scholarly journals Relevance of Factorization Method to Differential and Integral Equations Associated with Hybrid Class of Polynomials

2021 ◽  
Vol 6 (1) ◽  
pp. 5
Author(s):  
Naeem Ahmad ◽  
Raziya Sabri ◽  
Mohammad Faisal Khan ◽  
Mohammad Shadab ◽  
Anju Gupta
Keyword(s):  
Differential Equation ◽  
Integral Equation ◽  
Differential Equations ◽  
Integral Equations ◽  
Recurrence Relations ◽  
Factorization Method ◽  
New Class ◽  
Hybrid Class ◽  
Sheffer Polynomials ◽  
Differential And Integral Equations

This article has a motive to derive a new class of differential equations and associated integral equations for some hybrid families of Laguerre–Gould–Hopper-based Sheffer polynomials. We derive recurrence relations, differential equation, integro-differential equation, and integral equation for the Laguerre–Gould–Hopper-based Sheffer polynomials by using the factorization method.

Applications of Henstock-Kurzweil integrals on an unbounded interval to differential and integral equations

2018 ◽  
Vol 68 (1) ◽  
pp. 77-88
Author(s):  
Marcin Borkowski ◽  
Daria Bugajewska
Keyword(s):  
Integral Equation ◽  
Differential Equations ◽  
Integral Equations ◽  
Bounded Variation ◽  
Volterra Integral Equation ◽  
Functions Of Bounded Variation ◽  
Hammerstein Integral Equation ◽  
Linear Differential ◽  
Unbounded Intervals ◽  
Differential And Integral Equations

Abstract In this paper we are going to apply the Henstock-Kurzweil integrals defined on an unbounded intervals to differential and integral equations defined on such intervals. To deal with linear differential equations we examine convolution involving functions integrable in Henstock-Kurzweil sense. In the case of nonlinear Hammerstein integral equation as well as Volterra integral equation we look for solutions in the space of functions of bounded variation in the sense of Jordan.

scholarly journals A new class of nonlinear Gronwall–Bellman delay integral inequalities with power and its applications

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Bo Fang ◽  
Yujiao Liu ◽  
Run Xu
Keyword(s):  
Differential Equations ◽  
Integral Equations ◽  
Integral Inequalities ◽  
Qualitative Properties ◽  
New Class ◽  
Convenient Tool ◽  
Qualitative Properties Of Solutions ◽  
Differential And Integral Equations

AbstractIn this paper, we establish some new delay Gronwall–Bellman integral inequalities with power, which can be used as a convenient tool to study the qualitative properties of solutions to differential and integral equations. We also give some examples to illustrate the application of our results to obtain the estimation for the solution of the integral and differential equations.

scholarly journals Differential and integral equations for Legendre-Laguerre based hybrid polynomials

2021 ◽  
Vol 73 (3) ◽  
pp. 408-421
Author(s):  
S. Khan ◽  
M. Riyasat ◽  
Sh. A. Wani
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Integral Equations ◽  
Recurrence Relations ◽  
Volterra Integral Equations ◽  
Series Expansions ◽  
Appell Polynomials ◽  
Shift Operators ◽  
Genocchi Polynomials ◽  
Differential And Integral Equations

UDC 517.9 In this article, a hybrid family of three-variable Legendre – Laguerre – Appell polynomials is explored and their properties including the series expansions, determinant forms, recurrence relations, shift operators, followed by differential, integro-differential and partial differential equations are established. The analogous results for the three-variable Hermite – Laguerre – Appell polynomials are deduced. Certain examples in terms of Legendre – Laguerre – Bernoulli, –E uler and – Genocchi polynomials are constructed to show the applications of main results. A further investigation is performed by deriving homogeneous Volterra integral equations for these polynomials and for their relatives.

Differential and integral equations for the 2-iterated Bernoulli, 2-iterated Euler and Bernoulli–Euler polynomials

2020 ◽  
Vol 27 (3) ◽  
pp. 375-389 ◽  
Author(s):  
Subuhi Khan ◽  
Mumtaz Riyasat
Keyword(s):  
Numerical Computation ◽  
Integral Equations ◽  
Mixed Type ◽  
Recurrence Relations ◽  
Graphical Representation ◽  
Approximate Solutions ◽  
Factorization Method ◽  
Euler Polynomials ◽  
Real Zeros ◽  
Differential And Integral Equations

AbstractThe main goal of this article is to derive differential and associated integral equations for some 2-iterated and mixed type special polynomial families. By using the factorization method, the recurrence relations and differential equations are derived for the 2-iterated Bernoulli, 2-iterated Euler and Bernoulli–Euler polynomials. The Volterra integral equations for these polynomial families are also established. The graphical representation of these 2-iterated and mixed polynomials is presented. The zeros of these polynomials are investigated for certain values of the index n using numerical computation. The approximate solutions of the real zeros of these polynomials are also given.

