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1

New Numerical Approach for Solving Abel’s Integral Equations

Anapalı Şenel Ayşe, Öztürk Yalçın, Gülsu Mustafa

Foundations of Computing and Decision Sciences

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2021

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Vol. 46, No. 3

255--271

EN

In this article, we present an efficient method for solving Abel’s integral equations. This important equation is consisting of an integral equation that is modeling many problems in literature. Our proposed method is based on first taking the truncated Taylor expansions of the solution function and fractional derivatives, then substituting their matrix forms into the equation. The main character behind this technique’s approach is that it reduces such problems to solving a system of algebraic equations, thus greatly simplifying the problem. Numerical examples are used to illustrate the preciseness and effectiveness of the proposed method. Figures and tables are demonstrated to solutions impress. Also, all numerical examples are solved with the aid of Maple.

2

Comparison of the approximate solution with the solution of the differential equation for the static problem of deep-profiled sandwich panels

Pozorska Jolanta

Journal of Applied Mathematics and Computational Mechanics

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2020

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Vol. 19, nr 4

81--88

EN

The article presents an approximate method of solving the problem of statics of sandwich elements with deep-profiled facings. This method is compared with the widely accepted theory of sandwich beams. The example of uniformly loaded, simply-supported, single-span sandwich beam is presented. The results of displacements, internal forces and stresses obtained by both methods are compared. The results confirm the good compliance of the presented solutions, in particular in terms of displacements.

3

Iterative techniques with computer realization for initial value problems for Riemann-Liouville fractional differential equations

Agarwal Ravi, Golev A., Hristova S., O’Regan D.

Journal of Applied Analysis

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2020

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Vol. 26, nr 1

21--47

EN

The main aim of this paper is to suggest some algorithms and to use them in an appropriate computer environment to solve approximately the initial value problem for scalar nonlinear Riemann-Liouville fractional differential equations on a finite interval. The iterative schemes are based on appropriately defined lower and upper solutions to the given problem. A number of different cases depending on the type of lower and upper solutions are studied and various schemes for constructing successive approximations are provided. The suggested schemes are applied to some problems and their practical usefulness is illustrated.

4

Particle trajectories on manifolds : some remarks on harmonic oscillator on a sphere

Chyła Kacper S.

Mathematica Applicanda

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2019

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Vol. 47, no. 1

45--59

EN

The classical harmonic oscillator with spherical configuration space is studied. The equations of motion are derived from Hamiltonian formulation of mechanics and approximate solutions in some special cases are obtained. A reliable symplectic integrator is used to obtain numerical solutions which are then compared with analytical approximations. The overall impact of positive curvature on the solutions is discussed and illustrated with comparison of solutions on cylindrical and spherical surfaces.

PL

W pracy analizowany jest klasyczny oscylator harmoniczny z sferyczną przestrzenią konfiguracyjną. Równania ruchu pochodzą z formułowania mechaniki Hamiltona i uzyskiwane są przybliżone rozwiązania w niektórych szczególnych przypadkach. Porównano rozwiązania numeryczne z proponowanej metody z przybliżeniami analitycznymi. Omówiono ogólny wpływ dodatniej krzywizny na rozwiazania i zilustrowano przez porównanie rozwiązań na powierzchniach cylindrycznych i sferycznych.

5

Approximate solutions of a certain second order integrodifferential equation

Pachpatte B. G.

Fasciculi Mathematici

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2010

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Nr 45

99-109

EN

The main aim of this paper is to study the approximate solutions of a certain second order Volterra type integrodifferential equation with given initial values. A variant of a certain basic integral inequality with explicit estimate is used to establish the results.