scholarly journals A Continuous Analogue of Lattice Path Enumeration

10.37236/8788 ◽  
2019 ◽  
Vol 26 (3) ◽  
Author(s):  
Quang-Nhat Le ◽  
Sinai Robins ◽  
Christophe Vignat ◽  
Tanay Wakhare
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Lattice Paths ◽  
Lattice Path ◽  
Continuous Version ◽  
Continuous Analog ◽  
Continuous Analogue ◽  
Lattice Path Enumeration ◽  
Multinomial Coefficients ◽  
General Method

Following the work of Cano and Díaz, we consider a continuous analog of lattice path enumeration. This process allows us to define a continuous version of many discrete objects that count certain types of lattice paths. As an example of this process, we define continuous versions of binomial and multinomial coefficients, and describe some identities and partial differential equations that they satisfy. Finally, as an important byproduct of these continuous analogs, we illustrate a general method to recover discrete combinatorial quantities from their continuous analogs, via an application of the Khovanski-Puklikov discretizing Todd operators.  

scholarly journals Direct construction method for conservation laws of partial differential equations Part II: General treatment

2002 ◽  
Vol 13 (5) ◽  
pp. 567-585 ◽  
Author(s):  
STEPHEN C. ANCO ◽  
GEORGE BLUMAN
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Conservation Laws ◽  
General Treatment ◽  
Local Conservation ◽  
Partial Differential ◽  
Direct Construction ◽  
Computational Implementation ◽  
Direct Construction Method ◽  
General Method

This paper gives a general treatment and proof of the direct conservation law method presented in Part I (see Anco & Bluman [3]). In particular, the treatment here applies to finding the local conservation laws of any system of one or more partial differential equations expressed in a standard Cauchy-Kovalevskaya form. A summary of the general method and its effective computational implementation is also given.

scholarly journals On Professor Whittaker's solution of differential equations by definite integrals: Part I

1931 ◽  
Vol 2 (4) ◽  
pp. 205-219 ◽  
Author(s):  
W. O. Kermack ◽  
W. H. McCrea
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Differential Operators ◽  
Preceding Paper ◽  
Contact Transformation ◽  
Partial Differential ◽  
Definite Integrals ◽  
General Method

In the preceding paper Professor Whittaker has given a general method for the solution of differential equations by means of definite integrals. It depends on finding a solution χ (q, Q) of an auxiliary pair of simultaneous partial differential equations to be derived from an arbitrary contact transformation by changing the momentum variables into differential operators. The first object of the present paper is to arrive at a method for passing from the contact transformation in its algebraic form to these partial differential equations, in a manner which is unambiguous and which makes them compatible. We show too how to obtain any number of such, pairs of equations from any given contact transformation. Successive transformations are also discussed.

scholarly journals Symmetry properties of conservation laws

2016 ◽  
Vol 30 (28n29) ◽  
pp. 1640003 ◽  
Author(s):  
Stephen C. Anco
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Conservation Laws ◽  
Conservation Law ◽  
Conserved Quantity ◽  
Partial Differential ◽  
Standard Formula ◽  
Infinitesimal Symmetry ◽  
Symmetry Properties ◽  
General Method

Symmetry properties of conservation laws of partial differential equations are developed by using the general method of conservation law multipliers. As main results, simple conditions are given for characterizing when a conservation law and its associated conserved quantity are invariant (and, more generally, homogeneous) under the action of a symmetry. These results are used to show that a recent conservation law formula (due to Ibragimov) is equivalent to a standard formula for the action of an infinitesimal symmetry on a conservation law multiplier.

scholarly journals A general method to construct invariant PDEs on homogeneous manifolds

2021 ◽  
pp. 2050089
Author(s):  
Dmitri V. Alekseevsky ◽  
Jan Gutt ◽  
Gianni Manno ◽  
Giovanni Moreno
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Conformal Space ◽  
Prolonged Action ◽  
Linear Action ◽  
Partial Differential ◽  
Open Orbit ◽  
Independent Variables ◽  
The Stability ◽  
General Method

