scholarly journals Extension of Khan’s Homotopy Transformation Method via Optimal Parameter for Differential Difference Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Mohamed S. Mohamed ◽  
Khaled A. Gepreel ◽  
Faisal A. Al-Malki ◽  
Nouf Altalhi
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Numerical Solutions ◽  
Optimal Parameter ◽  
Transformation Method ◽  
Series Solutions ◽  
Transform Method ◽  
Homotopy Perturbation ◽  
Physical Problems ◽  
Differential Difference Equation

A new scheme, deduced from Khan’s homotopy perturbation transform method (HPTM) (Khan, 2014; Khan and Wu, 2011) via optimal parameter, is presented for solving nonlinear differential difference equations. Simple but typical examples are given to illustrate the validity and great potential of Khan’s homotopy perturbation transform method (HPTM) via optimal parameter in solving nonlinear differential difference equation. The numerical solutions show that the proposed method is very efficient and computationally attractive. It provides more realistic series solutions that converge very rapidly for nonlinear real physical problems. The results reveal that the method is very effective and simple. This method gives more reliable results as compared to other existing methods available in the literature. The numerical results demonstrate the validity and applicability of the method.

Exact Solutions for Fractional Differential-Difference Equations by (G'/G)-Expansion Method with Modified Riemann-Liouville Derivative

2016 ◽  
Vol 8 (2) ◽  
pp. 293-305 ◽  
Author(s):  
Ahmet Bekir ◽  
Ozkan Guner ◽  
Burcu Ayhan ◽  
Adem C. Cevikel
Keyword(s):  
Difference Equation ◽  
Exact Solutions ◽  
Difference Equations ◽  
Applied Mathematics ◽  
Expansion Method ◽  
New Exact Solutions ◽  
Liouville Derivative ◽  
Nonlinear Lattice ◽  
Fractional Differential ◽  
Differential Difference Equation

AbstractIn this paper, the (G'/G)-expansion method is suggested to establish new exact solutions for fractional differential-difference equations in the sense of modified Riemann-Liouville derivative. The fractional complex transform is proposed to convert a fractional partial differential difference equation into its differential difference equation of integer order. With the aid of symbolic computation, we choose nonlinear lattice equations to illustrate the validity and advantages of the algorithm. It is shown that the proposed algorithm is effective and can be used for many other nonlinear lattice equations in mathematical physics and applied mathematics.

A Fractional Model of Gas Dynamics Equations and its Analytical Approximate Solution Using Laplace Transform

2012 ◽  
Vol 67 (6-7) ◽  
pp. 389-396 ◽  
Author(s):  
Sunil Kumar ◽  
Huseyin Kocak ◽  
Ahmet Yıldırım
Keyword(s):  
Laplace Transform ◽  
Gas Dynamics ◽  
Numerical Solutions ◽  
Mass Conservation ◽  
Momentum Conservation ◽  
Gas Dynamics Equations ◽  
Transform Method ◽  
Conservation Of Energy ◽  
Homotopy Perturbation ◽  
He’S Polynomials

In this study, the homotopy perturbation transform method (HPTM) is performed to give approximate and analytical solutions of nonlinear homogenous and non-homogenous time-fractional gas dynamics equations. Gas dynamics equations are based on the physical laws of conservation, namely, the laws of conservation of mass, conservation of momentum, conservation of energy etc. The HPTM is a combined form of the Laplace transform, the homotopy perturbation method, and He’s polynomials. The nonlinear terms can be easily handled by the use of He’s polynomials. The numerical solutions obtained by the proposed method indicate that the approach is easy to implement and accurate. Some numerical illustrations are given. These results reveal that the proposed method is very effective and simple to perform

scholarly journals A transformation to linearity of some non-linear differential-difference equations

1982 ◽  
Vol 23 (4) ◽  
pp. 416-419
Author(s):  
A. J. Bracken
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Non Linear ◽  
Differential Difference Equation ◽  
Linear Differential

AbstractThe problem of solving a differential-difference equation with quadratic non-linearities of a certain type is reduced to the problem of solving an associated linear differential-difference equation.

scholarly journals Numerical Solutions of Nonlinear Fractional Partial Differential Equations Arising in Spatial Diffusion of Biological Populations

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Jagdev Singh ◽  
Devendra Kumar ◽  
Adem Kılıçman
Keyword(s):  
Homotopy Perturbation Method ◽  
Numerical Solutions ◽  
Spatial Diffusion ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Transform Method ◽  
Homotopy Perturbation ◽  
Breeding Territory ◽  
Sumudu Transform ◽  
User Friendly

