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1
Hasimoto surfaces for two classes of curve evolution in Minkowski 3-space
EN
In this work, we study Hasimoto surfaces for the second and third classes of curve evolution corresponding to a Frenet frame in Minkowski 3-space. Later, we derive two formulas for the differentials of the second and third Hasimoto-like transformations associated with the repulsive-type nonlinear Schrödinger equation.
EN
In this paper we present the mathematical background of the four most used numerical methods of solving equations and few examples of Python applications that find the approximations of the roots of the given equations. We also compare the exact and approximate solutions of polynomial equations of third degree. Exact solutions are obtained with usage of Cardano formulae by the help of Mathematica environment, the approximate ones – based on the selected numerical methods by the help of applications written in Python language.
3
EN
A fourth order nonlinear evolution equation, which is a good starting point for the study of nonlinear water waves as first pointed out by Dysthe (1979) is derived for gravity waves propagating at the interface of two superposed fluids of infinite depth in the presence of air flowing over water and a basic current shear. A stability analysis is then made for a uniform Stokes gravity wave train. Graphs are plotted for the maximum growth rate of instability and for wave number at marginal stability against wave steepness for different values of air flow velocity and basic current shears. Significant deviations are noticed from the results obtained from the third order evolution equation, which is the nonlinear Schrödinger equation.
4
EN
Fourth order nonlinear evolution equations, which are a good starting point for the study of nonlinear water waves, are derived for deep water surface capillary gravity waves in the presence of second waves in which air is blowing over water. Here it is assumed that the space variation of the amplitude takes place only in a direction along which the group velocity projection of the two waves overlap. A stability analysis is made for a uniform wave train in the presence of a second wave train. Graphs are plotted for the maximum growth rate of instability wave number at marginal stability and wave number separation of fastest growing sideband component against wave steepness. Significant improvements are noticed from the results obtained from the two coupled third order nonlinear Schrödinger equations.
PL
W artykule zaproponowano wykorzystanie algorytmów ewolucyjnych w celu przeprowadzania analizy oczkowych sieci hydraulicznych. Zadaniem algorytmu ewolucyjnego jest wyznaczenie wartości przepływów w poszczególnych gałęziach arbitralnie zadanej sieci hydraulicznej. W artykule zaproponowano sposób kodowania rozwiązań na materiale genetycznym ewoluujących osobników oraz zdefiniowano postać funkcji dopasowania pozwalającej na ocenę rozwiązań odnajdowanych w toku procesów ewolucyjnych.
EN
In the paper we propose to use evolutionary algorithms for the purpose of analysis of hydraulic networks. The aim of evolutionary algorithm is to determine the values of flow in the branches of arbitrarily given hydraulic network. In the paper we propose the way of coding of solutions on genetic material of evolving individuals and we define the fitness function to evaluate solutions found during the process of evolution.
PL
Artykuł opisuje projekt obwodowego modelu numerycznego indukcyjnego nadprzewodnikowego ogranicznika prądu. Przedstawiono założenia modelu matematycznego, charakterystykę modelowanego obiektu oraz wyniki symulacji.
EN
This paper describes project the peripheral numerical model of induction superconducting fault current limiter. The assumptions of a mathematical model, characteristics of a modeled device and the simulation results where presented.
7
EN
This paper presents a leader glowworm swarm optimization algorithm (LGSO) for solving nonlinear equations systems. Since glowworm swarm optimization algorithm has bad optimized ability at high dimension, proposing glowworm swarm optimization algorithm with leader mechanism to strengthen the global optimization ability. Through various types nonlinear equations testing, experiment results show that the proposed algorithm has strong global searching capability and quickly finding the solutions of the equations, thus obviously improving the optimization global ability.
PL
Zaprezentowano optymalizacyjny algorytm mrówkowy “świetlikowy” do rozwiązywania system równań nieliniowych. Ponieważ algorytm ten ma słabe możliwości optymalizacyjne przy dużych rozmiarach wprowadzono wspomagający mechanizm prowadzący „leader”.
8
Analytical solution of forced-convective boundary-layer flow over a flat plate
EN
In this letter, the problem of forced convection heat transfer over a horizontal flat plate is investigated by employing the Adomian Decomposition Method (ADM). The series solution of the nonlinear differential equations governing on the problem is developed. Comparison between results obtained and those of numerical solution shows excellent agreement, illustrating the effectiveness of the method. The solution obtained by ADM gives an explicit expression of temperature distribution and velocity distribution over a flat plate.
PL
W artykule przedstawiono zastosowanie metody dekompozycji Adomiana do wymuszonego, konwekcyjnie przepływu ciepła w poziomej, płaskiej płycie. Rozwiązania nieliniowych równań różniczkowych opisujących zagadnienie poszukiwana w postaci szeregów Adomiana. Z porównania otrzymanych wyników z wynikami innych metod numerycznych wynika doskonała ich zgodność, która potwierdza skuteczność zastosowanej metody. Otrzymane rozwiązanie pozwoliło jednoznacznie wyznaczyć rozkład i prędkości mian temperatury w analizowanej płycie.
