scholarly journals On local fractional operators View of computational complexity: Diffusion and relaxation defined on cantor sets

2016 ◽  
Vol 20 (suppl. 3) ◽  
pp. 755-767 ◽  
Author(s):  
Xiao-Jun Yang ◽  
Zhi-Zhen Zhang ◽  
Tenreiro Machado ◽  
Dumitru Baleanu
Keyword(s):  
Computational Complexity ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Complex Systems ◽  
Exact Solutions ◽  
Fractional Derivatives ◽  
Cantor Sets ◽  
Fractional Operators ◽  
Fractional Partial Differential Equations ◽  
Comparative Results

This paper treats the description of non-differentiable dynamics occurring in complex systems governed by local fractional partial differential equations. The exact solutions of diffusion and relaxation equations with Mittag-Leffler and exponential decay defined on Cantor sets are calculated. Comparative results with other versions of the local fractional derivatives are discussed.

scholarly journals An efficient new iterative method for finding exact solutions of nonlinear time-fractional partial differential equations

2011 ◽  
Vol 16 (4) ◽  
pp. 403-414 ◽  
Author(s):  
Hüseyin Koçak ◽  
Ahmet Yıldırım
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Iterative Method ◽  
Exact Solutions ◽  
Fractional Derivatives ◽  
Convergent Series ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
New Iterative Method

In this paper, a new iterative method (NIM) is used to obtain the exact solutions of some nonlinear time-fractional partial differential equations. The fractional derivatives are described in the Caputo sense. The method provides a convergent series with easily computable components in comparison with other existing methods.

scholarly journals Oscillation criteria of certain fractional partial differential equations

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Di Xu ◽  
Fanwei Meng
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Derivatives ◽  
Sufficient Conditions ◽  
Riccati Transformation ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Oscillation Criteria ◽  
Generalized Riccati Transformation

Abstract In this article, we regard the generalized Riccati transformation and Riemann–Liouville fractional derivatives as the principal instrument. In the proof, we take advantage of the fractional derivatives technique with the addition of interval segmentation techniques, which enlarge the manners to demonstrate the sufficient conditions for oscillation criteria of certain fractional partial differential equations.

Numerical solutions for the Robin time-fractional partial differential equations of heat and fluid flows based on the reproducing kernel algorithm

2018 ◽  
Vol 28 (4) ◽  
pp. 828-856 ◽  
Author(s):  
Omar Abu Arqub
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Reproducing Kernel ◽  
Numerical Solutions ◽  
Fluid Flows ◽  
Kernel Functions ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Content Type

Purpose The purpose of this study is to introduce the reproducing kernel algorithm for treating classes of time-fractional partial differential equations subject to Robin boundary conditions with parameters derivative arising in fluid flows, fluid dynamics, groundwater hydrology, conservation of energy, heat conduction and electric circuit. Design/methodology/approach The method provides appropriate representation of the solutions in convergent series formula with accurately computable components. This representation is given in the W(Ω) and H(Ω) inner product spaces, while the computation of the required grid points relies on the R(y,s) (x, t) and r(y,s) (x, t) reproducing kernel functions. Findings Numerical simulation with different order derivatives degree is done including linear and nonlinear terms that are acquired by interrupting the n-term of the exact solutions. Computational results showed that the proposed algorithm is competitive in terms of the quality of the solutions found and is very valid for solving such time-fractional models. Research limitations/implications Future work includes the application of the reproducing kernel algorithm to highly nonlinear time-fractional partial differential equations such as those arising in single and multiphase flows. The results will be published in forthcoming papers. Practical implications The study included a description of fundamental reproducing kernel algorithm and the concepts of convergence, and error behavior for the reproducing kernel algorithm solvers. Results obtained by the proposed algorithm are found to outperform in terms of accuracy, generality and applicability. Social implications Developing analytical and numerical methods for the solutions of time-fractional partial differential equations is a very important task owing to their practical interest. Originality/value This study, for the first time, presents reproducing kernel algorithm for obtaining the numerical solutions of some certain classes of Robin time-fractional partial differential equations. An efficient construction is provided to obtain the numerical solutions for the equations, along with an existence proof of the exact solutions based upon the reproducing kernel theory.

scholarly journals Exact Solutions with Variable Coefficient Function Forms for Conformable Fractional Partial Differential Equations by an Auxiliary Equation Method

2018 ◽  
Vol 2018 ◽  
pp. 1-8 ◽  
Author(s):  
Fanwei Meng ◽  
Qinghua Feng
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Exact Solutions ◽  
Variable Coefficient ◽  
Equation Method ◽  
Auxiliary Equation ◽  
Coefficient Function ◽  
Fractional Partial Differential Equations ◽  
Auxiliary Equation Method

