Using Variable Iterative Method to Calculate the Numerical Approximations Solution for Fractional Differential Equations

Realizing the behavior of the solution in the asymptotic situations is essential for repetitive applications in the control theory and modeling of the real-world systems. This study discusses a robust and definitive attitude to find the interval approximate asymptotic solutions of fractional differential equations (FDEs) with the Atangana-Baleanu (A-B) derivative. In fact, such critical tasks require to observe precisely the behavior of the noninterval case at first. In this regard, we initially shed light on the noninterval cases and analyze the behavior of the approximate asymptotic solutions, and then, we introduce the A-B derivative for FDEs under interval arithmetic and develop a new and reliable approximation approach for fractional interval differential equations with the interval A-B derivative to get the interval approximate asymptotic solutions. We exploit Laplace transforms to get the asymptotic approximate solution based on the interval asymptotic A-B fractional derivatives under interval arithmetic. The techniques developed here provide essential tools for finding interval approximation asymptotic solutions under interval fractional derivatives with nonsingular Mittag-Leffler kernels. Two cases arising in the real-world systems are modeled under interval notion and given to interpret the behavior of the interval approximate asymptotic solutions under different conditions as well as to validate this new approach. This study highlights the importance of the asymptotic solutions for FDEs regardless of interval or noninterval parameters.

The aim of the present analysis is to apply the Adomian decomposition method and He's variational method for the approximate analytical solution of a nonlinear ordinary fractional differential equation. The solutions obtained by the above two methods have been numerically evaluated and presented in the form of tables and also compared with the exact solution. It was found that the results obtained by the above two methods are in excellent agreement with the exact solution. Finally, a surface plot of the approximate solutions of the fractional differential equation by the above two methods is drawn for 0⩽t⩽2 and 1<α⩽2.

At present, the theory of fractional calculus is widely used in many fields of science for modeling various processes. Differential equations with fractional derivatives are used to model the migration of pollutants in porous inhomogeneous media and allow a more correct description of the behavior of pollutants at large distances from the source. The analytical solution of differential equations with fractional order derivatives is often very complicated or even impossible. There has been proposed a numerical method for solving fractional differential equations in partial derivatives with respect to time to describe the migration of pollutants in groundwater. An implicit difference scheme is developed for the numerical solution of a non-stationary fractional differential equation, which is an analogue of the well-known implicit Crank-Nicholson difference scheme. The system of difference equations is presented in matrix form. The solution of the problem is reduced to the multiple solution of a tridiagonal system of linear algebraic equations by the tridiagonal matrix algorithm. The results of evaluating the spread of pollutant in groundwater based on the numerical method for model examples are presented. The concentrations of the substance obtained on the basis of the analytical and numerical solutions of the unsteady one-dimensional fractional differential equation are compared. The results obtained using the proposed method and on the basis of the well-known analytical solution of the fractional differential equation are in fairly good agreement with each other. The relative error is on average 9%. In contrast to the well-known analytical solution, the developed numerical method can be used to model the spread of pollutants in groundwater, taking into account their biodegradation.

The approximate solution for solving a class of composition fractional order optimal control problems (FOCPs) is suggested and studied in detail. However, the properties of Caputo and Riemann-Liouville derivatives are also given with complete details on Chebyshev approximation function to approximate the solution of fractional differential equation with different approach. Also, the relation between Caputo and Riemann-Liouville of fractional derivative took a big role for simplifying the fractional differential equation that represents the constraints of optimal control problems. The approximate solutions are defined on interval [0,1] and are compared with the exact solution of order one which is an important condition to support the working method. Finally, illustrative examples are included to confirm the efficiency and accuracy of the proposed method.

In this paper we describe a method to solve the linear non-homogeneous fractional differential equations (FDE), composed with Jumarie type Fractional Derivative, and describe this method developed by us, to find out Particular Integrals, for several types of forcing functions. The solutions are obtained in terms of Mittag-Leffler functions, fractional sine and cosine functions. We have used our earlier developed method of finding solution to homogeneous FDE composed via Jumarie fractional derivative, and extended this to non-homogeneous FDE. We have demonstrated these developed methods with few examples of FDE, and also applied in fractional damped forced differential equation. This method proposed by us is useful as it is having conjugation with the classical methods of solving non-homogeneous linear differential equations, and also useful in understanding physical systems described by FDE.

