scholarly journals Numerical Solution for Complex Systems of Fractional Order

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Rabha W. Ibrahim
Keyword(s):  
Complex Systems ◽  
Numerical Solution ◽  
Fractional Order ◽  
Analytic Solution ◽  
Approximate Solutions ◽  
Fractional Operators ◽  
Homotopy Perturbation

By using a complex transform, we impose a system of fractional order in the sense of Riemann-Liouville fractional operators. The analytic solution for this system is discussed. Here, we introduce a method of homotopy perturbation to obtain the approximate solutions. Moreover, applications are illustrated.

scholarly journals A Modified Techniques of Fractional-Order Cauchy-Reaction Diffusion Equation via Shehu Transform

2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Mounirah Areshi ◽  
A. M. Zidan ◽  
Rasool Shah ◽  
Kamsing Nonlaopon
Keyword(s):  
Fractional Order ◽  
Reaction Diffusion ◽  
Approximate Solutions ◽  
Transformation Method ◽  
Reaction Diffusion Equations ◽  
Reaction Diffusion Equation ◽  
Closed Form Solutions ◽  
Homotopy Perturbation ◽  
In Series ◽  
The Given

In this article, the iterative transformation method and homotopy perturbation transformation method are applied to calculate the solution of time-fractional Cauchy-reaction diffusion equations. In this technique, Shehu transformation is combined of the iteration and the homotopy perturbation techniques. Four examples are examined to show validation and the efficacy of the present methods. The approximate solutions achieved by the suggested methods indicate that the approach is easy to apply to the given problems. Moreover, the solution in series form has the desire rate of convergence and provides closed-form solutions. It is noted that the procedure can be modified in other directions of fractional order problems. These solutions show that the current technique is very straightforward and helpful to perform in applied sciences.

scholarly journals Numerical Solution of Nonlinear Fractional Volterra Integro-Differential Equations via Bernoulli Polynomials

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Emran Tohidi ◽  
M. M. Ezadkhah ◽  
S. Shateyi
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Linear Combination ◽  
Fractional Order ◽  
Bernoulli Polynomials ◽  
Approximate Solutions ◽  
Computational Approach ◽  
Numerical Results ◽  
Algebraic Equations

This paper presents a computational approach for solving a class of nonlinear Volterra integro-differential equations of fractional order which is based on the Bernoulli polynomials approximation. Our method consists of reducing the main problems to the solution of algebraic equations systems by expanding the required approximate solutions as the linear combination of the Bernoulli polynomials. Several examples are given and the numerical results are shown to demonstrate the efficiency of the proposed method.

scholarly journals On Semianalytical Study of Fractional-Order Kawahara Partial Differential Equation with the Homotopy Perturbation Method

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Muhammad Sinan ◽  
Kamal Shah ◽  
Zareen A. Khan ◽  
Qasem Al-Mdallal ◽  
Fathalla Rihan
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Solitary Wave ◽  
Perturbation Method ◽  
Fractional Order ◽  
Homotopy Perturbation Method ◽  
Nonlinear Partial Differential Equation ◽  
Approximate Solutions ◽  
Partial Differential ◽  
Homotopy Perturbation

In this study, we investigate the semianalytic solution of the fifth-order Kawahara partial differential equation (KPDE) with the approach of fractional-order derivative. We use Caputo-type derivative to investigate the said problem by using the homotopy perturbation method (HPM) for the required solution. We obtain the solution in the form of infinite series. We next triggered different parametric effects (such as x, t, and so on) on the structure of the solitary wave propagation, demonstrating that the breadth and amplitude of the solitary wave potential may alter when these parameters are changed. We have demonstrated that He’s approach is highly effective and powerful for the solution of such a higher-order nonlinear partial differential equation through our calculations and simulations. We may apply our method to an additional complicated problem, particularly on the applied side, such as astrophysics, plasma physics, and quantum mechanics, to perform complex theoretical computation. Graphical presentation of few terms approximate solutions are given at different fractional orders.

scholarly journals Numerical solution of a fractal-fractional order chaotic circuit system