The main aim of this work is to present a user friendly numerical algorithm based on homotopy perturbation Sumudu transform method for nonlinear fractional partial differential arising in spatial diffusion of biological populations in animals. The movements are made generally either by mature animals driven out by invaders or by young animals just reaching maturity moving out of their parental territory to establish breeding territory of their own. The homotopy perturbation Sumudu transform method is a combined form of the Sumudu transform method and homotopy perturbation method. The obtained results are compared with Sumudu decomposition method. The numerical solutions obtained by the proposed method indicate that the approach is easy to implement and accurate. These results reveal that the proposed method is computationally very attractive.

scholarly journals Global Dynamics of Sixth-Order Fuzzy Difference Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-16
Author(s):  
Abdul Khaliq ◽  
Muhammad Adnan ◽  
Abdul Qadeer Khan
Keyword(s):  
Difference Equation ◽  
Equilibrium Point ◽  
Difference Equations ◽  
Numerical Solutions ◽  
Nonnegative Solution ◽  
Real Life ◽  
Characterization Theorem ◽  
Sixth Order ◽  
Global Dynamics ◽  
Life Problems

Across many fields, such as engineering, ecology, and social science, fuzzy differences are becoming more widely used; there is a wide variety of applications for difference equations in real-life problems. Our study shows that the fuzzy difference equation of sixth order has a nonnegative solution, an equilibrium point and asymptotic behavior. y i + 1 = D y i − 1 y i − 2 / E + F y i − 3 + G y i − 4 + H y i − 5 , i = 0,1,2 , … , where y i is the sequence of fuzzy numbers and the parameter D , E , F , G , H and the initial condition y − 5 , y − 4 , y − 3 , y − 2 , y − 1 , y 0 are nonnegative fuzzy number. The characterization theorem is used to convert each single fuzzy difference equation into a set of two crisp difference equations in a fuzzy environment. So, a pair of crisp difference equations is formed by converting the difference equation. The stability of the equilibrium point of a fuzzy system has been evaluated. By using variational iteration techniques and inequality skills as well as a theory of comparison for fuzzy difference equations, we investigated the governing equation dynamics, such as its boundedness, existence, and local and global stability analysis. In addition, we provide some numerical solutions for the equation describing the system to verify our results.

scholarly journals GDTM-Padé technique for the non-linear differential-difference equation

2013 ◽  
Vol 17 (5) ◽  
pp. 1305-1310
Author(s):  
Jun-Feng Lu
Keyword(s):  
Difference Equation ◽  
Solitary Wave ◽  
Difference Equations ◽  
Wave Solution ◽  
Solitary Wave Solution ◽  
Approximate Solutions ◽  
Lattice Equation ◽  
Non Linear ◽  
Differential Difference Equation ◽  
Linear Differential

This paper focuses on applying the GDTM-Pad? technique to solve the non-linear differential-difference equation. The bell-shaped solitary wave solution of Belov-Chaltikian lattice equation is considered. Comparison between the approximate solutions and the exact ones shows that this technique is an efficient and attractive method for solving the differential-difference equations.

scholarly journals Results on the solutions of several second order mixed type partial differential difference equations

2021 ◽  
Vol 7 (2) ◽  
pp. 1907-1924
Author(s):  
Wenju Tang ◽  
◽  
Keyu Zhang ◽  
Hongyan Xu ◽  
◽  
...  
Keyword(s):  
Difference Equation ◽  
Finite Order ◽  
Difference Equations ◽  
Mixed Type ◽  
Existence Of Solutions ◽  
Second Order ◽  
Entire Solutions ◽  
Partial Differential ◽  
Differential Difference Equation ◽  
Positive Integers

<abstract><p>This article is concerned with the existence of entire solutions for the following complex second order partial differential-difference equation</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \left(\frac{\partial^2 f(z_1, z_2)}{\partial z_1^2}+\frac{\partial^2 f(z_1, z_2)}{\partial z_2^2}\right)^{l}+f(z_1+c_1, z_2+c_2)^{k} = 1, $\end{document} </tex-math></disp-formula></p> <p>where $ c_1, c_2 $ are constants in $ \mathbb{C} $ and $ k, l $ are positive integers. In addition, we also investigate the forms of finite order transcendental entire solutions for several complex second order partial differential-difference equations of Fermat type, and obtain some theorems about the existence and the forms of solutions for the above equations. Meantime, we give some examples to explain the existence of solutions for some theorems in some cases. Our results are some generalizations of the previous theorems given by Qi <sup>[<xref ref-type="bibr" rid="b23">23</xref>]</sup>, Xu and Cao <sup>[<xref ref-type="bibr" rid="b35">35</xref>]</sup>, Liu, Cao and Cao <sup>[<xref ref-type="bibr" rid="b17">17</xref>]</sup>.</p></abstract>

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