The traditional linear generation fuzzy fractional differential equation parameter identification algorithm lacks the update of parameter identification process, has large amount of calculation, slow convergence speed of parameter identification and strong dependence on initial values. In this paper, a new intelligent recognition algorithm for linearly generated fuzzy fractional differential equations is proposed. The parameters of the equation are re expressed by the constant level set. The piecewise constant level set algorithm based on equation parameter intelligent identification is used to solve the steady-state solution of fractional differential equation, and the non convergence problem caused by too much calculation is solved. A new algorithm scheme for linearly generating fuzzy fractional differential equations is established, the constraints of the level set of the differential equations are calculated, and the updated algorithm for parameter identification of the equation is obtained. The evolutionary algorithm is used to solve the updating algorithm to realize the intelligent identification algorithm of linear fractional fuzzy differential equations. Experimental results show that the algorithm has the advantages of fast convergence speed, high calculation accuracy and low initial value.

The aim of this paper is to extend the iterative method based on the DGJM method of solving functional equations, to solve the fractional differential equations, where the order of derivative is taken in Caputo’s sense. The iterative procedure is explained and demonstrated by solving non-linear time fractional partial differential equations like Heat equation, Burger’s equation, Fokker Planck equation, Korteweg-de Vries (KdV) equation and Klien-Gordon equation. The scheme of iteration is also extended to solve the system of Drinfeld-Sokolov-Wilson equations and coupled Jaulent-Miodek equations. Graphs are used to depict the accuracy of the method and absolute errors between exact and approximate solutions are tabulated to ensure that the proposed scheme is both computationally intriguing and simple to implement.

Solving differential equations of fractional order using an optimization technique based on training artificial neural network

The solutions of fractional differential equations (FDEs) have a natural singularity at the initial point. The accuracy of their numerical solutions is lower than the accuracy of the numerical solutions of FDEs whose solutions are differentiable functions. In the present paper we propose a method for improving the accuracy of the numerical solutions of ordinary linear FDEs with constant coefficients which uses the fractional Taylor polynomials of the solutions. The numerical solutions of the two-term and three-term FDEs are studied in the paper.

&lt;abstract&gt;&lt;p&gt;In the past, the existence and uniqueness of the solutions of fractional differential equations have been investigated by many researchers theoretically in various approaches in the literature. In this paper, there is no discussion of the existence of solutions for the nonlinear differential equations with fractal fractional operators. The objective of this work is to present novel contraction approaches, notably the $ \varpropto $-$ \psi $-contraction $ \varpropto $-type of the $ \tilde{\texttt{F}} $-contraction, within the context of $ \hat{F} $-metric and orbital metric spaces. The aim of this study is to illustrate certain fixed point theorems that offer a new and direct approach to establish the existence and uniqueness of the solution to the general partial differential equations by employing the fractal fractional operators.&lt;/p&gt;&lt;/abstract&gt;

This paper is devoted to studying on the locations of zeros of related integral operators and the solutions of some ordinary and fractional differential equations. We generalize Sturm and Picone's theorems and Leighton and Levin's criteria. Moreover, we share some oscillation and disconjugacy criteria for the solutions of ordinary second‐order Sturm–Liouville and fractional differential equations. Finally, we introduce some properties of the solutions of fractional differential equations.

Herein, two numerical algorithms for solving some linear and nonlinear fractional-order differential equations are presented and analyzed. For this purpose, a novel operational matrix of fractional-order derivatives of Fibonacci polynomials was constructed and employed along with the application of the tau and collocation spectral methods. The convergence and error analysis of the suggested Fibonacci expansion were carefully investigated. Some numerical examples with comparisons are presented to ensure the efficiency, applicability and high accuracy of the proposed algorithms. Two accurate semi-analytic polynomial solutions for linear and nonlinear fractional differential equations are the result.

