scholarly journals p-adic confluence of q-difference equations

2008 ◽  
Vol 144 (4) ◽  
pp. 867-919 ◽  
Author(s):  
Andrea Pulita
Keyword(s):  
Differential Equation ◽  
Difference Equation ◽  
Differential Equations ◽  
Difference Equations ◽  
Root Of Unity

AbstractWe develop the theory of p-adic confluence of q-difference equations. The main result is the fact that, in the p-adic framework, a function is a (Taylor) solution of a differential equation if and only if it is a solution of a q-difference equation. This fact implies an equivalence, called confluence, between the category of differential equations and those of q-difference equations. We develop this theory by introducing a category of sheaves on the disk D−(1,1), for which the stalk at 1 is a differential equation, the stalk at q isa q-difference equation if q is not a root of unity, and the stalk at a root of unity ξ is a mixed object, formed by a differential equation and an action of σξ.

On obtaining limiting values of a class of infinite products

2018 ◽  
Vol 102 (555) ◽  
pp. 428-434
Author(s):  
Stephen Kaczkowski
Keyword(s):  
Difference Equation ◽  
Differential Equations ◽  
Difference Equations ◽  
Diffusion Theory ◽  
Numerical Technique ◽  
Applied Mathematics ◽  
Flow Analysis ◽  
Step Size ◽  
Infinite Products ◽  
Fluid Flow Analysis

Difference equations have a wide variety of applications, including fluid flow analysis, wave propagation, circuit theory, the study of traffic patterns, queueing analysis, diffusion theory, and many others. Besides these applications, studies into the analogy between ordinary differential equations (ODEs) and difference equations have been a favourite topic of mathematicians (e.g. see [1] and [2]). These applications and studies bring to light the similar character of the solutions of a difference equation with a fixed step size and a corresponding ODE.Also, an important numerical technique for solving both ordinary and partial differential equations (PDEs) is the method of finite differences [3], whereby a difference equation with a small step size is utilised to obtain a numerical solution of a differential equation. In this paper, elements of both of these ideas will be used to solve some intriguing problems in pure and applied mathematics.

Optimal Finite Analytic Methods

1982 ◽  
Vol 104 (3) ◽  
pp. 432-437 ◽  
Author(s):  
R. Manohar ◽  
J. W. Stephenson
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Partial Differential Equations ◽  
Finite Difference ◽  
Differential Equations ◽  
Linear Combination ◽  
Difference Equations ◽  
New Method ◽  
Linear Partial Differential Equations ◽  
Finite Difference Equations

A new method is proposed for obtaining finite difference equations for the solution of linear partial differential equations. The method is based on representing the approximate solution locally on a mesh element by polynomials which satisfy the differential equation. Then, by collocation, the value of the approximate solution, and its derivatives at the center of the mesh element may be expressed as a linear combination of neighbouring values of the solution.

Flows on centre manifolds for scalar functional differential equations

1985 ◽  
Vol 101 (3-4) ◽  
pp. 193-201 ◽  
Author(s):  
Jack K. Hale
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Difference Equations ◽  
Imaginary Axis ◽  
Functional Differential Equations ◽  
Zero Eigenvalue ◽  
Functional Differential ◽  
Centre Manifolds ◽  
Scalar Functional ◽  
Linear Scalar

SynopsisBy assuming that a linear scalar functional differential equation (FDE) has only the zero eigenvalue on the imaginary axis, it is shown that the flows on the centre manifolds of all Cr-perturbations of this equation coincide with the flows obtained from scalar ordinary differential equations (ODEs) of order m, where m is the multiplicity of the zero eigenvalue. Furthermore, it is shown that the above situation can be realized through differential difference equations with m – 1 fixed distinct delays.

scholarly journals Weierstrass elliptic difference equations

1987 ◽  
Vol 35 (1) ◽  
pp. 43-48 ◽  
Author(s):  
Renfrey B. Potts
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Difference Equations ◽  
Elliptic Function ◽  
Order Differential Equation ◽  
Second Order ◽  
Elliptic Difference ◽  
First Order ◽  
Second Order Differential Equation ◽  
Weierstrass Elliptic Function

The Weierstrass elliptic function satisfies a nonlinear first order and a nonlinear second order differential equation. It is shown that these differential equations can be discretized in such a way that the solutions of the resulting difference equations exactly coincide with the corresponding values of the elliptic function.

Ladder operators and a differential equation for varying generalized Freud-type orthogonal polynomials

2018 ◽  
Vol 07 (04) ◽  
pp. 1840005 ◽  
Author(s):  
Galina Filipuk ◽  
Juan F. Mañas-Mañas ◽  
Juan J. Moreno-Balcázar
Keyword(s):  
Differential Equation ◽  
Difference Equation ◽  
Differential Equations ◽  
Orthogonal Polynomials ◽  
Inner Product ◽  
Ladder Operators ◽  
First Order ◽  
Standard Family ◽  
Differential Difference Equation ◽  
First Order Differential Equations

In this paper, we introduce varying generalized Freud-type polynomials which are orthogonal with respect to a varying discrete Freud-type inner product. Our main goal is to give ladder operators for this family of polynomials as well as find a second-order differential–difference equation that these polynomials satisfy. To reach this objective, it is necessary to consider the standard Freud orthogonal polynomials and, in the meanwhile, we find new difference relations for the coefficients in the first-order differential equations that this standard family satisfies.