Let [Formula: see text] be an [Formula: see text]-dimensional homogeneous manifold and [Formula: see text] be the manifold of [Formula: see text]-jets of hypersurfaces of [Formula: see text]. The Lie group [Formula: see text] acts naturally on each [Formula: see text]. A [Formula: see text]-invariant partial differential equation of order [Formula: see text] for hypersurfaces of [Formula: see text] (i.e., with [Formula: see text] independent variables and [Formula: see text] dependent one) is defined as a [Formula: see text]-invariant hypersurface [Formula: see text]. We describe a general method for constructing such invariant partial differential equations for [Formula: see text]. The problem reduces to the description of hypersurfaces, in a certain vector space, which are invariant with respect to the linear action of the stability subgroup [Formula: see text] of the [Formula: see text]-prolonged action of [Formula: see text]. We apply this approach to describe invariant partial differential equations for hypersurfaces in the Euclidean space [Formula: see text] and in the conformal space [Formula: see text]. Our method works under some mild assumptions on the action of [Formula: see text], namely: A1) the group [Formula: see text] must have an open orbit in [Formula: see text], and A2) the stabilizer [Formula: see text] of the fiber [Formula: see text] must factorize via the group of translations of the fiber itself.

Symmetry-invariant conservation laws of partial differential equations

2017 ◽  
Vol 29 (1) ◽  
pp. 78-117 ◽  
Author(s):  
STEPHEN C. ANCO ◽  
ABDUL H. KARA
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Conservation Laws ◽  
Stokes Equations ◽  
Two Dimensions ◽  
Navier Stokes ◽  
Partial Differential ◽  
Navier Stokes Equations ◽  
General Method

A simple characterization of the action of symmetries on conservation laws of partial differential equations is studied by using the general method of conservation law multipliers. This action is used to define symmetry-invariant and symmetry-homogeneous conservation laws. The main results are applied to several examples of physically interest, including the generalized Korteveg-de Vries equation, a non-Newtonian generalization of Burger's equation, theb-family of peakon equations, and the Navier–Stokes equations for compressible, viscous fluids in two dimensions.

scholarly journals On the general random walk formulation for diffusion in media with Diffusivities

1985 ◽  
Vol 27 (1) ◽  
pp. 73-87 ◽  
Author(s):  
James M. Hill ◽  
Barry D. Hughes
Keyword(s):  
Random Walk ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Continuum Limit ◽  
Double Diffusion ◽  
Random Walk Model ◽  
Partial Differential ◽  
Continuous Version ◽  
Two State Model ◽  
General Random Walk

AbstractA general discrete multi-dimensional and multi-state random walk model is proposed to describe the phenomena of diffusion in media with multiple diffusivities. The model is a generalization of a two-state one-dimensional discrete random walk model (Hill [8]) which gives rise to the partial differential equations of double diffusion. The same partial differential equations are shown to emerge as a special case of the continuous version of the present general model. For two states a particular generalization of the model given in [8] is presented which is not restricted to nearest neighbour transitions. Under appropriate circumstances this two-state model still yields the partial differential equations of double diffusion in the continuum limit, but an example of circumstances leading to a radically different continuum limit is presented.

XXII.—An Operational Method for the Solution of Linear Partial Differential Equations

1932 ◽  
Vol 51 ◽  
pp. 176-189 ◽  
Author(s):  
W. O. Kermack ◽  
W. H. McCrea
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Contact Transformation ◽  
Operational Method ◽  
Partial Differential ◽  
Definite Integrals ◽  
Linear Partial Differential Equations ◽  
Image Position ◽  
General Method

1. A general method for the solution of differential equations by definite integrals has recently been given by Professor E. T. Whittaker. It is briefly that, if a contact transformation from variables (q, p) to (Q, P) be given by Q = Q(q, p), P = P(q, p), and if this transforms an expression G(Q, P) into F(q, p), then the solutions of the differential equationsare connected by a relation of the form

scholarly journals B-Spline Solutions of General Euler-Lagrange Equations

Mathematics ◽  
2019 ◽  
Vol 7 (4) ◽  
pp. 365
Author(s):  
Lanyin Sun ◽  
Chungang Zhu
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fourth Order ◽  
Optimization Problems ◽  
Lagrange Equations ◽  
Surface Design ◽  
B Spline ◽  
Special Cases ◽  
Pde Surfaces ◽  
General Method

The Euler-Lagrange equations are useful for solving optimization problems in mechanics. In this paper, we study the B-spline solutions of the Euler-Lagrange equations associated with the general functionals. The existing conditions of B-spline solutions to general Euler-Lagrange equations are given. As part of this work, we present a general method for generating B-spline solutions of the second- and fourth-order Euler-Lagrange equations. Furthermore, we show that some existing techniques for surface design, such as Coons patches, are exactly the special cases of the generalized Partial differential equations (PDE) surfaces with appropriate choices of the constants.

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