Solution Of Fractional Order Research Articles

Approximate solutions of Legendre’s differential equation by using fractional differential transform method

In this article, we examine the fractional order of Legendre’s equations by applying the Differential Transformation Method (DTM). This method is effective and sustainable in the investigation of Legendre’s equations. Series solution could be used to compare the Bessel and Legendre differential equations. If α = 1 then, to get the solution of fractional order of Legendre’s equation. Generalized fraction variation will use to calculate the fraction form of a special function. The results of the Legendre equation will compare with the exact solutions at α=1 and also shows that the method is quite precise and reliable.

Comparative Study of Solutions of Fractional Order Mixed KdV Burger’s Equation

Objective: The aim of this study is to obtain numerical solutions of fractional order mixed KdV Burger’s equation using iterative methods. Methodology: The study is centered on a time fractional mixed KdV Burger’s equation in the Caputo sense. The approximate solutions of this equation are obtained using Adomian decomposition method and Homotopy Perturbation method. Also, a Python code is developed to obtained numerical solutions and simulate graphically. Finding: The obtained solutions are compared with the exact solution and presented graphically using Python program for better analysis. Novelty: These methods shall be used to obtain approximate solution of such type of non-linear fractional order differential equations. Keywords: Fractional Differential Equation, Caputo Derivative, ADM, HPM, KdV Burger equation, Python

The analytical analysis of fractional differential system via different operators and normalization functions

This article is related to the utilization of the Laplace Adomian Decomposition Method (LADM) to analyze the solution of a fractional order system under different operators and normalized functions. In this connection, the three new and updated operators i.e., Caputo, Caputo–Fabrizio, and Atangana–Baleano are used to define the fractional order derivative with the help of four different normalization functions. It is observed that, among the four normalized functions, the fourth kind of normalized function is the most suitable function for the above-mentioned operators. The LADM series solutions are obtained for both integer and fractional orders, implying that the proposed approach has a greater accuracy and convergence rate. Figures and Tables are constructed to verify the validity and efficiency of the suggested technique. LADM simulations confirmed, that the more iterations of LADM reduced the associated absolute error. Moreover, fractional order solutions are obtained which are convergent towards integer order solutions.

Approximate Solution of Fractional Order of Partial Differential Equations Using Laplace-Adomian Decomposition Method in MATLAB

This article presents the application of the Laplace-Adomian Decomposition Method (LADM) for solving partial differential equations (PDEs) in the context of heat conduction and wave propagation. The LADM combines Laplace transform and Adomian decomposition to approximate solutions to PDEs efficiently in MATLAB. The procedure involves transforming the PDE into simpler differential equations, which are then solved iteratively using the Adomian decomposition method. The advantages of LADM include simplicity, flexibility, and applicability to a wide range of PDEs. We demonstrate the effectiveness of LADM through numerical experiments solving the heat equation and wave equation using MATLAB. The results show good agreement with analytical solutions and highlight the efficiency and accuracy of LADM for solving PDEs.

Analysis of Age-Structured Mathematical Model of Malaria Transmission Dynamics via Classical and ABC Fractional Operators

Malaria is a complex disease with many factors influencing the transmission dynamics, including age. This research analyzes the transmission dynamics of malaria by developing an age-structured mathematical model using the classical integer order and Atangana–Baleanu–Caputo fractional operators. The analysis of the model focused on several important aspects. The existence and uniqueness of solutions of fractional order were explored based on some fixed-point theorems,such as Banach and Krasnoselski. The Positivity and boundedness of the solutions were also investigated. Furthermore, through mathematical analysis techniques, we analyzed different types of stability results, and the results showed that the disease-free equilibrium point of the model is proved to be both locally and globally asymptotically stable if the basic reproduction number is less than one, whereas the endemic equilibrium point of the model is both locally and globally asymptotically stable if the basic reproduction number is greater than one. The findings from the sensitivity analysis revealed that the most sensitive parameters, essential for controlling or eliminating malaria are mosquito biting rate, density-dependent natural mortality rate, clinical recovery rate, and recruitment rate for mosquitoes. Numerical simulations are also performed to examine the behavior of the model for different values of the fractional-order alpha,and the result revealed that as the value α reduces from 1, the spread of the endemic grows slower. By incorporating these findings, this research helps to clarify the dynamics of malaria and provides information on how to create efficient control measures.

