit. In realistic scenarios, the manufacturing process cannot be perfect, and it delivers some faulty units due to many inevitable reasons. In literature, an imperfect production inventory problem under the memory effect has not been studied. Our study aims to investigate the memory effect on a production inventory system. In this article, a fractional order inventory control problem is formulated by considering an imperfect manufacturing process and price-sensitive demand. The faulty units are repaired through a rework process. Caputo fractional derivatives and integrals are used to consider the memory effect. Due to the nonlinear cost elements in the formulated problem, optimal pricing and production policies are investigated by using a quasi-Newton optimization algorithm and particle swarm optimization approach. The managerial implications of the proposed study are discussed with the help of numerical illustrations. The numerical outcomes suggest that consideration of memory in the inventory system boosts the profitability of the firm.

Compact ADI Difference Scheme for the 2D Time Fractional Nonlinear Schrödinger Equation

In this paper, we will introduce a compact alternating direction implicit (ADI) difference scheme for solving the two-dimensional (2D) time fractional nonlinear Schrödinger equation. The difference scheme is constructed by using the L1−2−3 formula to approximate the Caputo fractional derivative in time and the fourth-order compact difference scheme is adopted in the space direction. The proposed difference scheme with a convergence accuracy of O(τ1+α+hx4+hy4)(α∈(0,1)) is obtained by adding a small term, where τ, hx, hy are the temporal and spatial step sizes, respectively. The convergence and unconditional stability of the difference scheme are obtained. Moreover, numerical experiments are given to verify the accuracy and efficiency of the difference scheme.

Controllability of Impulsive Damped Fractional Order Systems Involving State Dependent Delay

In this article, the concept of controllability on fractional order impulsive systems involving state dependent delay and damping behavior is analysed by utilizing Caputo fractional derivative. The main motivation is to derive the sufficient conditions for the controllability of the considered systems. Based on the Laplace transform and inverse Laplace transform, the solution of fractional-order dynamical systems are obtained. The results are established by utilizing basic ideas of fractional calculus, Mittag-Leffler function and Banach fixed point theorem. Finally, an application is provided to illustrate the derived result.

On the Global Existence of Solution and Lyapunov Asymptotic Practical Stability for Nonlinear Impulsive Caputo Fractional Derivative via Comparison Principle

This paper examines the existence of maximal solution of the comparison differential system and also establishes sufficient conditions for the asymptotic practical stability of the trivial solution of a nonlinear impulsive Caputo fractional differential equations with fixed moments of impulse using the vector Lyapunov functions. First, it was discovered that the vector form of the Lyapunov function was majorized by the maximal solution of the comparison system. From the results obtained, it was also established that the main system is asymptotically practically stable in the sense of Lyapunov.

Temporal second-order two-grid finite element method for semilinear time-fractional Rayleigh-Stokes equations

In this paper, we have developed a temporal second-order two-grid FEM to solve the semilinear time-fractional Rayleigh-Stokes equations. The proposed two-grid FEM uses the L2-1σ scheme and second order scheme to approximate the Caputo fractional derivative and the time first-order derivative in temporal direction and the standard FEM in spatial direction. The L2-norm and H1-norm stability and error estimates for the standard finite element solution and the two-grid solution are derived. The results shown that as long as the mesh sizes satisfy H=h12 and H=hr2r+2 respectively, the two-grid algorithm can achieve asymptotically optimal approximation. Furthermore, the non-uniform L2-1σ scheme was applied for temporal discretization to handle the weak singularity of the solution. Finally, the theoretical findings were confirmed by numerical results, and the effectiveness of the two-grid algorithm was demonstrated.

The Impact of White Noise on Chaotic Behavior in a Financial Fractional System with Constant and VariableOrder: A Comparative Study

In this study, we investigated the impact of white noise on the chaotic dynamics of a financial system with Caputo fractional derivatives of constant- and variable-order. We solve thesefractional financial systems numerically. We analyze the chaotic behavior of these fractional-order hyperchaotic systems through simulations, study the effects of white noise on chaotic dynamics, and provide a comparison between fractional hyperchaotic systems of constant and variable orders. The study reveals that white noise can either amplify or suppress chaos, with significant differences observed between the constant- and variable-order cases. We obtained numerous intriguing graphical findings for the model by evaluating various scenarios. The findings provide useful information for better understanding financial market dynamics in the presence of noise.