EN
Fourth order nonlinear evolution equations, which are a good starting point for the study of nonlinear water waves as first pointed out by Dysthe (1979) and later elaborated by Janssen (1983), are derived for deep water capillary-gravity waves in the presence of air flowing over water. Stability analysis is then made for a uniform Stokes capillary gravity wave train. Graphs are plotted for the maximum growth rate of instability, the frequency at marginal stability and the frequency separation for fastest growing side-band component as a function of wave steepness. Significant deviations are noticed from the results obtained from the third-order evolution equation, which is the nonlinear Schrödinger equation.
EN
We study the initial-boundary value problem for a nonlinear wave equation given by [...] where p > 2, q > l, K, lambda are given constants and uo, u1, F are given functions, the unknown function u(x,t) and the unknown boundary value P (t) satisfy the following nonlinear integral equation [...] where K1, alpha, beta are given constants and g, k arę given functions. In Part 1 we prove a theorem of existence and uniqueness of a weak solution (u, P) of problem (1), (2). The proof is based on the Faedo-Galerkin method associated with a priori estimates, weak convergence and compactness techniques. In Part 3 we obtain an asymptotic expansion of the solution (u, P) of the problem (1), (2) up to order N+1 in three small parameters K, lambda, K1.
11
On existence of solutions to nonlinear equations with degenerate mappings
EN
The paper presents recent advances in p-regularity theory. The main result of this theory gives a detailed description of the structure of the zero set of an irregular nonlinear mapping. We illustrate the theory with an application to the problem of existence of the solutions to the nonlinear equations with singular mappings.
EN
In this paper, the iterative solution is studied for equation x+Tx =f with a Lipschitz K-subaccetive operator in arbitrary Banach spaces, some previously results are generalized.
13
Stability of iterative procedures for multivalued maps in metric spaces
EN
Let (X,d) be a metric space and T a self-map of X. Let Xn+i = f(T,Xn) denote some iterative procedure. Let {xn} be convergent to a fixed point u of T and {yn} be an arbitrary sequence in X. Set En=d[yn+i, f(T, yn)], n = 0,1,2,..., then the iterative procedure f(T,Xn) is T-stable provided that limEn = 0 implies that limnyn = u. This definition has been extended by Singh and Chadha [34] to discuss the problem of stability for multivalued operators on metric spaces. The purpose of this paper is to present a fixed point theorem for generalized multivalued contractions on a setting more general than metric spaces. The same is utilized to discuss the problem of stability of iterative procedures in multivalued analysis. Some special cases due to Stefan Czerwik and others are discussed as special cases.
PL
W pracy podano schemat oparty na metodzie elementów skończonych rozwiązywania pewnego nieliniowego równania, opisującego pole koncentracji cząsteczek domieszki w ośrodku ciągłym z uwzględnieniem kombinacji dyfuzji i konwekcji.
EN
In the paper on the basis of the finite element method the technique of solving non-linear equation for convection diffusion is presented.
EN
In this paper we shall construct the solution to the nonlinear polyparabolic problem for the cylindrical domain with limit conditions of Riquer type. To construct the solution we shall apply the convenient heat polyparabolic potentials with unknown densities and polyparabolic thermal potential compatible with the source function. We reduce the considered problem to a system of the nonlinear integro-differential equations examined on the base of the well-known Banach's fixed point theorem.
PL
Celem pracy jest dowód twierdzenia o istnieniu i o jednoznaczności rozwiązania nieliniowego równania poliparabolicznego Pmu(x, t) = ƒ(x, t, u(x, t), Pu(x, t), P2u(x, t),..., Pm-1(x, t)), gdzie x = (x1,x2), P = Δ- Dt,, Δ= D2/x1 + D2/x2, Pm = P(Pm-1) i 1 < m ∈ N jest ustaloną liczbą. Rozwiązanie powyższego równania jest konstruowane w obszarze D = {(x,t) : x ∈ D1,t ∈ (0,T)}, D1 = {(x,0 ): /x/ < R} i spełnia następujące warunki początkowe Piu(x,0) = ƒi(x), x ∈ D1, i = 0,1,...,m-1, m ∈ N i warunki brzegowe Piu(x, t) = hi(x, t), (x, t) ∈ S = {(x, t) : x ∈ B{D1) x (0,T)}, B(D1) = {{x,0) : /x/=R}, i = 0,1,2,...,m - 1, m ∈ N Funkcje ƒ ,ƒi, hi, i = 0,1,2,..., m-1, m ∈ N są dane. Powyższy problem początkowo-brzegowy jest zredukowany do nieliniowego układu równań różniczkowo-całkowych, który jest rozwiązany na bazie dobrze znanego twierdzenia Banacha o punkcie stałym.
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