Some new results for the multivariable Humbert polynomials

2014 ◽  
Vol 64 (6) ◽  
Author(s):  
Rabıa Aktaş
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Integral Representation ◽  
Recurrence Relations ◽  
Addition Formula ◽  
Partial Differential ◽  
Special Cases ◽  
Humbert Polynomials

AbstractIn this paper, we present some miscellaneous properties of the multivariable Humbert polynomials whose special cases include some well-known multivariable polynomials such as Chan-Chyan-Srivastava, Lagrange-Hermite and Erkus-Srivastava multivariable polynomials. We give recurrence relations, addition formula and integral representation for them. Then, we obtain some partial differential equations for the products of the multivariable Humbert polynomials and some other multivariable polynomials. Furthermore, some special cases of the results presented in this study are also indicated.

scholarly journals Ultrasonic Waves in Bubbly Liquids: An Analytic Approach

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1309
Author(s):  
P. R. Gordoa ◽  
A. Pickering
Keyword(s):  
Mathematical Model ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Acoustic Waves ◽  
Elliptic Functions ◽  
Coupled System ◽  
Ultrasonic Waves ◽  
Partial Differential ◽  
Special Cases ◽  
Spherical Bubbles

We consider the problem of the propagation of high-intensity acoustic waves in a bubble layer consisting of spherical bubbles of identical size with a uniform distribution. The mathematical model is a coupled system of partial differential equations for the acoustic pressure and the instantaneous radius of the bubbles consisting of the wave equation coupled with the Rayleigh–Plesset equation. We perform an analytic analysis based on the study of Lie symmetries for this system of equations, concentrating our attention on the traveling wave case. We then consider mappings of the resulting reductions onto equations defining elliptic functions, and special cases thereof, for example, solvable in terms of hyperbolic functions. In this way, we construct exact solutions of the system of partial differential equations under consideration. We believe this to be the first analytic study of this particular mathematical model.

scholarly journals On the solution of reaction-diffusion equations with double diffusivity

1987 ◽  
Vol 10 (1) ◽  
pp. 163-172
Author(s):  
B. D. Aggarwala ◽  
C. Nasim
Keyword(s):  
Heat Conduction ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Partial Differential ◽  
Coupled Partial Differential Equations ◽  
Special Cases ◽  
Two Temperatures

In this paper, solution of a pair of Coupled Partial Differential equations is derived. These equations arise in the solution of problems of flow of homogeneous liquids in fissured rocks and heat conduction involving two temperatures. These equations have been considered by Hill and Aifantis, but the technique we use appears to be simpler and more direct, and some new results are derived. Also, discussion about the propagation of initial discontinuities is given and illustrated with graphs of some special cases.

scholarly journals STOCHASTIC SOLUTIONS OF A CLASS OF HIGHER ORDER CAUCHY PROBLEMS IN ℝd

2010 ◽  
Vol 10 (03) ◽  
pp. 341-366 ◽  
Author(s):  
ERKAN NANE
Keyword(s):  
Partial Differential Equations ◽  
Markov Process ◽  
Differential Equations ◽  
Stable Process ◽  
Higher Order ◽  
Cauchy Problems ◽  
Partial Differential ◽  
Iterated Brownian Motion ◽  
Special Cases ◽  
First Time

We study solutions of a class of higher order partial differential equations in bounded domains. These partial differential equations appeared first time in the papers of Allouba and Zheng [4], Baeumer, Meerschaert and Nane [10], Meerschaert, Nane and Vellaisamy [37], and Nane [42]. We express the solutions by subordinating a killed Markov process by a hitting time of a stable subordinator of index 0 < β < 1, or by the absolute value of a symmetric α-stable process with 0 < α ≤ 2, independent of the Markov process. In some special cases we represent the solutions by running composition of k independent Brownian motions, called k-iterated Brownian motion for an integer k ≥ 2. We make use of a connection between fractional-time diffusions and higher order partial differential equations established first by Allouba and Zheng [4] and later extended in several directions by Baeumer, Meerschaert and Nane [10].

scholarly journals On the complex Hermite polynomials

Filomat ◽  
2020 ◽  
Vol 34 (2) ◽  
pp. 409-420
Author(s):  
Zhi-Guo Liu
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Generating Function ◽  
Poisson Kernel ◽  
Hermite Polynomials ◽  
Addition Formula ◽  
Partial Differential ◽  
Expansion Theorem ◽  
New Approach ◽  
Complex Hermite Polynomials

