scholarly journals Integrating linear ordinary fourth-order differential equations in the MAPLE programming environment

2021 ◽  
Vol 3 (4 (111)) ◽  
pp. 51-57
Author(s):  
Irina Belyaeva ◽  
Igor Kirichenko ◽  
Oleh Ptashnyi ◽  
Natalia Chekanova ◽  
Tetiana Yarkho
Keyword(s):  
Power Series ◽  
Differential Equations ◽  
Fourth Order ◽  
Eigenvalue Problems ◽  
Boundary Problems ◽  
Generalized Power Series ◽  
Electronic Computing ◽  
Santa Monica ◽  
Ordinary Power ◽  
Special Points

This paper reports a method to solve ordinary fourth-order differential equations in the form of ordinary power series and, for the case of regular special points, in the form of generalized power series. An algorithm has been constructed and a program has been developed in the MAPLE environment (Waterloo, Ontario, Canada) in order to solve the fourth-order differential equations. All types of solutions depending on the roots of the governing equation have been considered. The examples of solutions to the fourth-order differential equations are given; they have been compared with the results available in the literature that demonstrate excellent agreement with the calculations reported here, which confirms the effectiveness of the developed programs. A special feature of this work is that the accuracy of the results is controlled by the number of terms in the power series and the number of symbols (up to 20) in decimal mantissa in numerical calculations. Therefore, almost any accuracy allowed for a given electronic computing machine or computer is achievable. The proposed symbolic-numerical method and the work program could be successfully used for solving eigenvalue problems, in which controlled accuracy is very important as the eigenfunctions are extremely (exponentially) sensitive to the accuracy of eigenvalues found. The developed algorithm could be implemented in other known computer algebra packages such as REDUCE (Santa Monica, CA), MATHEMATICA (USA), MAXIMA (USA), and others. The program for solving ordinary fourth-order differential equations could be used to construct Green’s functions of boundary problems, to solve differential equations with private derivatives, a system of Hamilton’s differential equations, and other problems related to mathematical physics.

scholarly journals Global Bifurcation of Fourth-Order Nonlinear Eigenvalue Problems’ Solution

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Fatma Aydin Akgun
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Fourth Order ◽  
Eigenvalue Problems ◽  
Global Bifurcation ◽  
Nonlinear Eigenvalue Problems ◽  
Nodal Properties ◽  
Nonlinear Eigenvalue

In this paper, we study the global bifurcation of infinity of a class of nonlinear eigenvalue problems for fourth-order ordinary differential equations with nondifferentiable nonlinearity. We prove the existence of two families of unbounded continuance of solutions bifurcating at infinity and corresponding to the usual nodal properties near bifurcation intervals.

scholarly journals The first integrals and rational solutions of some fourth-order differential equations

2020 ◽  
Vol 56 (3) ◽  
pp. 318-327
Author(s):  
E. R. Babich ◽  
I. P. Martynov
Keyword(s):  
Differential Equations ◽  
Fourth Order ◽  
Order Differential Equation ◽  
Rational Solution ◽  
Resonance Method ◽  
First Integrals ◽  
Rational Solutions ◽  
Analytical Properties ◽  
Special Points ◽  
Two Parameter

The object of this research is fourth-order differential equations. The aim of the research is to study the analytical properties of the solutions of these differential equations. The general form of the considered equations is indicated, and also the choice of the research object is justified. Herein we studied fourth-order differential equations for which sets of resonances with all positive nontrivial resonances are absent. Besides, three of these equations satisfy the conditions of absence in the solutions of moving multivalued singular points. The solutions of the next three equations have movable special points of multivalued character. Moreover, we also investigated the analytical properties of one more fourth-order differential equation of another general form for which it is also possible to construct a two-parameter rational solution as there is a nontrivial negative resonance in the related set of resonances. The first integrals of the equations under study are found and their rational solutions are constructed from negative non-trivial resonances. The resonance method was used in this study. The obtained results can be used in the analytical theory of differential equations.

scholarly journals An Efficient Zero-Stable Numerical Method for Fourth-Order Differential Equations

2008 ◽  
Vol 2008 ◽  
pp. 1-10 ◽  
Author(s):  
S. J. Kayode
Keyword(s):  
Approximate Solution ◽  
Power Series ◽  
Differential Equations ◽  
Numerical Method ◽  
Fourth Order ◽  
Optimal Order ◽  
Direct Solution ◽  
Order Of Accuracy ◽  
First Order ◽  
Stable Numerical Method

The purpose of this paper is to produce an efficient zero-stable numerical method with the same order of accuracy as that of the main starting values (predictors) for direct solution of fourth-order differential equations without reducing it to a system of first-order equations. The method of collocation of the differential system arising from the approximate solution to the problem is adopted using the power series as a basis function. The method is consistent, symmetric, and of optimal order . The main predictor for the method is also consistent, symmetric, zero-stable, and of optimal order .

scholarly journals Two-loop integrals for planar five-point one-mass processes

2020 ◽  
Vol 2020 (11) ◽  
Author(s):  
Samuel Abreu ◽  
Harald Ita ◽  
Francesco Moriello ◽  
Ben Page ◽  
Wladimir Tschernow ◽  
...  
Keyword(s):  
Scattering Amplitudes ◽  
Power Series ◽  
Differential Equations ◽  
Finite Fields ◽  
W Boson ◽  
Boson Production ◽  
Important Ingredient ◽  
Feynman Integrals ◽  
Generalized Power Series ◽  
Numerical Precision

Abstract We present the computation of a full set of planar five-point two-loop master integrals with one external mass. These integrals are an important ingredient for two-loop scattering amplitudes for two-jet-associated W-boson production at leading color in QCD. We provide a set of pure integrals together with differential equations in canonical form. We obtain analytic differential equations efficiently from numerical samples over finite fields, fitting an ansatz built from symbol letters. The symbol alphabet itself is constructed from cut differential equations and we find that it can be written in a remarkably compact form. We comment on the analytic properties of the integrals and confirm the extended Steinmann relations, which govern the double discontinuities of Feynman integrals, to all orders in ϵ. We solve the differential equations in terms of generalized power series on single-parameter contours in the space of Mandelstam invariants. This form of the solution trivializes the analytic continuation and the integrals can be evaluated in all kinematic regions with arbitrary numerical precision.

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