2021 ◽  
Vol 67 (5 Sep-Oct) ◽  
Author(s):  
Muhammad Altaf Khan ◽  
Abdon Atangana ◽  
Taseer Muhammad ◽  
Ebraheem Alzahrani
Keyword(s):  
Numerical Solution ◽  
Fractional Order ◽  
Research Area ◽  
Fractional Operators ◽  
Chaotic Circuit ◽  
Order Model ◽  
Fractional Order Model ◽  
Chaotic Model ◽  
Important Research Area ◽  
Fractional Orders

The dynamical system has an important research area and due to its wide applications many researchers and scientists working to develop new model and techniques for their solution. We present in this work the dynamics of a chaotic model in the presence of newly introduced fractal-fractional operators. The model is formulated initially in ordinary differential equations and then we utilize the fractal-fractional (FF) in power law, exponential and Mittag-Leffler to generalize the model. For each fractal-fractional order model, we briefly study its numerical solution via the numerical algorithm. We present some graphical results with arbitrary order of fractal and fractional orders, and present that these operators provide different chaotic attractors for different fractal and fractional order values. The graphical results demonstrate the effectiveness of the fractal-fractional operators.

scholarly journals AN ACCURATE SOLUTION TO THE LOTKA-VOLTERRA EQUATIONS BY MODIFIED HOMOTOPY PERTURBATION METHOD

2012 ◽  
Vol 09 ◽  
pp. 326-333 ◽  
Author(s):  
M. S. H. CHOWDHURY ◽  
M. M. RAHMAN
Keyword(s):  
Perturbation Method ◽  
Numerical Solution ◽  
Nonlinear Equations ◽  
Homotopy Perturbation Method ◽  
Approximate Solutions ◽  
Accurate Solution ◽  
Volterra Equations ◽  
Algebraic Equations ◽  
Time Intervals ◽  
Homotopy Perturbation

In this paper, we suggest a method to solve the multispecies Lotka-Voltera equations. The suggested method, which we call modified homotopy perturbation method, can be considered as an extension of the homotopy perturbation method (HPM) which is very efficient in solving a varety of differential and algebraic equations. The HPM is modified in order to obtain the approximate solutions of Lotka-Voltera equation response in a sequence of time intervals. In particular, the example of two species is considered. The accuracy of this method is examined by comparison with the numerical solution of the Runge-Kutta-Verner method. The results prove that the modified HPM is a powerful tool for the solution of nonlinear equations.

scholarly journals NUMERICAL SOLUTION OF THE PARTIAL DIFFERENTIAL EQUATION USING RANDOMLY GENERATED FINITE GRIDS AND TWO-DIMENSIONAL FRACTIONAL-ORDER LEGENDRE FUNCTION

2021 ◽  
Author(s):  
Sanaullah Mastoi
Keyword(s):  
Finite Difference ◽  
Numerical Solution ◽  
Fractional Order ◽  
Numerical Solutions ◽  
Approximate Solutions ◽  
Polar Coordinates ◽  
Legendre Functions ◽  
Two Dimensional ◽  
Cartesian Coordinates ◽  
Uniform Grids

There are various methods to solve the physical life problem involving engineering, scientific and biological systems. It is found that numerical methods are approximate solutions. In this way, randomly generated finite difference grids achieve an approximation with fewer iterations. The idea of randomly generated grids in cartesian coordinates and polar form are compared with the exact, iterative method, uniform grids, and approximate solutions in a generalized expansion form of two-dimensional fractional-order Legendre functions. The most ideal and benchmarking method is the finite difference method over randomly generated grids on Cartesian coordinates, polar coordinates used for numerical solutions. This concept motivates the investigation of the effects of the randomly generated meshes. The two-dimensional equation is solved over randomly generated meshes to test randomly generated grids and the implementation. The feasibility of the numerical solution is analyzed by comparing simulation profiles.

scholarly journals Convergence of the Generalized Homotopy Perturbation Method for Solving Fractional Order Integro-Differential Equations

2014 ◽  
Vol 11 (4) ◽  
pp. 1637-1648
Author(s):  
Baghdad Science Journal
Keyword(s):  
Theoretical Analysis ◽  
Differential Equations ◽  
Perturbation Method ◽  
Fractional Order ◽  
Infinite Series ◽  
Homotopy Perturbation Method ◽  
Approximate Solutions ◽  
Powerful Method ◽  
Homotopy Perturbation ◽  
Fractional Order Integral

