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1

Optimal system of 1-d subalgebras and conserved quantities of a nonlinear wave equation in three dimensions arising in engineering physics

Adeyemo Oke Davies, Khalique Chaudry Masood

Acta Mechanica et Automatica

|

2024

|

Vol. 18, no. 2

177--192

EN

The construction of explicit structures of conserved vectors plays diverse crucial roles in the study of nonlinear science inclusive of the fact that they are invoked in developing appropriate numerical schemes and for other mathematical analyses. Therefore, in this paper, we examine the conserved quantities of a nonlinear wave equation, existing in three dimensions, and highlight their applications in physical sciences. The robust technique of the Lie group theory of differential equations (DEs) is invoked to achieve analytic solutions to the equation. This technique is used in a systematic way to generate the Lie point symmetries of the equation under study. Consequently, an optimal system of one-dimensional (1-D) Lie subalgebras related to the equation is obtained. Thereafter, we engage the formal Lagrangian of the nonlinear wave equation in conjunction with various gained subalgebras to construct conservation laws of the equation under study using Ibragimov’s theorem for conserved vectors.

2

Modifications of Evans price equilibrium model

Zabolotnii Serhii, Mogilei Sergii

Informatyka, Automatyka, Pomiary w Gospodarce i Ochronie Środowiska

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2023

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T. 13, nr 1

58--63

EN

The paper regards the classical Evans price equilibrium model in the free product market in the aspect of regarding the opportunities for expanding (modifying) the model given that is aimed at perfecting the accuracy of its mathematical formulating. As an accuracy criterion, we have chosen a summary quadratic deviation of the calculated indices from the given ones. One of the approaches of modifying the basic Evans model is suggesting there is a linear dependence between price function and time as well as its first and second derivatives. In this case, the model will be described through differential equation of second order with constant coefficients, revealing some oscillatory process. Besides, it is worth regarding a non-linear (polynomial) dependence between demand, supply and price. The paper proposes mathematical formulating for the modified Evans models that have been approbated for real indices of exchange rates fluctuations. It also proves that increase of the differential and/or polynomial order of the given model allows its essential accuracy perfection. Besides, the influence of arbitrary restricting circumstances of the model on its accuracy is regarded. Each expanded Evans model is accompanied by mathematically formulated price and time dependence.

PL

W artykule rozpatrzono klasyczny model równowagi cenowej Evansa na wolnym rynku produktów w aspekcie możliwości rozbudowy (modyfikacji) danego modelu, zmierzającej do udoskonalenia dokładności jego matematycznego sformułowania. Jako kryterium dokładności wybraliśmy sumaryczne, kwadratowe odchylenie obliczonych indeksów od zadanych. Jednym z podejść do modyfikacji podstawowego modelu Evansa jest sugerowanie istnienia liniowej zależności między funkcją ceny a czasem oraz jej pierwszą i drugą pochodną. W takim przypadku model będzie opisany równaniem różnicowym drugiego rzędu o stałych współczynnikach, ujawniającym pewien proces oscylacyjny. Ponadto warto zwrócić uwagę na nieliniową (wielomianową) zależność pomiędzy popytem, podażą i ceną. W artykule zaproponowano matematyczne sformułowania dla zmodyfikowanych modeli Evansa, które zostały zaaprobowane dla rzeczywistych indeksów wahań kursów walutowych. Udowodniono, że zwiększenie rzędu różniczkowego i/lub wielomianowego danego modelu pozwala na jego zasadniczą poprawę dokładności. Ponadto rozważany jest wpływ dowolnych okoliczności ograniczających model na jego dokładność. Każdemu rozszerzonemu modelowi Evansa towarzyszy matematycznie sformułowana zależność cenowa i czasowa.

3

Monodromy invariant Hermitian forms for second order Fuchsian differential equations with four singularities

Adachi Shunya

Opuscula Mathematica

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2022

|

Vol. 42, no. 3

361--391

EN

We study the monodromy invariant Hermitian forms for second order Fuchsian differential equations with four singularities. The moduli space of our monodromy representations can be realized by certain affine cubic surface. In this paper we characterize the irreducible monodromies having the non-degenerate invariant Hermitian forms in terms of that cubic surface. The explicit forms of invariant Hermitian forms are also given. Our result may bring a new insight into the study of the Painlevé differential equations.

