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EN
In the present paper, the fractional-order cubic nonlinear Schrödinger equation is considered. The Schrödinger equation with time and space fractional derivative is studied at the same time. Different types of travelling wave solutions including the kink solution, soliton solution, periodic solution, and singular solution for the mentioned equation are obtained by using the Jacobi elliptic functions expansion method. It is shown that the solutions turn into the exact solutions when the fractional orders go to 1. This method can be relied on gaining the solutions to time or space fractional order partial differential equations as well as ordinary ones. Throughout this work, the fractional derivative is given in a conformable sense.
EN
The initial/boundary value problem for the fourth-order homogeneous ordinary differential equation with constant coefficients is considered. In this paper, the particular solutions an ordinary differential equation with respect to the set of boundary conditions are studied. At least one of the boundary conditions is described by a fractional derivative. Finally, a few illustrative examples of particular solutions to the considered problem are shown.
EN
Finding the exact solution to dynamical systems in the field of mathematical modeling is extremely important and to achieve this goal, various integral transforms have been developed. In this research analysis, non-integer order ordinary differential equations are analytically solved via the Laplace-Carson integral transform technique, which is a technique that has not been previously employed to test the non-integer order differential systems. Firstly, it has proved that the Laplace-Carson transform for n-times repeated classical integrals can be computed by dividing the Laplace-Carson transform of the underlying function by n-th power of a real number p which later helped us to present a new result for getting the Laplace-Carson transform for d-derivative of a function under the Caputo operator. Some initial value problems based upon Caputo type fractional operator have been precisely solved using the results obtained thereof.
EN
This work develops a technique for constructing a reduced-order system that not only has low computational complexity, but also maintains the stability of the original nonlinear dynamical system. The proposed framework is designed to preserve the contractivity of the vector field in the original system, which can further guarantee stability preservation, as well as provide an error bound for the approximated equilibrium solution of the resulting reduced system. This technique employs a low-dimensional basis from proper orthogonal decomposition to optimally capture the dominant dynamics of the original system, and modifies the discrete empirical interpolation method by enforcing certain structure for the nonlinear approximation. The efficiency and accuracy of the proposed method are illustrated through numerical tests on a nonlinear reaction diffusion problem.
EN
In the present research analysis, linear fractional order ordinary differential equations with some defined condition (s) have been solved under the Caputo differential operator having order α > 0 via the Shehu integral transform technique. In this regard, we have presented the proof of finding the Shehu transform for a classical nth order integral of a piecewise continuous with an exponential nt h order function which leads towards devising a theorem to yield exact analytical solutions of the problems under investigation. Varying fractional types of problems are solved whose exact solutions can be compared with solutions obtained through existing transformation techniques including Laplace and Natural transforms.
EN
A method of solving the integro-differential equations is presented. The discussed equations will be solved by the Taylor differential transformation. By using appropriate properties of this transformation the integro-differential equation will be transformed to a respective recurrence equation. Unfortunately, the high degree of generality and complexity of such defined problem does not allow to obtain the solution in general form. Each equation requires a special method of solution.
7
Modern Taylor series method in numerical integration
EN
The paper deals with extremely exact, stable, and fast numerical solutions of systems of differential equations. It also involves solutions of problems that can be reduced to solving a system of differential equations. The approach is based on an original mathematical method, which uses the Taylor series method for solving differential equations in a non-traditional way. Even though this method is not much preferred in the literature, experimental calculations have verified that the accuracy and stability of the Taylor series method exceed the currently used algorithms for numerically solving differential equations. The Modern Taylor Series Method (MTSM) is based on a recurrent calculation of the Taylor series terms for each time interval. Thus, the complicated calculation of higher order derivatives (much criticised in the literature) need not be performed but rather the value of each Taylor series term is numerically calculated. An important part of the method is an automatic integration order setting, i.e. using as many Taylor series terms as the defined accuracy requires. The aim of our research is to propose the extremely exact, stable, and fast numerical solver for modelling technical initial value problems that offers wide applications in many engineering areas including modelling of electrical circuits, mechanics of rigid bodies, control loop feedback (controllers), etc.