In this paper, an auxiliary equation method is introduced for seeking exact solutions expressed in variable coefficient function forms for fractional partial differential equations, where the concerned fractional derivative is defined by the conformable fractional derivative. By the use of certain fractional transformation, the fractional derivative in the equations can be converted into integer order case with respect to a new variable. As for applications, we apply this method to the time fractional two-dimensional Boussinesq equation and the space-time fractional (2+1)-dimensional breaking soliton equation. As a result, some exact solutions including variable coefficient function solutions as well as solitary wave solutions for the two equations are found.

scholarly journals The Jacobi Elliptic Equation Method for Solving Fractional Partial Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Bin Zheng ◽  
Qinghua Feng
Keyword(s):  
Partial Differential Equations ◽  
Elliptic Equation ◽  
Differential Equations ◽  
Exact Solutions ◽  
Equation Method ◽  
Hyperbolic Function ◽  
Jacobi Elliptic Function ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Liouville Derivative

Based on a nonlinear fractional complex transformation, the Jacobi elliptic equation method is extended to seek exact solutions for fractional partial differential equations in the sense of the modified Riemann-Liouville derivative. For demonstrating the validity of this method, we apply it to solve the space fractional coupled Konopelchenko-Dubrovsky (KD) equations and the space-time fractional Fokas equation. As a result, some exact solutions for them including the hyperbolic function solutions, trigonometric function solutions, rational function solutions, and Jacobi elliptic function solutions are successfully found.

scholarly journals The generalized Kudryashov method for the nonlinear fractional partial differential equations with the beta-derivative

2020 ◽  
Vol 66 (6 Nov-Dec) ◽  
pp. 771
Author(s):  
Yusuf Gurefe
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Exact Solutions ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Generalized Kudryashov Method ◽  
Kudryashov Method

In this article, we consider the exact solutions of the Hunter-Saxton and Schrödinger equations defined by Atangana's comformable derivative using the general Kudryashov method. Firstly, Atangana's comformable fractional derivative and its properties are included. Then, by introducing the generalized Kudryashov method, exact solutions of nonlinear fractional partial differential equations (FPDEs), which can be expressed with the comformable derivative of Atangana, are classified. Looking at the results obtained, it is understood that the generalized Kudryashov method can yield important results in obtaining the exact solutions of FPDEs containing beta-derivatives.

scholarly journals New exact solutions of fractional Hirota-Satsuma coupled Korteweg-de Vries equations

2015 ◽  
Vol 19 (4) ◽  
pp. 1173-1176 ◽  
Author(s):  
Lian-Xiang Cui ◽  
Li-Mei Yan ◽  
Yan-Qin Liu
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Function Method ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Fractional Complex Transform ◽  
New Exact Solutions ◽  
Non Linear ◽  
De Vries

An improved extended tg-function method, which combines the fractional complex transform and the extended tanh-function method, is applied to find exact solutions of non-linear fractional partial differential equations. Generalized Hirota-Satsuma coupled Korteweg-de Vries equations are used as an example to elucidate the effectiveness and simplicity of the method.

scholarly journals A modified invariant subspace method for solving partial differential equations with non-singular kernel fractional derivatives

2020 ◽  
Vol 5 (2) ◽  
pp. 35-48 ◽  
Author(s):  
Kamal Ait Touchent ◽  
Zakia Hammouch ◽  
Toufik Mekkaoui
Keyword(s):  
Partial Differential Equations ◽  
Numerical Methods ◽  
Differential Equations ◽  
Exact Solutions ◽  
Invariant Subspace ◽  
Fractional Derivatives ◽  
Subspace Method ◽  
Partial Differential ◽  
Singular Kernels ◽  
Fractional Differential

AbstractIn this work, the well known invariant subspace method has been modified and extended to solve some partial differential equations involving Caputo-Fabrizio (CF) or Atangana-Baleanu (AB) fractional derivatives. The exact solutions are obtained by solving the reduced systems of constructed fractional differential equations. The results show that this method is very simple and effective for constructing explicit exact solutions for partial differential equations involving new fractional derivatives with nonlocal and non-singular kernels, such solutions are very useful to validate new numerical methods constructed for solving partial differential equations with CF and AB fractional derivatives.

scholarly journals Improved Fractional Subequation Method and Exact Solutions to Fractional Partial Differential Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-18
Author(s):  
Jun Jiang ◽  
Yuqiang Feng ◽  
Shougui Li
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Burgers Equation ◽  
Space Time ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Biological Population ◽  
Regularized Long Wave Equation ◽  
Generalized Time

In this paper, the improved fractional subequation method is applied to establish the exact solutions for some nonlinear fractional partial differential equations. Solutions to the generalized time fractional biological population model, the generalized time fractional compound KdV-Burgers equation, the space-time fractional regularized long-wave equation, and the (3+1)-space-time fractional Zakharov-Kuznetsov equation are obtained, respectively.

Sign in / Sign up

Export Citation Format

Share Document