4

Critical cases in neutral functional differential equations, arising from hydraulic engineering

Răsvan Vladimir

Opuscula Mathematica

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2022

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Vol. 42, no. 4

605--633

EN

This paper starts from several applications described by initial/boundary value problems for 1D (time and one space variable) hyperbolic partial differential equations whose basic properties and stability of equilibria are studied throughout the same properties for certain associated neutral functional differential equations. It is a common fact that asymptotic stability for neutral functional differential equations is normally obtained under the assumption of asymptotic stability of the difference operator associated to the aforementioned neutral functional differential equations. However the physically meaningful applications presented in the paper have the associated difference operator(s) in critical cases (their stability is, generally speaking, non-asymptotic). Consequently the stability of the considered application models is either non-asymptotic or fragile (in a sense introduced in the paper). The models represent an overview gathered from various fields, processed here in order to emphasize the associated neutral functional differential equations which, consequently, are a challenge to the usual approaches. In the concluding part there are suggested possible ways to overcome these difficulties.

5

Asymptotic behavior of even-order noncanonical neutral differential equations

Moaaz Osama, Muhib Ali, Abdeljawad Thabet, Santra Shyam S., Anis Mona

Demonstratio Mathematica

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2022

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Vol. 55, nr 1

28--39

EN

In this article, we study the asymptotic behavior of even-order neutral delay differential equation (a⋅(u+ρ⋅u∘τ)(n−1))′(ℓ)+h(ℓ)u(g(ℓ))=0,ℓ≥ℓ0, where n≥4, and in noncanonical case, that is, ∞∫a−1(s)ds<∞. To the best of our knowledge, most of the previous studies were concerned only with the study of n-order neutral equations in canonical case. By using comparison principle and Riccati transformation technique, we obtain new criteria which ensure that every solution of the studied equation is either oscillatory or converges to zero. Examples are presented to illustrate our new results.

6

$1000 SNOOK PRIZES FOR 2021 : The Information Dimensions of a Two-Dimensional Baker Map

Hoover Wm. G., Hoover C. G.

Computational Methods in Science and Technology

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2021

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Vol. 27, No. 2

93--94

EN

The fractal information dimension can be computed in three ways: (1) mapping points, (2) mapping regions (two-dimensional areas here), and (3) applying the Kaplan-Yorke conjecture. For the simplest nonequilibrium Baker N2 Map these three approaches can give different results. A pedagogical exploration and explanation of this situation is the 2021 Ian Snook Prize Problem.

7

Physics-guided neural networks (PGNNs) to solve differential equations for spatial analysis

Borzyszkowski Bartłomiej, Damaszke Karol, Romankiewicz Jakub, Świniarski Marcin, Moszyński Marek

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2021

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Vol. 69, nr 6

art. no. e139391

EN

Numerous examples of physically unjustified neural networks, despite satisfactory performance, generate contradictions with logic and lead to many inaccuracies in the final applications. One of the methods to justify the typical black-box model already at the training stage involves extending its cost function by a relationship directly inspired by the physical formula. This publication explains the concept of Physics-guided neural networks (PGNN), makes an overview of already proposed solutions in the field and describes possibilities of implementing physics-based loss functions for spatial analysis. Our approach shows that the model predictions are not only optimal but also scientifically consistent with domain specific equations. Furthermore, we present two applications of PGNNs and illustrate their advantages in theory by solving Poisson’s and Burger’s partial differential equations. The proposed formulas describe various real-world processes and have numerous applications in the area of applied mathematics. Eventually, the usage of scientific knowledge contained in the tailored cost functions shows that our methods guarantee physics-consistent results as well as better generalizability of the model compared to classical, artificial neural networks.