CS
Clánek se zabývá presným, stabilním a rychlým rešením soustav diferenciálních rovnic. Soustavou diferenciálních rovnic lze reprezentovat velké množství reálných problému. Numerické rešení je založeno na unikátní numerické metode, která netradicne využívá Taylorovu radu. I presto, že tato metoda není v literature príliš preferována, experimentální výpocty potvrdily, že presnost a stabilita této metody presahuje aktuálne používané numerické algoritmy pro numerické rešení diferenciálních rovnic. Moderní metoda Taylorovy rady je založena na rekurentním výpoctu clenu Taylorovy rady v každém casovém intervalu. Derivace vyšších rádu nejsou pro výpocet prímo využity, derivace jsou zahrnuty do clenu Taylorovy rady, které se pocítají rekurentne numericky. Duležitou vlastností metody je automatická volba rádu metody v závislosti na velikosti integracního kroku, tzn. je využito tolik clenu Taylorovy rady, kolik vyžaduje zadaná presnost výpoctu. Cílem výzkumu je navrhnout velmi presný, stabilní a rychlý nástroj pro modelování technických pocátecních problému využitých v praxi pri modelování elektrických obvodu, mechaniky tuhých teles, problematiky zpetnovazebního rízení a další.
EN
The paper presents the method of solving some problems belonging to the area of the calculus of variations, that is the problems of searching for the selected types of functionals which can be transformed to some, nonlinear in general, ordinary differential equations or systems of such equations. The obtained equations are solved on the basis of the Taylor differential transformation.
EN
A mathematical model for fluid and solute transport in peritoneal dialysis is constructed. The model is based on a three-component nonlinear system of two-dimensional partial differential equations for fluid, glucose and albumin transport with the relevant boundary and initial conditions. Our aim is to model ultrafiltration of water combined with inflow of glucose to the tissue and removal of albumin from the body during dialysis, by finding the spatial distributions of glucose and albumin concentrations as well as hydrostatic pressure. The model is developed in one spatial dimension approximation, and a governing equation for each of the variables is derived from physical principles. Under some assumptions the model can be simplified to obtain exact formulae for spatially non-uniform steady-state solutions. As a result, the exact formulae for fluid fluxes from blood to the tissue and across the tissue are constructed, together with two linear autonomous ODEs for glucose and albumin concentrations in the tissue. The obtained analytical results are checked for their applicability for the description of fluid-glucose-albumin transport during peritoneal dialysis.
10
Teaching, modeling and visualisation of ordinary differential equations
EN
Advances in computer technology and increased interest in dynamical systems influence the way of teaching ordinary differential equations. The paper presents inquiry oriented teaching, usage of modeling, visualisation and interactive web services. Last chapter describes the ways of using MATLAB or public domain software (e.g. Octave) to solve ordinary differential equations.
PL
Postęp w technologii komputerowej oraz wzrost zainteresowania modelowaniem systemów dynamicznych wpływa na sposób nauczania matematyki, w tym równań różniczkowych zwyczajnych. Przedstawiono podejścia: nauczania przez zadawanie pytań, wykorzystanie modelowania, wizualizacji i interaktywnych usług sieci web. Ostatni rozdział opisuje sposoby wykorzystania środowiska MATLAB lub oprogramowania dostępnego jako public domain (np. Octave) do rozwiązywania równań różniczkowych zwyczajnych.
EN
We prove a comparison theorem for an ODE and DAE system which arises from the method of lines. Under a Perron comparison condition on the functional dependence and a specific Lipschitz and (W+) condition on the classical argument, we obtain strong uniqueness criteria.