Fractional-Order Mathematical Modeling of Breast Cancer: Comparing Adaptive Immune Responses and Estrogen Dynamics with Experimental Data

Breast cancer is a significant global health concern that requires innovative approaches to understand its behavior and improve treatment strategies. This paper introduces a new fractional-order mathematical model to explain the complex dynamics of breast cancer progression, including adaptive immune responses and estrogen dynamics. Utilizing Caputo fractional derivatives, our model reveals insights into the impact of fractional-order dynamics on cancer cell populations. Simulation results demonstrate a notable increase in cell populations with higher fractional orders, suggesting heightened aggressiveness, while lower orders correspond to subdued progression. Unlike traditional integer-order models, fractional-order derivatives offer a more nuanced depiction of nonlinear dynamics, crucial for capturing the complexities of cancer progression. Importantly, our findings underscore the potential clinical relevance of fractional-order models in informing personalized treatment strategies, particularly through the modulation of estrogen levels. By integrating treatment considerations, such as hormone therapy, our model holds promise for advancing precision medicine approaches tailored to individual patient characteristics.

A numerical algorithm based on extended cubic B-spline functions for solving time-fractional convection-diffusion-reaction equation with variable coefficients

In this research, we develop an innovative and efficient numerical method for solving a nonlinear one-dimensional time-fractional convection-diffusion-reaction equation (TFCDRE) with a variable coefficient. This method integrates the Crank-Nicolson finite difference scheme with extended cubic B-spline (ExCBS) basis functions. The Caputo fractional derivative is applied for temporal discretization, while the ExCBS functions are utilized for spatial discretization. The stability of the method was discussed by the von Neumann method, which shows unconditional stability; and the convergence analysis secure an order of $O(h^2+\triangle t^{2-\alpha})$. We demonstrated the efficiency and simplicity of our method through three numerical experiments and verified its accuracy using the absolute error $(L_2)$ and maximum error $(L_\infty)$ norms in temporal and spatial dimensions. The results are also graphically represented and show substantial agreement with the approximated and exact solution. We confirmed the effectiveness and accuracy of our approach over the local discontinuous Galerkin finite element and the compact finite difference methods, respectively, from the literature.

FRACTIONAL ANALYSIS OF ONE-DIMENSIONAL SCHRÖDINGER MODEL IN THE CONTEXT OF CAPUTO-TYPE FRACTIONAL DERIVATIVES BASED ON RESIDUAL POWER SERIES METHOD

This research presents a novel analytical approach to explore the fractional analysis of the one-dimensional time-fractional Schrödinger model (TFSM) using Caputo fractional derivatives. By integrating the Mohand transform (MT) with the residual power series method (RPSM), we develop the Mohand residual power series method (MT-RPSM) that provides results in the form of convergent series without assumptions on variables. Initially, we employ MT to reduce the fractional order, and then we transfer the fractional problem into the Mohand space formulation. Second, we use the RPSM concept to derive the iterative series formula for the Mohand space formulation. We analyze these findings using visual layouts to show the physical representation of the TFSM, which matches the precise results very well. The results indicate that the proposed technique is a reliable and practical method for identifying and analyzing various nonlinear models of physical phenomena.

A Global Method for Approximating Caputo Fractional Derivatives—An Application to the Bagley–Torvik Equation

In this paper, we propose a global numerical method for approximating Caputo fractional derivatives of order α(Dαf)(y)=1Γ(m−α)∫0y(y−x)m−α−1f(m)(x)dx,y>0, with m−1<α≤m,m∈N. The numerical procedure is based on approximating f(m) by the m-th derivative of a Lagrange polynomial, interpolating f at Jacobi zeros and some additional nodes suitably chosen to have corresponding logarithmically diverging Lebsegue constants. Error estimates in a uniform norm are provided, showing that the rate of convergence is related to the smoothness of the function f according to the best polynomial approximation error and depending on order α. As an application, we approximate the solution of a Volterra integral equation, which is equivalent in some sense to the Bagley–Torvik initial value problem, using a Nyström-type method. Finally, some numerical tests are presented to assess the performance of the proposed procedure.