8

Practical parallelization of Gear-Nordsieck and Brayton-Gustavson-Hatchel stiff ODE solver

Stabrowski Marek

Annals of Computer Science and Information Systems

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2021

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Vol. 25

313--316

EN

The paper compares two ODE solvers using an example of a heat transfer equation. The sequential version of Brayton-Gustavson-Hatchel solver has been slightly inferior to Gear-Nordsieck solver. Algorithms profiling has led to the decision of parallelizing linear equation solving section and function evaluation. The first approach (parallelizing linear equations) improves performance of both algorithms. Second approach (parallelizing function evaluation) boosts BGH solver performance. Finally, it has been proved that wholly parallel version of BGH solver is more efficient with respect to processing time.

9

Governmental combat of migration between competing terrorist organisations

Hausken Kjell

Operations Research and Decisions

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2020

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Vol. 30, No. 3

21--46

EN

Each terrorist organisation is modelled with four coupled differential time equations for the evolution of ideologues, criminal mercenaries, captive participants, and capital sponsoring. Emigration of ideologues may cause unbounded growth of the organisation receiving ideologues. The organisation losing ideologues may reach a stationary state where ideologues are supported by capital sponsors and mercenaries. Emigration of mercenaries may cause the organisation losing mercenaries to experience growth. The organisation receiving mercenaries may lose capital sponsors permanently, allowing for the presence of mercenaries, or capital sponsors may rebound deterring mercenaries. Emigration of ideologues from one organisation to another requires more government intervention into the latter to ensure termination. Emigration of mercenaries from one organisation to another may require more government intervention into the latter, since mercenaries support ideologues. Competing terrorist organisations may facilitate their mutual extinction. Various intervention strategies are considered: the most threatening organisation is eliminated first, aided by competition from the least threatening, after which the remaining organisation is eliminated. The government’s instantaneous and accumulated utilities are analysed through time and compared, depending on emigration, competition, and government intervention strategies.

10

Metody rozwiązywania równań różnicowych oparte na analogii do równań różniczkowych

Augustyn Dariusz Rafał

MINUT Matematyka i Informatyka na Uczelniach Technicznych

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2020

|

Nr 2

163-170

PL

Artykuł prezentuje mało znane metody rozwiązywania liniowych równań różnicowych (rekurencyjnych), opisujących klasę dynamicznych układów dyskretnych. Metody te nie wykorzystują pojęcia transformaty Z czy Laplace'a-Carsona. Oparte są natomiast na pewnych analogiach do dobrze znanych metod rozwiązywania równań różniczkowych, opisujących dynamiczne układy ciągłe. W artykule zaprezentowano m.in. metody przewidywania i uzmienniania stałych, zastosowane do rozwiązywania równań różnicowych.

11

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.

12

A review: differential transform method for semi-analytical solution of differential equations

Rashidi M. M., Rabiei F., Naser Nia S., Abbasbandy S.

International Journal of Applied Mechanics and Engineering

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2020

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Vol. 25, no. 2

122--129

EN

In this article, the semi-analytical method known as the Differential Transform Method (DTM) for solving different types of differential equations is reviewed. First, basic definitions and formulas of DTM and Differential Transform-Padé approximation (DTM-Padé), which are used to increase the convergence and accuracy of DTM approximations, are discussed. Then both techniques of DTM and DTM-Padé, which have been successfully applied to partial differential equations, as well as the application of these methods in fluid mechanic and heat transfer are presented. In addition, the extension of DTM for integral differential equations and the fuzzy differential transformation method (FDTM) for fuzzy problems are discussed.

13

Differential equation of y = f(y') and interpretation of the solution in Mathematica program

Czajkowski Andrzej Antoni, Oleszak Wojciech Kazimierz, Frączak Piotr Stanisław

Problemy Nauk Stosowanych

|

2019

|

T. 10

25--34

EN

Introduction and aims: The paper presents a method of solving y=f(y') equations. The main aim of the work is to show how to solve this type of differential equations. In addition, the purpose of the discussion is to present the appropriate algorithms in Mathematica program, which are used to present the geometric interpretation of the obtained solutions. Material and methods: The sources contain material on the subject of differential equations. The method of mathematical analysis has been used. Results: In the analysis of selected examples, the method of substitution of new variable t has been used and the solution of the studied differential equation has been obtained in the form of the system of equations x=x(t) and y=y(t). Conclusion The solution of the differential equation of the type y=f(y') in the form of a system of equations x=x(t) and y=y(t) can be interpreted graphically using an appropriately used algorithm in Mathematica numerical program.