This paper presents a numerical scheme for the solutions of Fractional Differential Equations (FDEs) of order α, 1 &amp;lt; α &amp;lt; 2 which have been expressed in terms of Caputo Fractional Derivative (FD). In this scheme, the properties of the Caputo derivative are used to reduce an FDE into a Volterra type integral equation. The entire domain is divided into several small domains, and the distribution of the unknown function over the domain is expressed in terms of the function values and its slopes at the node points. These approximations are then substituted into the Volterra type integral equation to reduce it to algebraic equations. Since the method enforces the continuity of variables at the node points, it provides a solution that is continuous and with a slope that is also continuous over the entire domain. The method is used to solve a simple FDE using two different types of polynomials, namely cubic order and fractional order. Results obtained using both types of polynomials agree well with the analytical results. However, the fractional order polynomials give more accurate results than the cubic order polynomials do. This suggests that for the numerical solutions of FDEs fractional order polynomials may be more suitable than the integer order polynomials. A series of numerical studies suggests that the algorithm is stable.

Part I: Transmutations, Integral Equations and Special Functions.- Some Recent Developments in the Transmutation Operator Approach.- Transmutation Operators and Their Applications.- Hankel Generalized Convolutions with the Associated Legendre Functions in the Kernel and Their Applications.- Second Type Neumann Series Related to Nicholson's and to Dixon-Ferrar Formula.- On Some Generalizations of the Properties of the Multidimensional Generalized Erdelyi-Kober Operators and Their Applications.- Alternative Approach to Miller-Paris Transformations and Their Extensions.- Transmutation Operators For Ordinary Dunkl-Darboux Operators.- Theorems on Restriction of Fourier-Bessel and Multidimensional Bessel Transforms to Spherical Surfaces.- Necessary Condition for the Existence of an Intertwining Operator and Classification of Transmutations on Its Basis.- Polynomial Quantization on Line Bundles.- Fourier-Bessel Transforms of Measures and Qualitative Properties of Solutions of Singular Differential Equations.- Inversion of Hyperbolic B-Potentials.- One-Dimensional and Multi-Dimensional Integral Transforms of Buschman-Erdelyi Type with Legendre Functions in Kernels.- Distributions, Non-smooth Manifolds, Transmutations and Boundary Value Problems.- Part II: Transmutations in ODEs, Direct and Inverse Problems.- On a Transformation Operator Approach in the Inverse Spectral Theory of Integral and Integro-Differential Operators.- Expansion in Terms of Appropriate Functions and Transmutations.- Transmutation Operators as a Solvability Concept of Abstract Singular Equations.- On the Bessel-Wright Operator and Transmutation with Applications.- On a Method of Solving Integral Equation of Carleman Type on the Pair of Segments.- Transmutation Operators Boundary Value Problems.- Solution of Inverse Problems for Differential Operators with Delay.- Part III: Transmutations for Partial and Fractional Differential Equations.- Transmutations of the Composed Erdelyi-Kober Fractional Operators and Their Applications.- Distributed Order Equations in Banach Spaces with Sectorial Operators.- Transformation Operators for Fractional Order Ordinary Differential Equations and Their applications.- Strong Solutions of Semilinear Equations with Lower Fractional Derivatives.- Mean Value Theorems and Properties of Solutions of Linear Differential Equations.- Transmutations for Multi-Term Fractional Operators.- Fractional Bessel Integrals and Derivatives on Semi-axes.- The Fractional Derivative Expansion Method in Nonlinear Dynamics of Structures: A Memorial Essay.- Boundary Value Problem with Integral Condition for the Mixed Type Equation with a Singular Coefficient.

This paper presents a numerical scheme for the solutions of Fractional Differential Equations (FDEs) of order α, 1&lt;α&lt;2 which have been expressed in terms of Caputo Fractional Derivative (FD). In this scheme, the properties of the Caputo derivative are used to reduce an FDE into a Volterra-type integral equation. The entire domain is divided into several small domains, and the distribution of the unknown function over the domain is expressed in terms of the function values and its slopes at the node points. These approximations are then substituted into the Volterra-type integral equation to reduce it to algebraic equations. Since the method enforces the continuity of variables at the node points, it provides a solution that is continuous and with a slope that is also continuous over the entire domain. The method is used to solve two problems, linear and nonlinear, using two different types of polynomials, cubic order and fractional order. Results obtained using both types of polynomials agree well with the analytical results for problem 1 and the numerical results obtained using another scheme for problem 2. However, the fractional order polynomials give more accurate results than the cubic order polynomials do. This suggests that for the numerical solutions of FDEs fractional order polynomials may be more suitable than the integer order polynomials. A series of numerical studies suggests that the algorithm is stable.