scholarly journals Approximate Solution of Generalized Modified b-Equation by Optimal Auxiliary Function Method

2022 ◽  
Vol 1 (1) ◽  
Keyword(s):  
Approximate Solution ◽  
Function Method ◽  
Auxiliary Function

THE EXACT TRAVELING WAVE SOLUTIONS AND THEIR BIFURCATIONS IN THE GARDNER AND GARDNER–KP EQUATIONS

2012 ◽  
Vol 22 (05) ◽  
pp. 1250126 ◽  
Author(s):  
FANG YAN ◽  
CUNCAI HUA ◽  
HAIHONG LIU ◽  
ZENGRONG LIU
Keyword(s):  
Traveling Wave ◽  
Function Method ◽  
Traveling Wave Solutions ◽  
Analytic Solutions ◽  
Auxiliary Function ◽  
Kp Equation ◽  
Gardner Equation ◽  
Exact Traveling Wave Solutions ◽  
Wave Solutions ◽  
Kp Equations

By using the method of dynamical systems, this paper studies the exact traveling wave solutions and their bifurcations in the Gardner equation. Exact parametric representations of all wave solutions as well as the explicit analytic solutions are given. Moreover, several series of exact traveling wave solutions of the Gardner–KP equation are obtained via an auxiliary function method.

scholarly journals Analytical Solution of System of Volterra Integral Equations Using OHAM

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Muhammad Akbar ◽  
Rashid Nawaz ◽  
Sumbal Ahsan ◽  
Dumitru Baleanu ◽  
Kottakkaran Sooppy Nisar
Keyword(s):  
Numerical Methods ◽  
Analytical Solution ◽  
Integral Equations ◽  
Function Method ◽  
Auxiliary Function ◽  
Volterra Integral Equations ◽  
Large Parameter ◽  
Reliable Technique ◽  
Present Technique ◽  
Optimal Homotopy Asymptotic Method

In this work, a reliable technique is used for the solution of a system of Volterra integral equations (VIEs), called optimal homotopy asymptotic method (OHAM). The proposed technique is successfully applied for the solution of different problems, and comparison is made with the relaxed Monto Carlo method (RMCM) and hat basis function method (HBFM). The comparisons show that the present technique is more suitable and reliable for the solution of a system of VIEs. The presented technique uses auxiliary function containing auxiliary constants, which control the convergence. Moreover, OHAM does not require discretization like other numerical methods and is also free from small or large parameter.

scholarly journals A new auxiliary function method for general constrained global optimization

Optimization ◽  
2013 ◽  
Vol 62 (2) ◽  
pp. 193-210 ◽  
Author(s):  
Z.Y. Wu ◽  
F.S. Bai ◽  
Y.J. Yang ◽  
M. Mammadov
Keyword(s):  
Global Optimization ◽  
Function Method ◽  
Auxiliary Function ◽  
Constrained Global Optimization

scholarly journals MEEF: A Minimum-Elimination-Escape Function Method for Multimodal Optimization Problems

2015 ◽  
Vol 2015 ◽  
pp. 1-16
Author(s):  
Lei Fan ◽  
Yuping Wang ◽  
Xiyang Liu ◽  
Liping Jia
Keyword(s):  
Function Method ◽  
Optimization Problems ◽  
Auxiliary Function ◽  
Local Optimum ◽  
Multimodal Optimization ◽  
Optimal Solutions ◽  
Mexican Hat ◽  
Global Optimal ◽  
Elimination Function ◽  
Made In

Auxiliary function methods provide us effective and practical ideas to solve multimodal optimization problems. However, improper parameter settings often cause troublesome effects which might lead to the failure of finding global optimal solutions. In this paper, a minimum-elimination-escape function method is proposed for multimodal optimization problems, aiming at avoiding the troublesome “Mexican hat” effect and reducing the influence of local optimal solutions. In the proposed method, the minimum-elimination function is constructed to decrease the number of local optimum first. Then, a minimum-escape function is proposed based on the minimum-elimination function, in which the current minimal solution will be converted to the unique global maximal solution of the minimum-escape function. The minimum-escape function is insensitive to its unique but easy to adopt parameter. At last, an minimum-elimination-escape function method is designed based on these two functions. Experiments on 19 widely used benchmarks are made, in which influences of the parameter and different initial points are analyzed. Comparisons with 11 existing methods indicate that the performance of the proposed algorithm is positive and effective.

scholarly journals A new auxiliary function method for systems of nonlinear equations

2015 ◽  
Vol 11 (2) ◽  
pp. 345-364 ◽  
Author(s):  
Zhiyou Wu ◽  
◽  
Fusheng Bai ◽  
Guoquan Li ◽  
Yongjian Yang ◽  
...  
Keyword(s):  
Nonlinear Equations ◽  
Function Method ◽  
Auxiliary Function ◽  
Systems Of Nonlinear Equations

scholarly journals A NEW TECHNIQUE FOR APPROXIMATE SOLUTION OF FRACTIONAL-ORDER PARTIAL DIFFERENTIAL EQUATIONS

Fractals ◽  
2021 ◽  
Author(s):  
LAIQ ZADA ◽  
RASHID NAWAZ ◽  
MOHAMMAD A. ALQUDAH ◽  
KOTTAKKARAN SOOPPY NISAR
Keyword(s):  
Approximate Solution ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Order ◽  
Auxiliary Function ◽  
Partial Differential ◽  
Convergence Control ◽  
A New Technique ◽  
Fifth Order ◽  
First Time

In the present paper, the optimal auxiliary function method (OAFM) has been extended for the first time to fractional-order partial differential equations (FPDEs) with convergence analysis. To find the accuracy of the OAFM, we consider the fractional-order KDV-Burger and fifth-order Sawada–Kotera equations as a test example. The proposed technique has auxiliary functions and convergence control parameters, which accelerate the convergence of the method. The other advantage of this method is that there is no need for a small or large parameter assumption, and it gives an approximate solution after only one iteration. Further, the obtained results have been compared with the exact solution through different graphs and tables, which shows that the proposed method is very effective and easy to implement for different FPDEs.

scholarly journals An extension of optimal auxiliary function method to fractional order high dimensional equations

2021 ◽  
Vol 60 (5) ◽  
pp. 4809-4818
Author(s):  
Rashid Nawaz ◽  
Laiq Zada ◽  
Farman Ullah ◽  
Hijaz Ahmad ◽  
Muhammad Ayaz ◽  
...  
Keyword(s):  
Fractional Order ◽  
Function Method ◽  
Auxiliary Function ◽  
High Dimensional
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