Related papers: Differential equations associated with Legendre po…
Often a non-linear mechanical problem is formulated as a non-linear differential equation. A new method is introduced to find out new solutions of non-linear differential equations if one of the solutions of a given non-linear differential…
A class of second-order differential equations commonly arising in physics applications are considered, and their explicit hypergeometric solutions are provided. Further, the relationship with the Generalized and Universal Associated…
A symbolic computational algorithm which detects " linear "` solutions of nonlinear polynomial differential equations of single functions, is developed throughout this paper.
In this short note we are presenting a method of finding particular solutions of nonhomegeneous linear equations. This approach is different from methods of undetermined coefficients or variation of parameters presented in virtually every…
Differential equations are a powerful tool to tackle Feynman integrals. In this talk we discuss recent progress, where the method of differential equations has been applied to Feynman integrals which are not expressible in terms of multiple…
Differential equations with constant and variable coefficients over octonions are investigated. It is found that different types of differential equations over octonions can be resolved. For this purpose non-commutative line integration is…
In this paper, we consider several special polynomials related to associated sequences of polynomials. Finally, we give some new and interesting identities of those polynomials arising from transfer formula for the associated sequences.
Some connections between classical and nonclassical symmetries of a partial differential equation (PDE) are given in terms of determining equations of the two symmetries. These connections provide additional information for determining…
The construction of stochastic solutions for nonlinear partial differential equations is a powerful method to obtain new exact results and to develop efficient numerical algorithms, in particular when domain decomposition techniques are…
In this paper, we study the degenerate Eulerian polynomials and numbers and give some new and interesting identities associated with several special numbers and polynomials.
Orthogonal polynomials and multiple orthogonal polynomials are interesting special functions because there is a beautiful theory for them, with many examples and useful applications in mathematical physics, numerical analysis, statistics…
Given the Fourier-Legendre expansions of $f$ and $g$, and mild conditions on $f$ and $g$, we derive the Fourier-Legendre expansion of their product in terms of their corresponding Fourier-Legendre coefficients. In this way, expansions of…
A novel method, connecting the space of solutions of a linear differential equation, of arbitrary order, to the space of monomials, is used for exploring the algebraic structure of the solution space. Apart from yielding new expressions for…
In this paper, we study some properties of associated sequences of special polynomials. From the properties of associated sequences of polynomials, we derive some interesting identities of special polynomials.
Let $\{Q_{n}(x)\}$ be a system of integral Legendre polynomials of degree exactly n,and let $\{P_{n}(x)\}$ be polar polynomials primitives of integral Legendre polynomials. We derive some identities and relations and extremal problems and…
Partial ordinary Bell polynomials are used to formulate and prove a version of the Fa\`{a} di Bruno's formula which is convenient for handling nonlinear terms in the differential transformation. Applicability of the result is shown in two…
We present some results in the analysis of non-compact differential equations on unbounded domains.
An extensive table of pairs of functions linked by the Legendre transformation is presented. Many special functions and formulas that are used in the sciences are included in the pairs. Formulations are provided for finding the Legendre…
In the present paper, we deal mainly with arithmetic properties of Legendre polynomials by using their orthogonality property. We show that Legendre polynomials are proportional with Bernoulli, Euler, Hermite and Bernstein polynomials.
In this paper, we study the existence and non-existence of entire solutions of certain non-linear delay-differential equations.