On some classes of differential equations and associated integral equations for the Laguerre–Appell polynomials

2018 ◽  
Vol 9 (3) ◽  
pp. 185-194 ◽  
Author(s):  
Subuhi Khan ◽  
Mumtaz Riyasat ◽  
Shahid Ahmad Wani
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Integral Equations ◽  
Recurrence Relations ◽  
Laguerre Polynomials ◽  
Factorization Method ◽  
Volterra Integral Equations ◽  
Appell Polynomials ◽  
Partial Differential ◽  
Genocchi Polynomials

Abstract The article aims to explore some new classes of differential equations and associated integral equations for some hybrid families of Laguerre polynomials. The recurrence relations and differential, integro-differential and partial differential equations for the hybrid Laguerre–Appell polynomials are derived via the factorization method. An analogous study of these results for the hybrid Laguerre–Bernoulli, Euler and Genocchi polynomials is presented. Further, the Volterra integral equations for the hybrid Laguerre–Appell polynomials and for their corresponding members are also explored.

scholarly journals Enhancing Accuracy of Runge–Kutta-Type Collocation Methods for Solving ODEs

Mathematics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 174
Author(s):  
Janez Urevc ◽  
Miroslav Halilovič
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Integration ◽  
Integral Form ◽  
Collocation Methods ◽  
Nonlinear Odes ◽  
New Class ◽  
New Methods ◽  
Runge Kutta

In this paper, a new class of Runge–Kutta-type collocation methods for the numerical integration of ordinary differential equations (ODEs) is presented. Its derivation is based on the integral form of the differential equation. The approach enables enhancing the accuracy of the established collocation Runge–Kutta methods while retaining the same number of stages. We demonstrate that, with the proposed approach, the Gauss–Legendre and Lobatto IIIA methods can be derived and that their accuracy can be improved for the same number of method coefficients. We expressed the methods in the form of tables similar to Butcher tableaus. The performance of the new methods is investigated on some well-known stiff, oscillatory, and nonlinear ODEs from the literature.

scholarly journals The General Solution of Differential Equations with Caputo-Hadamard Fractional Derivatives and Noninstantaneous Impulses

2017 ◽  
Vol 2017 ◽  
pp. 1-11 ◽  
Author(s):  
Xianzhen Zhang ◽  
Zuohua Liu ◽  
Hui Peng ◽  
Xianmin Zhang ◽  
Shiyong Yang
Keyword(s):  
Differential Equation ◽  
Integral Equation ◽  
General Solution ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Order System ◽  
Impulsive Systems ◽  
Fractional Differential ◽  
Noninstantaneous Impulses

Based on some recent works about the general solution of fractional differential equations with instantaneous impulses, a Caputo-Hadamard fractional differential equation with noninstantaneous impulses is studied in this paper. An equivalent integral equation with some undetermined constants is obtained for this fractional order system with noninstantaneous impulses, which means that there is general solution for the impulsive systems. Next, an example is given to illustrate the obtained result.

scholarly journals The existence of solutions for Sturm–Liouville differential equation with random impulses and boundary value problems

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Zihan Li ◽  
Xiao-Bao Shu ◽  
Tengyuan Miao
Keyword(s):  
Differential Equation ◽  
Integral Equation ◽  
Differential Equations ◽  
Boundary Value Problems ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Lower Solution ◽  
Iterative Sequence ◽  
Random Impulses ◽  
Sturm Liouville

AbstractIn this article, we consider the existence of solutions to the Sturm–Liouville differential equation with random impulses and boundary value problems. We first study the Green function of the Sturm–Liouville differential equation with random impulses. Then, we get the equivalent integral equation of the random impulsive differential equation. Based on this integral equation, we use Dhage’s fixed point theorem to prove the existence of solutions to the equation, and the theorem is extended to the general second order nonlinear random impulsive differential equations. Then we use the upper and lower solution method to give a monotonic iterative sequence of the generalized random impulsive Sturm–Liouville differential equations and prove that it is convergent. Finally, we give two concrete examples to verify the correctness of the results.

scholarly journals An integral equation associated with linear homogeneous differential equations

1986 ◽  
Vol 9 (2) ◽  
pp. 405-408 ◽  
Author(s):  
A. K. Bose
Keyword(s):  
Differential Equation ◽  
Integral Equation ◽  
Differential Equations ◽  
Real Line ◽  
Homogeneous Differential Equation ◽  
The Real ◽  
Linear Homogeneous Differential Equation

Associated with each linear homogeneous differential equationy(n)=∑i=0n−1ai(x)y(i)of ordernon the real line, there is an equivalent integral equationf(x)=f(x0)+∫x0xh(u)du+∫x0x[∫x0uGn−1(u,v)a0(v)f(v)dv]duwhich is satisfied by each solutionf(x)of the differential equation.

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