Problems For Ordinary Differential Equations Research Articles

Inverse Hybrid Linear Multistep Methods for Solving the Second Order Initial Value Problems in Ordinary Differential Equations

Inverse linear multistep methods (ILMMs) for first and second order differential equations have been proved to be suitable numerical methods for the solution of inverse initial value problems (IVPs). This paper presents the hybrid version of the ILMMs for the numerical solution of second order inverse IVPs. The stability of the proposed methods is represented in the boundary locus graph. The applicability of the schemes is demonstrated herein for the solution of linear and nonlinear problems. Computational results on the problems are compared with those from the existing method and ode45 (the explicit Runge–Kutta method).

Lipschitz stability of nonlinear ordinary differential equations with non-instantaneous impulses in ordered Banach spaces

Abstract We consider Lipschitz stability of zero solutions to the initial value problem of nonlinear ordinary differential equations with non-instantaneous impulses on ordered Banach spaces. Using Lyapunov function, Lipschitz stability of zero solutions to nonlinear ordinary differential equation with non-instantaneous impulses is obtained.

Novel Fixed-Time Stability Criteria for Discontinuous Nonautonomous Systems: Lyapunov Method With Indefinite Derivative.

This article considers a general class of nonautonomous discontinuous ordinary differential equations (ODE). By constructing the Filippov multimap, the fixed-time stability (FTS) problem of discontinuous ODE is transformed into that of differential inclusion (DI). In order to establish the FTS criteria of the zero solution for DI, the generalized Lyapunov function (LF) method is developed. The generalized LF of this article is relaxed to have an indefinite derivative for almost everywhere along the state trajectories of the system. However, the traditional LF is required to possess negative definite or semi-negative definite derivative for everywhere. As a result, several novel sufficient conditions for FTS are given. Moreover, the settling time of FTS is provided. Then, the theoretical results are applied to solve the fixed-time stabilization control problems of ball motion model and neural networks (NNs) with discontinuities. The developed LF method of FTS is extremely significant in the field of control engineering.

FORMULATION OF BLOCK SCHEMES WITH LINEAR MULTISTEP METHOD FOR THE APPROXIMATION OF FIRST-ORDER IVPS

ABSTRACT
 In this paper, formulation of an efficient numerical schemes for the approximation first-order initial value problems (IVPs) of ordinary differential equations (ODE) is presented. The method is a block scheme for some k-step linear multi-step methods (and) using the Hermite Polynomials a basis function. The continuous and discrete linear multi-step methods (LMM) are formulated through the technique of collocation and interpolation. Numerical examples of ODE have been examined and results obtained show that the proposed scheme can be efficient in solving initial value problems of first order ODE.

Unique Solvability of Problems for a Mixed Parabolic Equation in Unbounded Domain

In the work, parabolic equations with perpendicular time directions considered together and the Gevrey problem with generalized gluing conditions, and a problem with the Bitsadze–Samarskii conditions have been studied. The uniqueness and existence of the solution of the considered problems were proved. The Gevrey problem was equivalently reduced to the two-point problem for ordinary differential equation involving fractional differential operators. The uniqueness of the solution of the obtained problem was proved by the principle of extremum and the existence was proved by the method of integral equations. The theory of the second kind Volterra integral equations and properties of two-parameter Mittag-Leffler function are applied at studying a unique solvability of the problem with Bitsadze–Samarskii condition. Necessary conditions for given functions for solvability of considered problems have been found.

Numerical solving of initial value problems for ordinary differential equations with internal limiting singular points

In the present paper, we investigate the numerical methods for solving initial value problems for systems of nonlinear ordinary differential equations with several limiting singular points. Similar tasks have great practical importance in such areas as hydrodynamics, chemical kinetics, combustion theory, computational geometry, and nuclear reactors. However, the exact solution may be constructed to nonlinear initial value problems only in exceptional cases. Numerical methods also do not give the desired result. Explicit methods make it possible to obtain a solution only if there is one limiting singular point on the right boundary of the considered interval of the argument change. Implicit methods also have usage restrictions. They consist of the difficulty or impossibility of overcoming the internal limiting singular points. For solving the considered class of problems, the paper proposes to use the method of solution continuation, which allows passing any limiting singular points effectively. Features of the application of various numerical methods are shown by solving one test initial value problem. The reliability of the obtained numerical results is confirmed by comparison with the finding analytical solution of the considered test problem.