Cauchy discretization method for solve fractional linear optimal control problems

This paper presents a numerical scheme designed for solving fractional linear optimal control problems (FLOCPs) governed by Caputo fractional derivatives (CFDs) of order 0<α≤1. The method transforms the original fractional control problem into an equivalent linear programming problem (LPP), enabling a more straightforward solution process. To evaluate the performance and accuracy of this approach, several numerical experiments were conducted using MATLAB. These experiments highlight the method’s computational efficiency and its simplicity in terms of implementation. The proposed approach offers accurate results, particularly in scenarios where traditional methods may fall short due to the complexities introduced by fractional derivatives. A detailed numerical example is presented to further illustrate the method’s effectiveness in addressing FLOCPs. The outcomes suggest that this approach is not only practical but also provides valuable insights for researchers dealing with fractional calculus in optimal control theory. This contribution is expected to be beneficial for applications in engineering and the physical sciences, where fractional-order systems play a significant role.

On the Existence of Maximal Solution and Lyapunov Practical Stability of Nonlinear Impulsive Caputo Fractional Derivative via Comparison Principle

This paper examines the existence of maximal solution of the comparison differential system and also establishes sufficient conditions for the practical stability of the trivial solution of a nonlinear impulsive Caputo fractional differential equations with fixed moments of impulse using the vector Lyapunov functions. First, it was discovered that the vector form of the Lyapunov function was majorized by the maximal solution of the comparison system. From the results obtained, it was established that the main system is practically stable in the sense of Lyapunov.

Numerical simulation of the two-dimensional fractional Schrödinger equation for describing the quantum dynamics on a comb with the absorbing boundary conditions

This paper mainly contributes to investigating a distinctive fractional Schrödinger equation (FSE) for describing the quantum dynamics on a comb. The special characteristic of the comb is that the diffusion versus the x-direction only occurs on the x-axis. We replace the infinite boundary with the absorbing boundary conditions (ABCs), which can be obtained using the Laplace transform. We utilize the integration property to deal with the singularity function in the FSE and propose the finite difference method. The stability and convergence of this scheme are investigated. The L1 scheme to discretize the Caputo fractional derivative needs a particularly expensive computational and storage cost. To increase the calculation speed, a fast algorithm is used to improve the operation efficiency. By introducing a source term, the exact solution and the numerical solution are compared. The comparison of the CPU time for the L1 scheme and fast scheme is given. The comparison between the FSE on a comb and the classical one is also analyzed and discussed. The transport process versus the x-axis and the y-axis with various parameters are shown. Finally, the mean square displacement (MSD) is investigated. We find that the solution with the ABCs has a better agreement with the exact expression compared to the solution with truncated zero boundary conditions.

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.

A Novel Fourth-Order Finite Difference Scheme for European Option Pricing in the Time-Fractional Black–Scholes Model

This paper addresses the valuation of European options, which involves the complex and unpredictable dynamics of fractal market fluctuations. These are modeled using the α-order time-fractional Black–Scholes equation, where the Caputo fractional derivative is applied with the parameter α ranging from 0 to 1. We introduce a novel, high-order numerical scheme specifically crafted to efficiently tackle the time-fractional Black–Scholes equation. The spatial discretization is handled by a tailored finite point scheme that leverages exponential basis functions, complemented by an L1-discretization technique for temporal progression. We have conducted a thorough investigation into the stability and convergence of our approach, confirming its unconditional stability and fourth-order spatial accuracy, along with (2−α)-order temporal accuracy. To substantiate our theoretical results and showcase the precision of our method, we present numerical examples that include solutions with known exact values. We then apply our methodology to price three types of European options within the framework of the time-fractional Black–Scholes model: (i) a European double barrier knock-out call option; (ii) a standard European call option; and (iii) a European put option. These case studies not only enhance our comprehension of the fractional derivative’s order on option pricing but also stimulate discussion on how different model parameters affect option values within the fractional framework.

A fully discrete GL-ADI scheme for 2D time-fractional reaction-subdiffusion equation

Alternating direction implicit (ADI) difference method for solving a 2D reaction-subdiffusion equation whose solution behaves a weak singularity at t=0 is studied in this paper. A Grünwald-Letnikov (GL) approximation is used for the discretization of Caputo fractional derivative (of order α, with 0<α<1) on a uniform mesh. Stability and convergence of the fully discrete ADI scheme are rigorously established. With the help of a discrete fractional Gronwall inequality, we get the sharp error estimate. The stability in L2 norm and the convergence of the GL-ADI scheme are strictly proved, where the convergent order is O(τtsα−1+τ2α+h12+h22). Numerical experiments are given to verify the theoretical analysis.