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1

Application of LDG scheme to solve semi-differential equations

Izadi Mohammad

Journal of Applied Mathematics and Computational Mechanics

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2019

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Vol. 18, nr 4

27--39

EN

In the current work, we investigate a technique based on discontinuous Galerkin method for the numerical approximation of semi-differential equations with Caputo’s fractional derivative. In this approach, using the natural upwind fluxes enables us to solve the model problem element by element locally in each subintervals and there is no need to solve a full global matrix. Numerical experiments are given to verify the efficiency and accuracy of the proposed method. Numerical solutions are compared with the exact solutions as well as the numerical solutions obtained by other available well-established computational procedures. The results show that the LDG method is more accurate for solving this class of differential equation with relatively low degrees of polynomials and number of elements.

2

Metoda równań różnicowych dla opisu stanu nieustalonego obwodu szeregowego RLC załączonego do napięcia sinusoidalnego z użyciem programu MathCAD

Frączak Piotr Stanisław, Czajkowski Andrzej Antoni

Problemy Nauk Stosowanych

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2018

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T. 9

33--40

PL

Wstęp i cel: W pracy przedstawiono opis i symulacje stanu nieustalonego w obwodzie elektrycznym szeregowym RLC. Pokazano zastosowanie metody równań różnicowych do rozwiązywania równania różniczkowego drugiego rzędu w programie MathCAD. Materiał i metody: W wyniku zastosowania metody równań różnicowych wskazano na możliwość przejścia od równań różniczkowych liniowych drugiego rzędu o stałych współczynnikach. Zastosowano metodę analityczno-numeryczną. W analizie numerycznej użyto program MathCAD. Wyniki: Otrzymano jednakowy kształt przebiegu krzywej prądu nieustalonego przy wyznaczaniu metodą równań różnicowych drugiego rzędu i równaniem różniczkowo-całkowym z wykorzystaniem przekształcenia odwrotnego Laplace’a. Ponadto otrzymane kształty prądów nieustalonych w rozpatrywanym obwodzie elektrycznym zweryfikowano w programie numerycznym PSpice Wniosek: Stosując zarówno metodę równań różnicowych i metodę przekształceń Laplace’a otrzymuje się jednakowe przebiegi prądu nieustalonego w funkcji czasu.

EN

Introduction and aim: Some description and simulation of the transient in RLC circuit have been presented in this paper. Also has been shown the application of the Laplace transform to solve the differential equation. Material and methods: By using the Laplace transformation to the option of the transition from linear differential equations of the second order with constant coefficients to the algebraic equations. The analytical and numerical methods have been used in the considerations. In numerical analysis, a reversed Laplace transform was applied by using the MathCAD program. Results: It has been obtained the same curve shape of the transient current at the determination by the second-order differential equation (classical solution) and the different-integral equation by using the inverse Laplace transform. In addition, the obtained shapes of transients in the considered electrical circuit were verified in the numerical program PSpice Conclusion: By applying both the Laplace transform method and the analytical method, the same transient currents are obtained as a function of time.

3

Persistence and global attractivity for a discretized version of a general model of glucose-insulin interaction

Huong D. C.

Demonstratio Mathematica

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2016

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Vol. 49, nr 3

302--318

EN

In this paper, we construct a non-standard finite difference scheme for a general model of glucose-insulin interaction. We establish some new sufficient conditions to ensure that the discretized model preserves the persistence and global attractivity of the continuous model. One of the main findings in this paper is that we derive two important propositions (Proposition 3.1 and Proposition 3.2) which are used to prove the global attractivity of the discretized model. Furthermore, when investigating the persistence and, in some cases, the global attractivity of the discretized model, the nonlinear functions f and h are not required to be differentiable. Hence, our results are more realistic because the statistical data of glucose and insulin are collected and reported in discrete time. We also present some numerical examples and their simulations to illustrate our results.

4

An explicit formula for solutions of some system of linear delay difference equations

Kwapisz M., Łachacz M.

Journal of Applied Analysis

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2002

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Vol. 8, nr 1

23-32

EN

The aim of this paper is to provide an explicit formula for solutions of the following system of delay difference equations (wzór) where (wzór) ;αn= [n/k] (the symbol [x] stands for entire part of the real number x and k is a fixed positive integer). (An), (Bn), n∈ N, are sequences of square matrices of order m, (fn) is a sequence of vectors from Rm. From this formula conditions for the stability and asymptotic stability of solutions are derived.

5

On the structure of the set of solutions of some boundary value problems for differential equations

Bielawski D.

Demonstratio Mathematica

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1998

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Vol. 31, nr 1

135-142

6

On the behaviour of rational first order difference equation with constant coefficients

Magnucka-Blandzi E., Popenda J.

Demonstratio Mathematica

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1998

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Vol. 31, nr 2

405-416

7

On row-finite systems of differential equations in Banach spaces

Herzog G.

Demonstratio Mathematica

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1998

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Vol. 31, nr 4

835-840

EN

We prove an existence theorem for initial value problems in Banach spaces, including a wide class of row-finite systems of ordinary differential equations.