Parallel solution of telegraph line

This paper deals with a model of the telegraph line that consists of system of ordinary differential equations, rather than partial differential telegraph equation. Numerical solution is then based on an original mathematical method. This method uses the Taylor series for solving ordinary differential equations with initial condition - initial value problems in a non-traditional way. Systems of ordinary differential equations are solved using variable order, variable step-size Modern Taylor Series Method. The Modern Taylor Series Method is based on a recurrent calculation of the Taylor series terms for each time interval. The second part of paper presents the solution of linear problems which comes from the model of telegraph line. All experiments were performed using MATLAB software, the newly developed linear solver that uses Modern Taylor Series Method. Linear solver was compared with the state of the art solvers in MATLAB and SPICE software.

Using the Lie symmetry approach, the authors have examined exact and numerical solutions of coupled short pulse equation with time-dependent coefficients. The method reduces the system of partial differential equations to a system of ordinary differential equations according to the Lie symmetry admitted. In particular, we found the relevant system of ordinary differential equations for all optimal subgroups. The system of ordinary differential equations is further studied in general to obtain exact and numerical solutions. Several new physically important families of exact and numerical solutions are derived.

Using the Lie symmetry approach, we have examined herein the system of partial differential equations corresponding to the Einstein–Maxwell equations for a static axially symmetric spacetime. The method used reduces the system of partial differential equations to a system of ordinary differential equations according to the Lie symmetry admitted. In particular, we found the relevant system of ordinary differential equations is all optimal subgroups. The system of ordinary differential equations is further solved in general to obtain exact solutions. Several new physically important families of exact solutions are derived.

It is well known that a model of any electrical circuit can be translated into a system of ordinary differential equations. Usually, the simulation is performed in the time-domain using classical discrete approaches convenient for both linear and non-linear systems. An other method consists in the quantization on the variable magnitude which is based on discrete events. The discrete event system specification formalism is adapted to simulate any electrical circuit described with a system of ordinary differential equations using the system state quantification. In this paper, a solution to make more efficient the magnitude quantification for a system of linear ordinary differential equations is given. The proposed technique is used to simulate a simple three phase coupled power systems and it is compared to other solutions given by the explicit integration using formal language and the time-domain discretization using fixed-step fifth order Runge-Kutta integration method.

We consider boundary value problems with periodic boundary conditions for first-order linear systems of impulsive ordinary differential equations with unknown right-hand sides and jumps of solutions at the impulse points entering into the statement of these problems which are assumed to be subjected to some quadratic restrictions. From indirect noisy observations of their solutions on a finite system of intervals, we obtain the optimal, in certain sense, estimates of images of their right-hand sides under linear continuous operators. Under the condition that the unknown correlation functions of noises in observations belong to some special sets, it is established that such estimates and estimation errors are expressed explicitly via solutions of special periodic boundary value problems for linear impulsive systems of ordinary differential equations.

3 - LINEAR SYSTEMS

71 - Integrable Combinations

The material presented below amounts roughly to the so-called fundamental theorems of Lie and their implications concerning Lie algebras, Lie subgroups and subalgebras. By straightforward and fairly elementary steps, we shall extend the concept of Lie group to include groups admitting coordinate systems in which the functions defining ab, the group product of elements a, b, possess continuous first order derivatives in the coordinates of a and satisfy a Lipschitz condition in those of b. That such groups are equivalent to classical (i.e. analytical) Lie groups was announced in 1936 by van Kampen, although apparently van Kampen published nothing by way of proof beyond a certain decisive uniqueness theorem concerning systems of ordinary differential equations. (This is contained in our theorem (12.1); the proof given below is essentially that of van Kampen [3]). It is conceivable that the condition relative to coordinates of b could be weakened or dispensed with entirely. For, the only part the condition in question plays in our development is to make it certain that the inverse of a certain mapping-the mapping into canonical coordinates-is single-valued. In this connection mention should be made of the paper [1J of Garrett Birkhoff in which it is shown that the existence of continuous first order derivatives of ab with respect to the coordinates of both the a's and the b's is sufficient for defining a Lie group. We shall consider only groups with a finite number of real parameters (whereas the groups considered by Birkhoff are not necessarily finite dimensional). This makes it possible to use the standard existence theorems for systems of ordinary real differential equations. We make no use whatever of the theory of partial differential equations. We maintain throughout a purely local point of view in the sense that we consider only what happens in the neighborhood of the identity. Historical notes. The proof of the uniqueness of square roots (8.5) is due to Claude Chevalley (unpublished). The proof given below of Lie's theorem that a linear system of vector fields which is closed under commutation defines a Lie group is, we believe, due to van der Waerden [4]. We have supplied a number of preliminary lemmas which make the proof rigorously applicable to the case in which the vector fields are only assumed to possess continuous first order derivatives. The theorem in ?17 that every local subgroup of a Lie group is a Lie group is due essentially to E. Cartan [2]. 481

