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1

A general study of fundamental solutions in aniotropicthermoelastic media with mass diffusion and voids

Chawla Vijay, Kamboj Deepmala

International Journal of Applied Mechanics and Engineering

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2020

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Vol. 25, no. 4

22--41

EN

The present paper deals with the study of a fundamental solution in transversely isotropic thermoelastic media with mass diffusion and voids. For this purpose, a two-dimensional general solution in transversely isotropic thermoelastic media with mass diffusion and voids is derived first. On the basis of the obtained general solution, the fundamental solution for a steady point heat source on the surface of a semi-infinite transversely isotropic thermoelastic material with mass diffusion and voids is derived by nine newly introduced harmonic functions. The components of displacement, stress, temperature distribution, mass concentration and voids are expressed in terms of elementary functions and are convenient to use. From the present investigation, some special cases of interest are also deduced and compared with the previous results obtained, which prove the correctness of the present result.

2

On General Solution for Fractional Differential Equations with Not Instantaneous Impulses

Zhang X., Zhang X., Cao H.

Fundamenta Informaticae

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2017

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Vol. 151, nr 1/4

355--369

EN

In this paper we mainly study a kind of fractional differential equations with not instantaneous impulses, and find the equivalent equations of the impulsive system. The obtained result discovers that there exist general solution for the impulsive system. Next, an example is given to illustrate the obtained result.

3

The computer method of construction of the general solution of the linear differential equation of the third order

Stepaniuk G., Chychuryn A.

Studia i Materiały / Europejska Uczelnia Informatyczno-Ekonomiczna w Warszawie

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2014

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Nr 2(8)

17--26

EN

In the paper we consider the third order linear differential equation, coefficients of which satisfies the system of two differential equations of the first order. By means of the computer method which is based on the usage of Schwarzian derivative the general solution of the linear equation is found. Two examples which contain the solutions of the initial problem is considered. For the obtained solutions the visualization in the real domain is given. All the calculations and visualization are realized in Mathematica 9 system.

PL

W pracy rozważamy nieliniowe równanie różniczkowe trzeciego rzędu, którego współczynniki spełniają układ dwóch równań różniczkowych pierwszego rzędu. Za pomocą metody komputerowej, która opiera się na wykorzystaniu pochodnej Schwarzego, znaleziono ogólne rozwiązanie tego równania liniowego. Rozpatrywane są dwa przykłady zawierające rozwiązanie zagadnienia Cauchiego. Dla znalezionych rozwiązań podana jest wizualizacja w dziedzinie rzeczywistej. Wszystkie obliczenia i wizualizacje realizowane są w systemie Mathematica 9.

4

The computer method of construction of the general solution of the nonlinear differential equation of the fourth order

Chichurin A.

Studia i Materiały / Europejska Uczelnia Informatyczno-Ekonomiczna w Warszawie

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2014

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Nr 1(7)

39--47

EN

In the paper we consider the fourth order nonlinear differential equation which arises when investigating the fourth order linear differential equations using the Schwarzian derivative. As a result the general solution is obtained. An example of a solution of the Cauchy problem is given as well as visualization on the real line and the complex plane. All the calculations and visualization are realized in Mathematica 9 system.

PL

W pracy rozważamy nieliniowe równanie różniczkowe czwartego rzędu, które powstaje podczas badania czwartego rzędu liniowych równan różniczkowych przy pomocy pochodnej Schwarza. W rezultacie uzyskujemy ogólne rozwiązanie. Podany jest przykład rozwiązania zagadnienia Cauchy’ego, jak również wizualizacji na płaszczyznach prostej i zespolonej. Wszystkie obliczenia i wizualizacje sa realizowane w kodach programu Mathematica 9.

5

Sur la méthode de l'intégrale particuliere et sur ses conséquences pour l'équation de Riccati et pour les équations différentielles linéaires et homogenes d'ordre supérieur

Kapcia A.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

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2007

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Vol. 47, [Z] 2

255-283

EN

This paper presents the method of particular solution for solving the Riccati equation and linear homogenous equations of second and third order, as well as its certain application to linear homogenous equations of n-th order. The conditions of effective integrability for equations (0.1) and (0.2) are expressed in symbolic (operator) form and also for equation (0.3) in fully expanded form. There have been proved three theorems which state the following: for any subclass of differential equations of the form (0.1), (0.2), (0.3), if there are known, respectively: a particular solution y0, a particular solution u0, two linearly independent particular solutions u1, u2, then it is possible to construct superclasses of differential equations of the given class, using classes cited in [6, 7, 8, 9]. Moreover, one may obtain their effectively integrable generalizations. Numerous examples provided illustrate the above results. The article presents also a practical way of applying the method of particular solution to linear equations of n-th order. This method enables us to integrate more general equations than those described in [4, 5, 14] of the form (0.1), (0.2), (0.3), (0.4) for which the particular solutions are cited therein.