PL

Wstęp i cele: W pracy przedstawiono metodę rozwiązywania równań typu y=f(y'). Głównym celem pracy jest pokazanie sposobu rozwiązywania tego typu równań różniczkowych. Ponadto celem rozważań jest przestawienie odpowiednich algorytmów w programie Mathematica, które służą do przedstawienia interpretacji geometrycznej otrzymanych rozwiązań. Materiały i metody: Źródła zawierają materiał dotyczący tematyki równań różniczkowych. Zastosowano metodę analizy matematycznej. Wyniki: W analizie wybranych przykładów zastosowano metodę podstawienia nowej zmiennej t i otrzymano rozwiązanie badanego równania różniczkowego w postaci układu równań x=x(t) i y=y(t). Wniosek: Rozwiązanie równania różniczkowego typu y=f(y') w postaci układu równań x=x(t) i y=y(t) można zinterpretować graficznie stosując odpowiednio zastosowany algorytm w programie numerycznym Mathematica.

14

Differential equation of x = f(y') and interpretation of the solution in Mathematica program

Czajkowski Andrzej Antoni, Oleszak Wojciech Kazimierz, Frączak Piotr Stanisław

Problemy Nauk Stosowanych

|

2019

|

T. 10

15--24

EN

Introduction and aims: The paper presents a method of solving x=f(y') equations. The main aim of the work is to show how to solve this type of differential equations. In addition, the purpose of the discussion is to present the appropriate algorithms in Mathematica program, which are used to present the geometric interpretation of the obtained solutions. Material and methods: The sources contain material on the subject of differential equations. The method of mathematical analysis has been used. Results: In the analysis of selected examples, the method of substitution of new variable t has been used and the solution of the studied differential equation has been obtained in the form of the system of equations x=x(t) and y=y(t). Conclusion: The solution of the differential equation of the type x=f(y') in the form of a system of equations x=x(t) and y=y(t) can be interpreted graphically using an appropriately used algorithm in Mathematica numerical program.

PL

Wstęp i cele: W pracy przedstawiono metodę rozwiązywania równań typu y=f(y'). Głównym celem pracy jest pokazanie sposobu rozwiązywania tego typu równań różniczkowych. Ponadto celem rozważań jest przestawienie odpowiednich algorytmów w programie Mathematica, które służą do przedstawienia interpretacji geometrycznej otrzymanych rozwiązań. Materiały i metody: Źródła zawierają materiał dotyczący tematyki równań różniczkowych. Zastosowano metodę analizy matematycznej. Wyniki: W analizie wybranych przykładów zastosowano metodę podstawienia nowej zmiennej t i otrzymano rozwiązanie badanego równania różniczkowego w postaci układu równań x=x(t) i y=y(t). Wniosek: Rozwiązanie równania różniczkowego typu y=f(y') w postaci układu równań x=x(t) i y=y(t) można zinterpretować graficznie stosując odpowiednio zastosowany algorytm w programie numerycznym Mathematica.

15

The NF-κB network as an example of a regulatory network with a positive and negative feedback loop

Grajek Julia, Bodnar Marek

Mathematica Applicanda

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2019

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Vol. 47, no. 2

165--176

EN

The family of transcription factors NF-κB plays a crucial role in immune response regulation, cell proliferation and cell survival; therefore, deregulated NF-κB activation results in severe health problems. However, its elaborate regulatory network is not yet fully understood. In this paper, we propose and analyze modifications of a mathematical model of the regulatory network that considers the positive feedback between NF-κB and cytokines and the negative feedback between NF-κB and its inhibitors. This mathematical framework captures the transient dynamics of NF-κB while remaining simple enough to obtain a stability condition of the equilibria. We anticipate that a deeper understanding of the NF-κB framework will increase the effectiveness of therapeutic strategies based on NF-κB inhibition. Moreover, the modified model is generic enough to prove useful in modelling different biological processes.