EN
Purpose: The aim of this article is focused on providing numerical solutions for system of second order robot arm problem using the Runge-Kutta Sixth order algorithm. Design/methodology/approach: The parameters governing the arm model of a robot control problem have also been discussed through RK-sixth-order algorithm. The precised solution of the system of equations representing the arm model of a robot has been compared with the corresponding approximate solutions at different time intervals. Findings: Results and comparison show the efficiency of the numerical integration algorithm based on the absolute error between the exact and approximate solutions. The stability polynomial for the test equation γ=λγ (�γ is a complex Number) using RK-butcher algorithm obtained by Murugesan et. al. [1] and Park et. al. [2,3] is not correct and the stability regions for RK-Butcher methods have been absurdly presented. They have made a blunder in determining the range for real parts of �λh (h is a step size) involved in the test equation for RK-Butcher algorithms. Further, they have abruptly drawn the stability region for STWS method assuming that it is based on the Taylor's series technique. Research limitations/implications: It is noticed that STWS algorithm is not based on the Taylor�'s series method and it is an A-stable method. In the present paper, a corrective measure has been taken to obtain the stability polynomial for the case of RK-Butcher algorithm, the ranges for the real part of �λh and to present graphically the stability regions of the RK-Butcher methods. Originality/value: Based on the numerical results and graphs, a thorough comparison is carried out between the numerical algorithms.
13
A dynamic prictionless contact problem with adhesion and damage
EN
We consider a dynamic frictionless contact problem for a viscoelastic material with damage. The contact is modeled with normal compliance condition. The adhesion of the contact surfaces is considered and is modeled with a surface variable, the bonding field, whose evolution is described by a first order differential equation. We establish a variational formulation for the problem and prove the existence and uniqueness of the solution. The proofs are based on the theory of evolution equations with monotone operators, a classical existence and uniqueness result for parabolic inequalities, and fixed point arguments.
EN
The paper deals with the anti-periodic boundary value problem for impulsive ordinary differential equations. The impulsive differential inequalities generated by this problem are considered and a uniqueness criterion is obtained.
16
Modelling Tumour-Immunity Interactions With Different Stimulation Functions
EN
Tumour immunotherapy is aimed at the stimulation of the otherwise inactive immune system to remove, or at least to restrict, the growth of the original tumour and its metastases. The tumour-immune system interactions involve the stimulation of the immune response by tumour antigens, but also the tumour induced death of lymphocytes. A system of two non-linear ordinary differential equations was used to describe the dynamic process of interaction between the immune system and the tumour. Three different types of stimulation functions were considered: (a) Lotka-Volterra interactions, (b) switching functions dependent on the tumour size in the Michaelis-Menten form, and (c) Michaelis-Menten switching functions dependent on the ratio of the tumour size to the immune capacity. The linear analysis of equilibrium points yielded several different types of asymptotic behaviour of the system: unrestricted tumour growth, elimination of tumour or stabilization of the tumour size if the initial tumour size is relatively small, otherwise unrestricted tumour growth, global stabilization of the tumour size, and global elimination of the tumour. Models with switching functions dependent on the tumour size and the tumour to the immune capacity ratio exhibited qualitatively similar asymptotic behaviour.
EN
Free vibrations of one-dimensional rheological structure have been described by the system of the conjugated partial differential equations. A vector form of this system of equations allows to identify the self adjoint linear operators of inertia, damping and stiffness. These operators are not homothetic, hence the method of a separation of variables for the considered system of equations is applicable only in the introduced complex Hilbert space. Such a separation of variables leads to the system of ordinary differential equations in time and to the system of three ordinary differential equations with respect to spatial variables. Solution of the obtained boundary-value problems proceeds in a classical way, however, the results are of a complex conjugated type. Applying the fundamental principle of the general orthogonality of complex eigenvectors, the problem of free vibrations of the system with arbitrary initial conditions was solved in exact form.
EN
A nonlinear integro-differential flutter equation of a thin airfoil placed in an incompressible flow is solved by two different methods. The first method involves the center-manifold reduction and gives the asymptotic limit cycle amplitude and frequency in terms of power series expansions. The second method replaces the integro-differential equation by an approximate set of first-order ordinary differential equations which are solved by using bifurcation and continuation software package. A comparison of these two methods shows that the domain of a good agreement between them varies significantly depending on the parameters of the problem.
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