Multiple Mittag-Leffler Stability of Almost Periodic Solutions for Fractional-Order Delayed Neural Networks: Distributed Optimization Approach.

This article proposes new theoretical results on the multiple Mittag-Leffler stability of almost periodic solutions (APOs) for fractional-order delayed neural networks (FDNNs) with nonlinear and nonmonotonic activation functions. Profited from the superior geometrical construction of activation function, the considered FDNNs have multiple APOs with local Mittag-Leffler stability under given algebraic inequality conditions. To solve the algebraic inequality conditions, especially in high-dimensional cases, a distributed optimization (DOP) model and a corresponding neurodynamic solving approach are employed. The conclusions in this article generalize the multiple stability of integer-or fractional-order NNs. Besides, the consideration of the DOP approach can ameliorate the excessive consumption of computational resources when utilizing the LMI toolbox to deal with high-dimensional complex NNs. Finally, a simulation example is presented to confirm the accuracy of the theoretical conclusions obtained, and an experimental example of associative memories is shown.

A non-singular fractional-order logistic growth model with multi-scaling effects to analyze and forecast population growth in Bangladesh

This paper is primarily concerned with data analysis employing the nonlinear least squares curve fitting method and the mathematical prediction of future population growth in Bangladesh. Available actual and adjusted census data (1974–2022) of the Bangladesh population were applied in the well-known autonomous logistic population growth model and found that all data sets of the logistic (exact), Atangana-Baleanu-Caputo (ABC) fractional-order derivative approach, and logistic multi-scaling approximation fit with good agreement. Again, the existence and uniqueness of the solution for fractional-order and Hyers-Ulam stability have been studied. Generally, the growth rate and maximum environmental support of the population of any country slowly fluctuate with time. Including an approximate closed-form solution in this analysis confers several advantages in assessing population models for single species. Prior studies predominantly employed constant growth rates and carrying capacity, neglecting the investigation of fractional-order methods. Thus, the current study fills a crucial gap in the literature by introducing a more formal approach to analyzing population dynamics. Therefore, we bank on the findings of this article to contribute to accurate population forecasting and planning, national development, and national progress.

Retracted: Existence and Uniqueness of Mild Solution for Fractional-Order Controlled Fuzzy Evolution Equation

Retracted: Existence and Uniqueness of Mild Solution for Fractional-Order Controlled Fuzzy Evolution Equation

Solution of Fractional Heat-Like and Fractional Wave-Like Equation by Using Modern Strategy

Abstract This paper introduces a novel form of the Adomian decomposition (ADM) method for solving fractional-order heat-like and wave-like equations with starting and boundary value problems. The derivations are provided in the sense of Caputo. In order to help understanding, the generalised formulation of the current approach is provided. Several numerical examples of fractional-order diffusion-wave equations (FDWEs) are solved using the suggested method in this context. In addition to examining the applicability of the suggested method to the solving of fractional-order heat-like and wave-like equations, a graphical depiction of the solutions to three instructive cases was constructed. Solution graphs were arrived at for integer and fractional-order problems. The derived and exact solutions to integer-order problems were found to be in excellent agreement. The subject of the present research endeavour is the convergence of fractional-order solutions. This strategy is considered to be the most successful way of addressing fractional-order initial-boundary value issues in science and engineering. This strategy is presented here.

Dynamical analysis of nonlinear fractional order Lorenz system with a novel design of intelligent solution predictive radial base networks

In this research work, the convective flow of Lorenz attractor in the fractional domain is modeled with a nonlinear flexible structure of radial basis neural network (RBNN). The physical parameters of the complex fractional-order system are initially computed with fractional order Runge–Kutta solver for different stochastic scenarios of control constraints and designed its parametric model with RBFN under various initial conditions of the chaotic Lorenz system. Multiple chaos of the Lorenz system with the dynamical neural network structure are computed for different fractional order solutions to analyze the sensitivity of chaotic behavior by means of Lyapunov exponents. Phase diagrams of the chaotic pattern show the fractional-order Lorenz system has compact dynamical behavior and has the potential for the design application of a real dynamical system. Embedded dimension and time delay of the fractional Lorenz system are computed using the average mutual information (AMI) technique to evaluate the suitability of the time delay chaotic pattern in the fractional domain, which is affected by the numerical algorithm and fluctuated on minute time step size. The proposed design of RBFN has been confirmed as an exceptionally high-performance tool for soft computing and dynamics analysis of chaotic systems in the fractional order domain.

