scholarly journals Resurgence of formal series solutions of nonlinear differential and difference equations

2016 ◽  
Vol 92 (8) ◽  
pp. 92-95
Author(s):  
Shingo Kamimoto
Keyword(s):  
Difference Equations ◽  
Formal Series ◽  
Series Solutions

scholarly journals Special formal series solutions of linear operator equations

2000 ◽  
Vol 210 (1-3) ◽  
pp. 3-25 ◽  
Author(s):  
Sergei A. Abramov ◽  
Marko Petkovšek ◽  
Anna Ryabenko
Keyword(s):  
Linear Operator ◽  
Formal Series ◽  
Series Solutions ◽  
Operator Equations ◽  
Linear Operator Equations

scholarly journals Application of the Variational Iteration Method to Strongly Nonlinear -Difference Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-12
Author(s):  
Hsuan-Ku Liu
Keyword(s):  
Approximate Solution ◽  
Time Scales ◽  
Difference Equations ◽  
Iteration Method ◽  
Variational Iteration Method ◽  
Numerical Results ◽  
Series Solutions ◽  
Nonlinear Difference Equations ◽  
Strongly Nonlinear ◽  
Variational Iteration

The theory of approximate solution lacks development in the area of nonlinear -difference equations. One of the difficulties in developing a theory of series solutions for the homogeneous equations on time scales is that formulas for multiplication of two -polynomials are not easily found. In this paper, the formula for the multiplication of two -polynomials is presented. By applying the obtained results, we extend the use of the variational iteration method to nonlinear -difference equations. The numerical results reveal that the proposed method is very effective and can be applied to other nonlinear -difference equations.

scholarly journals INVERSE FACTORIAL-SERIES SOLUTIONS OF DIFFERENCE EQUATIONS

2004 ◽  
Vol 47 (2) ◽  
pp. 421-448 ◽  
Author(s):  
A. B. Olde Daalhuis
Keyword(s):  
Hypergeometric Function ◽  
Difference Equations ◽  
Asymptotic Expansions ◽  
Subject Classification ◽  
Series Solutions ◽  
Linear Difference Equations ◽  
Secondary 33C05 ◽  
Rank One ◽  
Factorial Series ◽  
Primary 34E05

AbstractWe obtain inverse factorial-series solutions of second-order linear difference equations with a singularity of rank one at infinity. It is shown that the Borel plane of these series is relatively simple, and that in certain cases the asymptotic expansions incorporate simple resurgence properties. Two examples are included. The second example is the large $a$ asymptotics of the hypergeometric function ${}_2F_1(a,b;c;x)$.AMS 2000 Mathematics subject classification: Primary 34E05; 39A11. Secondary 33C05

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.

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