scholarly journals Constructing Artistic Surface Modeling Design Based on Nonlinear Over-limit Interpolation Equation

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Tan Cheng ◽  
Madini O. Alassafi ◽  
Bishr Muhamed Muwafak
Keyword(s):  
Differential Equation ◽  
Difference Equation ◽  
Parameter Optimization ◽  
Difference Equations ◽  
3D Modeling ◽  
Surface Modeling ◽  
Curved Surfaces ◽  
Physical Methods ◽  
Modeling Methods ◽  
Fractional Differential

Abstract The digital and physical methods of establishing minimal curved surfaces are the basis for realizing the design of the minimal curved surface modeling structure. Based on this research background, the paper showed an artistic surface modeling method based on nonlinear over-limit difference equations. The article combines parameter optimization and 3D modeling methods to model the constructed surface modeling. The research found that the nonlinear out-of-limit difference equation proposed in the paper is more accurate than the standard fractional differential equation algorithm. For this reason, the method can be extended and applied to the design of artistic surface modeling.

scholarly journals Asymptotic behaviour of solutions to neutral functional differential equations

1989 ◽  
Vol 40 (3) ◽  
pp. 345-355
Author(s):  
Shaozhu Chen ◽  
Qingguang Huang
Keyword(s):  
Differential Equation ◽  
Difference Equation ◽  
Differential Equations ◽  
Asymptotic Behaviour ◽  
Functional Differential Equation ◽  
Necessary Conditions ◽  
Functional Differential Equations ◽  
Neutral Functional Differential Equation ◽  
Neutral Functional Differential Equations ◽  
Functional Differential

Sufficient or necessary conditions are established so that the neutral functional differential equation [x(t) − G(t, xt)]″ + F(t, xt) = 0 has a solution which is asymptotic to a given solution of the related difference equation x(t) = G(t, xt) + a + bt, where a and b are constants.

scholarly journals The Chebyshev Difference Equation

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 74
Author(s):  
Tom Cuchta ◽  
Michael Pavelites ◽  
Randi Tinney
Keyword(s):  
Difference Equation ◽  
Differential Equations ◽  
Difference Equations ◽  
Chebyshev Polynomials ◽  
Hypergeometric Series ◽  
Recurrence Relations ◽  
New Class

We define and investigate a new class of difference equations related to the classical Chebyshev differential equations of the first and second kind. The resulting “discrete Chebyshev polynomials” of the first and second kind have qualitatively similar properties to their continuous counterparts, including a representation by hypergeometric series, recurrence relations, and derivative relations.

scholarly journals A class of big (p,q)-Appell polynomials and their associated difference equations

Filomat ◽  
2019 ◽  
Vol 33 (10) ◽  
pp. 3085-3121
Author(s):  
H.M. Srivastava ◽  
B.Y. Yaşar ◽  
M.A. Özarslan
Keyword(s):  
Differential Equation ◽  
Difference Equation ◽  
Recurrence Relation ◽  
Difference Equations ◽  
Recurrence Relations ◽  
Bernoulli Polynomials ◽  
Equivalence Theorem ◽  
Euler Polynomials ◽  
Appell Polynomials ◽  
Special Case

In the present paper, we introduce and investigate the big (p,q)-Appell polynomials. We prove an equivalance theorem satisfied by the big (p, q)-Appell polynomials. As a special case of the big (p,q)- Appell polynomials, we present the corresponding equivalence theorem, recurrence relation and difference equation for the big q-Appell polynomials. We also present the equivalence theorem, recurrence relation and differential equation for the usual Appell polynomials. Moreover, for the big (p; q)-Bernoulli polynomials and the big (p; q)-Euler polynomials, we obtain recurrence relations and difference equations. In the special case when p = 1, we obtain recurrence relations and difference equations which are satisfied by the big q-Bernoulli polynomials and the big q-Euler polynomials. In the case when p = 1 and q ? 1-, the big (p,q)-Appell polynomials reduce to the usual Appell polynomials. Therefore, the recurrence relation and the difference equation obtained for the big (p; q)-Appell polynomials coincide with the recurrence relation and differential equation satisfied by the usual Appell polynomials. In the last section, we have chosen to also point out some obvious connections between the (p; q)-analysis and the classical q-analysis, which would show rather clearly that, in most cases, the transition from a known q-result to the corresponding (p,q)-result is fairly straightforward.

scholarly journals Developing a Series Solution Method of -Difference Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-4
Author(s):  
Hsuan-Ku Liu
Keyword(s):  
Difference Equation ◽  
Differential Equations ◽  
Difference Equations ◽  
Series Solution ◽  
Solution Method ◽  
Generalized Polynomials ◽  
Order Difference ◽  
Multiplication Rule ◽  
Second Order Difference Equation ◽  
Order Difference Equation

The series solution is widely applied to differential equations on but is not found in -differential equations. Applying the Taylor and multiplication rule of two generalized polynomials, we develop a series solution of linear homogeneous -difference equations. As an example, the series solution method is used to find a series solution of the second-order -difference equation of Hermite’s type.

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