In this paper,the homtopy perturbation method (HPM) was applied to obtain the approximate solutions of the fractional order integro-differential equations . The fractional order derivatives and fractional order integral are described in the Caputo and Riemann-Liouville sense respectively. We can easily obtain the solution from convergent the infinite series of HPM . A theorem for convergence and error estimates of the HPM for solving fractional order integro-differential equations was given. Moreover, numerical results show that our theoretical analysis are accurate and the HPM can be considered as a powerful method for solving fractional order integro-diffrential equations.

scholarly journals Application of Asymptotic Homotopy Perturbation Method to Fractional Order Partial Differential Equation

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2215
Author(s):  
Haji Gul ◽  
Sajjad Ali ◽  
Kamal Shah ◽  
Shakoor Muhammad ◽  
Thanin Sitthiwirattham ◽  
...  
Keyword(s):  
Perturbation Method ◽  
Fractional Order ◽  
Homotopy Perturbation Method ◽  
Approximate Solutions ◽  
Order Partial Differential Equation ◽  
Two Dimensional ◽  
Numerical Treatment ◽  
Helmholtz Equations ◽  
Homotopy Perturbation ◽  
Linear Differential

In this article, we introduce a new algorithm-based scheme titled asymptotic homotopy perturbation method (AHPM) for simulation purposes of non-linear and linear differential equations of non-integer and integer orders. AHPM is extended for numerical treatment to the approximate solution of one of the important fractional-order two-dimensional Helmholtz equations and some of its cases . For probation and illustrative purposes, we have compared the AHPM solutions to the solutions from another existing method as well as the exact solutions of the considered problems. Moreover, it is observed that the symmetry or asymmetry of the solution of considered problems is invariant under the homotopy definition. Error estimates for solutions are also provided. The approximate solutions of AHPM are tabulated and plotted, which indicates that AHPM is effective and explicit.

scholarly journals A Novel Homotopy Perturbation Method with Applications to Nonlinear Fractional Order KdV and Burger Equation with Exponential-Decay Kernel

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Shabir Ahmad ◽  
Aman Ullah ◽  
Ali Akgül ◽  
Manuel De la Sen
Keyword(s):  
Perturbation Method ◽  
Fractional Order ◽  
Homotopy Perturbation Method ◽  
Approximate Solutions ◽  
Order Partial Differential Equation ◽  
The Novel ◽  
Homotopy Perturbation ◽  
Novel Approach ◽  
In Series ◽  
Novel Method

In this paper, we introduce the Yang transform homotopy perturbation method (YTHPM), which is a novel method. We provide formulae for the Yang transform of Caputo-Fabrizio fractional order derivatives. We derive an algorithm for the solution of Caputo-Fabrizio (CF) fractional order partial differential equation in series form and show its convergence to the exact solution. To demonstrate the novel approach, we include some examples with detailed solutions. We use tables and graphs to compare the exact and approximate solutions.

ANALYTICAL COMPARISON BETWEEN THE HOMOTOPY PERTURBATION METHOD AND VARIATIONAL ITERATION METHOD FOR DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER

2008 ◽  
Vol 22 (23) ◽  
pp. 4041-4058 ◽  
Author(s):  
ZAID ODIBAT ◽  
SHAHER MOMANI
Keyword(s):  
Differential Equations ◽  
Perturbation Method ◽  
Fractional Order ◽  
Iteration Method ◽  
Homotopy Perturbation Method ◽  
Variational Iteration Method ◽  
Approximate Solutions ◽  
Variational Iteration ◽  
Homotopy Perturbation ◽  
Different Types

Comparison of homotopy perturbation method (HPM) and variational iteration method (VIM) is made, revealing that the two methods can be used as alternative and equivalent methods for obtaining analytic and approximate solutions for different types of differential equations of fractional order. Furthermore, the former is more general and powerful than the latter. Numerical results show that the two approaches are easy to implement and accurate when applied to differential equations of fractional order.

Sign in / Sign up

Export Citation Format

Share Document