Orthogonal Polynomials Satisfying Sixth Order Differential Equations

2002 ◽  
pp. 281-290
Author(s):  
Allan M. Krall
Keyword(s):  
Differential Equations ◽  
Orthogonal Polynomials ◽  
Sixth Order

scholarly journals Non-local Solvable Birth–Death Processes

2021 ◽  
Author(s):  
Giacomo Ascione ◽  
Nikolai Leonenko ◽  
Enrica Pirozzi
Keyword(s):  
Differential Equations ◽  
Orthogonal Polynomials ◽  
Limit Distribution ◽  
Spectral Decomposition ◽  
Strong Solutions ◽  
Stochastic Representation ◽  
Discrete Approximations ◽  
Local Difference ◽  
Non Local ◽  
Birth Death

AbstractIn this paper, we study strong solutions of some non-local difference–differential equations linked to a class of birth–death processes arising as discrete approximations of Pearson diffusions by means of a spectral decomposition in terms of orthogonal polynomials and eigenfunctions of some non-local derivatives. Moreover, we give a stochastic representation of such solutions in terms of time-changed birth–death processes and study their invariant and their limit distribution. Finally, we describe the correlation structure of the aforementioned time-changed birth–death processes.

scholarly journals Diagonally Implicit Runge–Kutta Type Method for Directly Solving Special Fourth-Order Ordinary Differential Equations with Ill-Posed Problem of a Beam on Elastic Foundation

Algorithms ◽  
2018 ◽  
Vol 12 (1) ◽  
pp. 10 ◽  
Author(s):  
Nizam Ghawadri ◽  
Norazak Senu ◽  
Firas Adel Fawzi ◽  
Fudziah Ismail ◽  
Zarina Ibrahim
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Elastic Foundation ◽  
Fourth Order ◽  
Sixth Order ◽  
Test Problems ◽  
Type Method ◽  
Ill Posed ◽  
Runge Kutta ◽  
Fifth Order

In this study, fifth-order and sixth-order diagonally implicit Runge–Kutta type (DIRKT) techniques for solving fourth-order ordinary differential equations (ODEs) are derived which are denoted as DIRKT5 and DIRKT6, respectively. The first method has three and the another one has four identical nonzero diagonal elements. A set of test problems are applied to validate the methods and numerical results showed that the proposed methods are more efficient in terms of accuracy and number of function evaluations compared to the existing implicit Runge–Kutta (RK) methods.

RESULTS FOR SIXTH ORDER POSITIVELY HOMOGENEOUS EQUATIONS

2009 ◽  
Vol 14 (1) ◽  
pp. 25-32 ◽  
Author(s):  
Tatjana Garbuza
Keyword(s):  
Differential Equations ◽  
Linear Equations ◽  
Sixth Order ◽  
Extremal Solutions ◽  
Conjugate Points ◽  
Comparison Results

We consider positively homogeneous the sixth order differential equations of the type x (6) = h(t, x), where hpossesses the property that h(t, cx) = ch(t, x) for c ≥ 0. This class includes the linear equations x (6) = p(t)x and piece‐wise linear ones x (6) = k 2 x+ - k 1 x− . We consider conjugate points and angles associated with extremal solutions and prove some comparison results.

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