In this paper we use a set of partial differential equations to prove an expansion theorem for multiple complex Hermite polynomials. This expansion theorem allows us to develop a systematic and completely new approach to the complex Hermite polynomials. Using this expansion, we derive the Poisson Kernel, the Nielsen type formula, the addition formula for the complex Hermite polynomials with ease. A multilinear generating function for the complex Hermite polynomials is proved.

scholarly journals New Families of Three-Variable Polynomials Coupled with Well-Known Polynomials and Numbers

Symmetry ◽  
2019 ◽  
Vol 11 (2) ◽  
pp. 264 ◽  
Author(s):  
Can Kızılateş ◽  
Bayram Çekim ◽  
Naim Tuğlu ◽  
Taekyun Kim
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Generating Functions ◽  
Explicit Representation ◽  
Partial Differential ◽  
Generalized Polynomials ◽  
Special Cases

In this paper, firstly the definitions of the families of three-variable polynomials with the new generalized polynomials related to the generating functions of the famous polynomials and numbers in literature are given. Then, the explicit representation and partial differential equations for new polynomials are derived. The special cases of our polynomials are given in tables. In the last section, the interesting applications of these polynomials are found.

scholarly journals Integral representation of solution manifolds for over determined systems with one singular line

2021 ◽  
pp. 5-9
Author(s):  
Boitura Shoimkulov ◽  
Keyword(s):  
Partial Differential Equations ◽  
Initial Data ◽  
Differential Equations ◽  
Integral Representation ◽  
Compatibility Condition ◽  
Second Order ◽  
Singular Line ◽  
Cauchy Type ◽  
Partial Differential ◽  
Partial Differential System

In this paper, an over determined system of second-order partial differential equations with one singular line is investigated. A compatibility condition is found for over determined systems of second-order partial differential equations with one singular line. Under the condition of compatibility, introducing a new function, we come to a over determined system of partial differential equations of the second order with one singular line of a simpler form. The integral representation of the manifold of solutions of the redefined second-order partial differential system with one singular line is found explicitly through three arbitrary constants, for which initial data problems (Cauchy type problems) can be posed.

scholarly journals Müntz sturm-liouville problems: Theory and numerical experiments

2021 ◽  
Vol 24 (3) ◽  
pp. 775-817
Author(s):  
Hassan Khosravian-Arab ◽  
Mohammad Reza Eslahchi
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Error Bounds ◽  
Spectral Properties ◽  
Numerical Experiments ◽  
Recurrence Relations ◽  
Basis Functions ◽  
Partial Differential ◽  
Orthogonal Projections ◽  
Sturm Liouville

Abstract This paper presents two new classes of Müntz functions which are called Jacobi-Müntz functions of the first and second types. These newly generated functions satisfy in two self-adjoint fractional Sturm-Liouville problems and thus they have some spectral properties such as: orthogonality, completeness, three-term recurrence relations and so on. With respect to these functions two new orthogonal projections and their error bounds are derived. Also, two new Müntz type quadrature rules are introduced. As two applications of these basis functions some fractional ordinary and partial differential equations are considered and numerical results are given.

Solution of some Engineering Partial Differential Equations Governed by the Minimal of a Functional by Global Optimization Method

2013 ◽  
Vol 29 (3) ◽  
pp. 507-516 ◽  
Author(s):  
Y. M. Cheng ◽  
D. Z. Li ◽  
N. Li ◽  
Y. Y Lee ◽  
S. K. Au
Keyword(s):  
Finite Element ◽  
Global Optimization ◽  
Approximate Solution ◽  
Partial Differential Equations ◽  
Finite Difference ◽  
Differential Equations ◽  
Optimization Method ◽  
Global Optimization Method ◽  
Partial Differential ◽  
Special Cases

AbstractMany engineering problems are governed by partial differential equations which can be solved by analytical as well as numerical methods, and examples include the plasticity problem of a geotechnical system, seepage problem and elasticity problem. Although the governing differential equations can be solved by either iterative finite difference method or finite element, there are however limitations to these methods in some special cases which will be discussed in the present paper. The solutions of these governing differential equations can all be viewed as the stationary value of a functional. Using an approximate solution as the initial solution, the stationary value of the functional can be obtained easily by modern global optimization method. Through the comparisons between analytical solutions and fine mesh finite element analysis, the use of global optimization method will be demonstrated to be equivalent to the solutions of the governing partial differential equations. The use of global optimization method can be an alternative to the finite difference/ finite element method in solving an engineering problem, and it is particularly attractive when an approximate solution is available or can be estimated easily.

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