A Comparative Analysis of Fractional-Order Fokker–Planck Equation

The importance of partial differential equations in physics, mathematics and engineering cannot be emphasized enough. Partial differential equations are used to represent physical processes, which are then solved analytically or numerically to examine the dynamical behaviour of the system. The new iterative approach and the Homotopy perturbation method are used in this article to solve the fractional order Fokker–Planck equation numerically. The Caputo sense is used to characterize the fractional derivatives. The suggested approach’s accuracy and applicability are demonstrated using illustrations. The proposed method’s accuracy is expressed in terms of absolute error. The proposed methods are found to be in good agreement with the exact solution of the problems using graphs and tables. The results acquired using the given approaches are also obtained at various fractional orders of the derivative. It is observed from the graphs and tables that fractional order solutions converge to an integer solution when the fractional orders approach the integer-order of the problems. The tabular and graphical view for the given problems is obtained through Maple. The presented approaches can be applied to existing non-linear fractional partial differential equations due to their accurate, simple and straightforward implementation.

Analysis and Numerical Simulation of System of Fractional Partial Differential Equations with Non-Singular Kernel Operators

The exact solution to fractional-order partial differential equations is usually quite difficult to achieve. Semi-analytical or numerical methods are thought to be suitable options for dealing with such complex problems. To elaborate on this concept, we used the decomposition method along with natural transformation to discover the solution to a system of fractional-order partial differential equations. Using certain examples, the efficacy of the proposed technique is demonstrated. The exact and approximate solutions were shown to be in close contact in the graphical representation of the obtained results. We also examine whether the proposed method can achieve a quick convergence with a minimal number of calculations. The present approaches are also used to calculate solutions in various fractional orders. It has been proven that fractional-order solutions converge to integer-order solutions to problems. The current technique can be modified for various fractional-order problems due to its simple and straightforward implementation.

CAPUTO TIME FRACTIONAL MODEL BASED ON GENERALIZED FOURIER’S AND FICK’S LAWS FOR BRINKMAN-TYPE FLUID: EXACT SOLUTION VIA INTEGRAL TRANSFORM

This paper proposes a new method for the development of the Caputo time fractional model. The method relies on generalized Fourier’s and Fick’ laws to describe the flow behavior of Brinkman-type fluids. An analysis of the free convection flow through a channel is carried out using a new transformation method. This transformation affects fluid energy and concentration equations. The specific governing equations are solved using a Laplace transform and Fourier sine transform. We obtain the solutions of the governing partial differential equations (PDEs) in terms of the Mittag–Leffler function. Mathematical software has been used for both graphical and numerical computation in order to examine the effects of embedded parameters. From graphical and tabular analysis, fractional-order solution provides more than one layer for fluid behavior, thermal, and concentration distribution in the channel. Experimentalists and engineers can choose from many best-fitted layers to compare their data and results. A deviation in the velocity profile’s behavior is also seen for larger values of the Brinkman parameter.

The Fractional Investigation of Some Nonlinear Partial Differential Equations by Using an Efficient Procedure

The nonlinearity in many problems occurs because of the complexity of the given physical phenomena. The present paper investigates the non-linear fractional partial differential equations’ solutions using the Caputo operator with Laplace residual power series method. It is found that the present technique has a direct and simple implementation to solve the targeted problems. The comparison of the obtained solutions has been done with actual solutions to the problems. The fractional-order solutions are presented and considered to be the focal point of this research article. The results of the proposed technique are highly accurate and provide useful information about the actual dynamics of each problem. Because of the simple implementation, the present technique can be extended to solve other important fractional order problems.