PL

Rodzina czynników transkrypcyjnych NF-κB pełni ważną rolę w regulacji odpowiedzi immunologicznej oraz profliferacji i przeżyciu komórek, w związku z czym niepoprawna aktywacja NF-κB prowadzi do poważnych problemów zdrowotnych. Jednak skomplikowany sposób aktywacji i regulacji NF-κB nie jest jeszcze w pełni zbadany. W tej pracy proponujemy i analizujemy dwie wersje modelu matematycznego uwzględniającego dodatnie sprzężenie zwrotne między NF-κB i cytokinami oraz ujemne sprzężenie zwrotne między NF-κB i jego inhibitorami. Ten model matematyczny obrazuje przejściową aktywację NF-κB, i jest jednocześnie na tyle prosty, aby pozwolić na otrzymanie analitycznych warunków na stabilność stanów stacjonarnych. Oczekujemy, że lepsze zrozumienie systemu regulacji NF-κB zwiększy efektywność terapii opartych na inhibicji tego czynnika transkrypcyjnego. Ponadto, zmodyfikowany model jest na tyle ogólny, że jego analiza może się okazać przydatna w modelowaniu innych procesów biologicznych.

16

On certain modification of age-dependent predator-prey model

Dawidowicz Antoni Leon, Poskrobko Anna

Mathematica Applicanda

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2019

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Vol. 47, no. 1

117--125

EN

This work proposes a new model of coexistence of a predator population with a population of prey. In works [5] and [7] it is assumed that the number of prey aged x eaten by predators aged y is directly proportional to the number of prey aged x and the number of predators aged y. This paper presents a more general model. First of all, the dependency is functional, i.e. the chances of being eaten are affected by the structure of the whole population. In addition, this dependence is not bili near because the predator, after satisfying its hunger, will give up the hunt for prey.

PL

Praca niniejsza proponuje nowy model koegzystencji populacji drapieżcy z populacją ofiar. W pracach [5] i [7] założono, że liczba ofiar w wieku x zjadanych przez drapieżcę w wieku y jest wprost proporcjonalna do liczby ofiar w wieku x i do liczby drapieżców w wieku y. W niniejszej pracy zaprezentowany jest ogólniejszy model. Przede wszystkim zależność jest funkcjonalna, czyli na szanse zjedzenia ma wpływ struktura całej populacji. Poza tym zależność ta nie jest dwuliniowa, gdyż drapieżca po zaspokojeniu głodu zrezygnuje z polowania na ofiary.

17

Convergence of finite-dimensional approximations for mixed-integer optimization with differential equations

Hante Falk M., Schmidt Martin

Control and Cybernetics

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2019

|

Vol. 48, No. 2

209--226

EN

We consider a direct approach to solving the mixedinteger nonlinear optimization problems with constraints depending on initial and terminal conditions of an ordinary differential equation. In order to obtain a finite-dimensional problem, the dynamics are approximated using discretization methods. In the framework of general one-step methods, we provide sufficient conditions for the convergence of this approach in the sense of the corresponding optimal values. The results are obtained by considering the discretized problem as a parametric mixed-integer nonlinear optimization problem in finite dimensions, where the step size for discretization of the dynamics is the parameter. In this setting, we prove the continuity of the optimal value function under a stability assumption for the integer feasible set and second-order conditions from nonlinear optimization. We address the necessity of the conditions on the example of pipe sizing problems for gas networks.

18

Regular classes of linear extensions of dynamical systems on a torus

Morawiak M.

Silesian Journal of Pure and Applied Mathematics

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2018

|

Vol. 8, iss. 1

21--31

EN

The paper presents new methods of construction of the Lyapunov function for some sets of linear extensions of dynamic systems on torus.The paper is divided into two parts. The first part contains a brief theoretical introduction. In the second part are presented results of study and examples.

19

On regularity of linear systems of differential equations

Morawiak M.

Silesian Journal of Pure and Applied Mathematics

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2018

|

Vol. 8, iss. 1

5--20

EN

The paper presents the theorems about regularity of systems of differential equations. The paper is divided into two parts. The first part contains a theoretical introduction. The second part contains theorems which allows to determine the regularity of the system using the generalized Lyapunov function.

20

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

|

2018

|

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.