We consider mathematical models of infectious diseases built by G. I. Marchuk in his well-known book on immunology. These models are in the form of systems of ordinary delay differential equations. We add a distributed control in one of the equations describing the dynamics of the antibody concentration rate. Distributed control looks here naturally since the change of this concentration rather depends on the corresponding average value of the difference of the current and normal antibody concentrations on the time interval than on their difference at the point t only. Choosing this control in a corresponding form, we propose some ideas of the stabilization in the cases, where other methods do not work. The main idea is to reduce the stability analysis of a given integro-differential system of the order n, to one of the auxiliary systems of the order \(n+m\), where m is a natural number, which is “easy” for this analysis in a corresponding sense. Results for these auxiliary systems allow us to make conclusions for the given integro-differential system of the order n. We concentrate our attempts in the analysis of the distributed control in an integral form. An idea of reducing integro-differential systems to systems of ordinary differential equations is developed. We present results about the exponential stability of stationary points of integro-differential systems using the method based on the presentation of solution with the help of the Cauchy matrix. Various properties of integro-differential systems are studied by this way. Methods of the general theory of functional differential equations developed by N. V. Azbelev and his followers are used. One of them is the Azbelev W-transform. We propose ideas allowing to achieve faster convergence to stationary point using a distributed control. We obtain estimates of solutions using estimates of the Cauchy matrices.

For each associative, commutative, and unitary algebra over the field of real or complex numbers and an integrable nonlinear ordinary differential equation we can to construct integrable systems of ordinary differential equations and integrable systems of partial differential equations. In this paper we consider in some sense the inverse problem. Determine the conditions under which a given system of ordinary differential equations can be represented as a differential equation in some associative, commutative and unitary algebra. It is also shown that associativity is not a necessary condition.

The brief presentation of the method of fast expansions is given to solve nonlinear differential equations. Application rules of the operator of fast expansions are specified for solving differential equations. According to the method of fast expansions, an unknown function can be represented as the sum of the boundary function and Fourier series sines and cosines for one variable. The special construction of the boundary functions leads to reasonably fast convergence of the Fourier series, so that for engineering calculations, it is sufficient to consider only the first three members. The method is applicable both to linear and nonlinear integro-differential systems. By means of applying the method of fast expansions to nonlinear Navier-Stokes equations the problem is reduced to a closed system of ordinary differential equations, which solution doesn't represent special difficulties. We can reapply the method of fast expansions to the resulting system of differential equations and reduce the original problem to a system of algebraic equations. If the problem is n -dimensional, then after n -fold application of the method of fast expansions the problem will be reduced to a closed algebraic system. Finally, we obtain an analytic-form solution of complicated boundary value problem in partial derivatives. The flow of an incompressible viscous fluid of Navier–Stokes is considered in a curvilinear pipe. The problem is reduced to solving a closed system of ordinary differential equations with boundary conditions by the method of fast expansions. The article considers peculiarities of finding the coefficients of boundary functions and Fourier coefficients for the zero-order and first-order operators of fast expansions. Obtaining the analytic-form solution is of great interest, because it allows to analyze and to investigate the influence of various factors on the properties of the viscous fluid in specific cases.