A new numerical approach for solving initial value problems with exponential growth integral curves

Some physical processes raised in aerodynamics, hydrodynamics, and theory of catalytic reactions are modeled by stiff initial value problems. This paper considers a class of stiff problems such as singularly perturbed problems for ordinary differential equations. Singularly perturbed problems have regions of rapid change in the solution, which are called contrast structures. Most often, boundary layer exhibits in such problems, but several pieces of research devoted to inner layers too. There are many numerical and asymptotic methods for solving singularly perturbed initial value problems with contrast structures. However, it is still an open field for researches because the universal method does not exist. Problems with an exponential growth rate of integral curves have a particular interest for researchers because most of the methods for solving singularly perturbed problems cannot handle these. We are going to discuss methods for solving singularly perturbed initial value problems with contrast structures. Also, we are proposing a new technique based on the method of solution continuation for problems with the exponential growth of integral curves. This technique allows explicit numerical methods to get the solution of the considered problem when it seems impossible for them. The efficiency of the new approach will be shown on the test initial value problem.

On the Computation of Approximate Solution to Ordinary Differential Equations by the Chebyshev Series Method and Estimation of Its Error

An approximate method for solving the Cauchy problem for nonlinear first-order ordinary differential equations is considered. The method is based on using the shifted Chebyshev series and a Markov quadrature formula. Some approaches are given to estimate the error of an approximate solution expressed by a partial sum of a certain order series. The error is estimated using the second approximation of the solution expressed by a partial sum of a higher order series. An algorithm for partitioning the integration interval into elementary subintervals to ensure the computation of the solution with a prescribed accuracy is constructed on the basis of the proposed approaches to error estimation.

An Exponential Spline Difference Scheme for Solving a Class of Boundary Value Problems of Second-Order Ordinary Differential Equations

In this paper, we mainly study an exponential spline function space, construct a basis with local supports, and present the relationship between the function value and the first and the second derivative at the nodes. Using these relations, we construct an exponential spline-based difference scheme for solving a class of boundary value problems of second-order ordinary differential equations (ODEs) and analyze the error and the convergence of this method. The results show that the algorithm is high accurate and conditionally convergent, and an accuracy of 1/240h6 was achieved with smooth functions.

A computational method for solving a problem with parameter for linear systems of integro-differential equations

This article presents a computational method for solving a problem with parameter for a system of Fredholm integro-differential equations. Some additional parameters are introduced and the problem under consideration is reduced to solving a system of linear algebraic equations. The coefficients and right-hand side of the system are calculated by solving the Cauchy problems for ordinary differential equations. We establish a criterion for the unique solvability of the problem under consideration. A numerical algorithm is offered for solving the problem with parameter. The results are illustrated by numerical examples.

Global Bifurcation from Zero in Some Fourth-Order Nonlinear Eigenvalue Problems

In this paper, we study the nonlinear eigenvalue problem for ordinary differential equations of fourth order with a spectral parameter in the boundary condition. Global bifurcation of nontrivial solutions of this problem is investigated. We prove the existence of two families of unbounded continua of the set of solutions to this problem bifurcating from points and intervals of the line of trivial solutions. Moreover, it is shown that these continua are contained in classes of functions possessing oscillating properties of the eigenfunctions of the corresponding linear problem and their derivatives.

An Examination of a Second Order Numerical Method for Solving Initial Value Problems

This paper presents an examination of a Second Order Convergence Numerical Method (SOCNM) for solving Initial Value Problems (IVPs) in Ordinary Differential Equations (ODEs). The SOCNM has been derived via the interpolating function comprises of polynomial and exponential forms. The analysis and the properties of SOCNM were discussed. Three numerical examples have been solved successfully to examine the performance of SOCNM in terms of accuracy and stability. The comparative study of SOCNM, Improved Modified Euler Method (IMEM), Fadugba and Olaosebikan Scheme (FOS) and the Exact Solution (ES) is presented. By varying the step length, the absolute relative errors at the final nodal point of the associated integration interval are computed. Furthermore, the analysis of the properties of SOCNM shows that the method is consistent, stable, convergent and has second order accuracy. Moreover, the numerical results show that SOCNM is more accurate than IMEM and FOS and also compared favourably with the ES. By varying the step length, there are two-order decrease in the values of the final absolute relative errors generated via SOCNM. Hence, SOCNM is found to be accurate, stable and a good tool for the numerical solutions of IVPs of different characteristics in ODEs.