THE FRACTIONAL-ORDER SYSTEM OF SINGULAR AND NON-SINGULAR THERMO-ELASTICITY SYSTEM IN THE SENSE OF HOMOTOPY PERTURBATION TRANSFORM METHOD

This paper investigates the mathematical result of the nonlinear fractional scheme of equations representing the singular and non-singular thermoelastic system. The suggested method is a graceful combination of Shehu transformation with homotopy perturbation method and fractional derivative described with the Caputo operator. To show and confirm the effectiveness of the future method, I examined three cases and evaluated in terms of fractional-order the projected model. Furthermore, the physical behavior of the fractional-order solution achieved has been shown in graphs, and the mathematical model is proved to confirm the accuracy. The solutions achieved clarify that the suggested methodology is simple to apply, highly organized, and accurate to study the attitude of nonlinear system differential equations of arbitrary order arising in the related fields of engineering and science.

Analytical Solutions of the Nonlinear Time-Fractional Coupled Boussinesq-Burger Equations Using Laplace Residual Power Series Technique

In this paper, we present the series solutions of the nonlinear time-fractional coupled Boussinesq-Burger equations (T-FCB-BEs) using Laplace-residual power series (L-RPS) technique in the sense of Caputo fractional derivative (C-FD). To assert the efficiency, simplicity, performance, and reliability of our proposed method, an attractive and interesting numerical example is tested analytically and graphically. In addition, our obtained results show that this algorithm is compatible and accurate for investigating the fractional-order solutions of engineering and physical applications. Finally, Mathematica software 14 is applied to compute the numerical and graphical results.

Analysis of some new wave solutions of fractional order generalized Pochhammer-chree equation using exp-function method

Analysis of some new wave solutions of fractional order generalized Pochhammer-chree equation using exp-function method

The solutions of nonlinear fractional partial differential equations by using a novel technique

Abstract In this article, the solutions of higher nonlinear partial differential equations (PDEs) with the Caputo operator are presented. The fractional PDEs are modern tools to model various phenomena more accurately. The residual power series method (RPSM) is used for the solution analysis of fractional partial differential equations (FPDEs), which has direct implementation for the solutions of fractional partial differential equations. In this work, the solutions to a few nonlinear FPDEs are handled by the proposed technique. The general and particular schemes of RPSM are constructed and implemented successfully. The fractional solutions of PDEs have provided many useful dynamics of the targeted problems. The RPSM results for both integer and fractional-order FPDEs are further explained and elaborated by using graphs and tables. It is observed that the higher accuracy of RPSM is achieved with fewer calculations. Graphs and tables for fractional-order solutions are presented, which confirm the convergence phenomena of fractional solutions toward integer order solutions of each problem. The suggested method can be extended to the solutions of other nonlinear fractional partial differential equations.

Fractional-View Analysis of Fokker-Planck Equations by ZZ Transform with Mittag-Leffler Kernel

This work combines a ZZ transformation with the Adomian decomposition method to solve the fractional-order Fokker-Planck equations. The fractional derivative is represented in the Atangana-Baleanu derivative. It is looked at with graphs that show that the accurate and estimated results are close to each other, indicating that the method works. Fractional-order solutions are the most in line with the dynamics of the targeted problems, and they provide an endless number of options for an optimal mathematical model solution for a particular physical phenomenon. This analytical approach produces a series type result that quickly converges to actual answers. The acquired outcomes suggest that the novel analytical solution method is simple to use and very successful at assessing complicated equations that occur in related research and engineering fields.

Dynamics of COVID-19 epidemic via two different fractional derivatives

In December 2019, the novel Coronavirus, also known as 2019-nCoV or SARS-CoV-2 or COVID-19, was first recognized as a deadly disease in Wuhan, China. In this paper, we analyze two different nonclassical Coronavirus models to observe the outbreaks of this disease. Caputo and Caputo–Fabrizio (C–F) fractional derivatives are considered to simulate the given epidemic models by using two separate methods. We perform all required graphical simulations with the help of real data to demonstrate the behavior of the proposed systems. We observe that the given schemes are highly effective and suitable to analyze the dynamics of Coronavirus. We find different natures of the given model classes for both Caputo and C-F derivative sense. The main contribution of this study is to propose a novel framework of modeling to show how the fractional-order solutions can describe disease dynamics much more clearly as compared to integer-order operators. The motivation to use two different fractional derivatives, Caputo (singular-type kernel) and Caputo–Fabrizio (exponential decay-type kernel) is to explore the model dynamics under different kernels. The applications of two various kernel properties on the same model make this study more effective for scientific observations.