The aim of this study is to provide numerical solutions to magnetohydrodynamic (MHD) cross flow with heat and mass transfer over a stretching sheet in a hybrid nanofluid with suction/injection and convective boundary conditions. This will be accomplished by presenting numeric solutions. Nanoparticles of graphene oxide and silver were suspended in kerosene oil as the base fluid taken into consideration is a nanofluid hybrid. After that, the physical flow problem is modeled as a set of partial differential equations, which are then transformed into an equivalent system of nonlinear ordinary differential equations (ODEs) by making use of the relevant similarity transformations. To gain different graphical and numerical results for analyzing the influence of numerous physical restrictions on velocity, heat, and mass profiles, the system of ODEs is solved using the computing power of the Lobatto IIIA technique. The plot of the velocity profile, temperature, concentration, entropy generation and Bejan number with separating magnitude like that power index number ( 0.1 < n < 2.0 0.1\lt n\lt 2.0 ), Weissenberg number ( 0.1 < We < 1.0 0.1\lt {\rm{We}}\lt 1.0 ), Suction/injection ( 0.1 < S < 1.0 0.1\lt S\lt 1.0 ), Magnetic parameter (0.1 < M < 1.0), Biot number ( 0.1 < Bi < 0.3 0.1\lt {\rm{Bi}}\lt 0.3 ), Radiation parameter ( 0.1 < Rd < 2.0 0.1\lt {\rm{Rd}}\lt 2.0 ), Schmidt number ( 1.1 < Sc < 2.0 1.1\lt {\rm{Sc}}\lt 2.0 ), and Chemical reaction parameter ( 1.1 < Sc < 2.0 1.1\lt {\rm{Sc}}\lt 2.0 ). We noted that when the Weissenberg parameter ( We {\rm{We}} ) increases, the flow speed will increase as well. Also, the increasing values of M M slows the flow speed but the opposite effect is observed in the temperature profile. Moreover, the greater values of M M , We, and S reduced in the local skin friction. It is concluded that hybrid nanofluid is better than ordinary nanofluid by percentage values. The results are used in industry and technology in fluid-based systems with shrinkable or flexible materials, MHD flow meters, pumps, and the drawing of plastic strips. Because hybrid nanoparticles speed up the cooling process, they are important in cutting and production.

A Hopfield neural network is described by a system of nonlinear ordinary differential equations. We develop a broad range of numerical schemes that are applicable for a wide range of computational problems. We review here our study on an approximate solution of the Fredholm integral equation, and linear and nonlinear singular and hypersingular integral equations, using a continuous method for solving operator equations. This method assumes that the original system is associated with a Cauchy problem for systems of ordinary differential equations on Hopfield neural networks. We present sufficient conditions for the Hopfield networks’ stability defined via coefficients of systems of differential equations.

The non-standard Hamiltonian system, also referred to as a partial Hamiltonian system in the literature, of the form q ˙ i = ∂ H ∂ p i , p ˙ i = − ∂ H ∂ q i + Γ i ( t , q i , p i ) ${\dot q^i} = \frac{{\partial H}}{{\partial {p_i}}},{\text{ }}{\dot p^i} = - \frac{{\partial H}}{{\partial {q_i}}} + {\Gamma ^i}(t,{\text{ }}{q^i},{\text{ }}{p_i})$ appears widely in economics, physics, mechanics, and other fields. The non-standard (partial) Hamiltonian systems arise from physical Hamiltonian structures as well as from artificial Hamiltonian structures. We introduce the term ‘artificial Hamiltonian’ for the Hamiltonian of a model having no physical structure. We provide here explicitly the notion of an artificial Hamiltonian for dynamical systems of ordinary differential equations (ODEs). Also, we show that every system of second-order ODEs can be expressed as a non-standard (partial) Hamiltonian system of first-order ODEs by introducing an artificial Hamiltonian. This notion of an artificial Hamiltonian gives a new way to solve dynamical systems of first-order ODEs and systems of second-order ODEs that can be expressed as a non-standard (partial) Hamiltonian system by using the known techniques applicable to the non-standard Hamiltonian systems. We employ the proposed notion to solve dynamical systems of first-order ODEs arising in epidemics.

Numerical use of {[formula omitted]}-approximation method thermoelastic anisotropic thin shell theory equations represented in a special form