Improved Two-Point Block Backward Differentiation Formulae for Solving First Order Stiff Initial Value Problems of Ordinary Differential Equations

Two numerical methods- I2BBDF2 and I22BBDF2 that compute two points simultaneously at every step of integration by first providing a starting value via fourth order Runge-Kutta method are derived using Taylor series expansion. The two-point block schemes are derived by modifying the existing I2BBDF (5) method of Mohamad et al., (2018). Convergence and stability analysis of the new methods are established with the methods being of order two and A-stable in both cases. Despite the very low order of the new methods, the accuracy of these methods on some stiff initial value problems in the literature proves their superiority over existing methods of higher orders such as I2BBDF(5), BBDF(5), E2OSB(4) among others.

Two-point boundary value problems for ordinary differential equations, uniqueness implies existence

We consider a family of three point n − 2, 1, 1 conjugate boundary value problems for nth order nonlinear ordinary differential equations and obtain conditions in terms of uniqueness of solutions imply existence of solutions. A standard hypothesis that has proved effective in uniqueness implies existence type results is to assume uniqueness of solutions of a large family of n−point boundary value problems. Here, we replace that standard hypothesis with one in which we assume uniqueness of solutions of large families of two and three point boundary value problems. We then close the paper with verifiable conditions on the nonlinear term that in fact imply global uniqueness of solutions of the large family of three point boundary value problems.

A NEW CONTINUATION METHOD FOR THE PERIODICITY PROBLEMS OF ORDINARY DIFFERENTIAL EQUATIONS

A NEW CONTINUATION METHOD FOR THE PERIODICITY PROBLEMS OF ORDINARY DIFFERENTIAL EQUATIONS

Об оценке погрешности приближенного решения обыкновенных дифференциальных уравнений, определенного с помощью рядов Чебышёва

Рассматривается приближенный метод решения задачи Коши для нелинейных обыкновенных дифференциальных уравнений первого порядка, основанный на применении смещенных рядов Чебышёва и квадратурной формулы Маркова. Приведены способы оценки погрешности приближенного решения, выраженного в виде частичной суммы ряда некоторого порядка. Погрешность оценивается с помощью второго приближенного решения, вычисленного специальным образом и представленного частичной суммой ряда более высокого порядка. На основе предложенных способов оценки погрешности построен алгоритм автоматического разбиения промежутка интегрирования на элементарные сегменты, делающие возможным вычисление приближенного решения с наперед заданной точностью. Работа метода проиллюстрирована примерами, в том числе примером из небесной механики. An approximate method of solving the Cauchy problem for nonlinear first-order ordinary differential equations is considered. The method is based on using the shifted Chebyshev series and a Markov quadrature formula. Some approaches are given to estimate the error of an approximate solution expressed by a partial sum of a certain order series. The error is estimated using the second approximation of the solution expressed by a partial sum of a higher order series. An algorithm of partitioning the integration interval into elementary subintervals to ensure the computation of the solution with a prescribed accuracy is discussed on the basis of the proposed approaches to error estimation.

Enzymatic Degradation of 2,4,6-Trichlorophenol in a Microreactor using Soybean Peroxidase

Soybean peroxidase is an enzyme extracted from soybean seed hulls. In the presence of hydrogen peroxide, the enzyme has the potential to catalyze the biodegradation of toxic substances like chlorophenols. For this reason, its use in wastewater treatment processes is environmentally friendly since the enzyme can be obtained from a renewable and abundant raw material. In this work, enzymatic biodegradation of 2,4,6-trichlorophenol performed by soybean peroxidase in a microreactor was studied experimentally and theoretically. The experimental data set was obtained with a volume of 250 μL by using different soybean peroxidase concentrations and different reaction times. The fluid dynamics of the microreactor was modeled as well, using ANSYS CFX. The simulations exhibited secondary flows, which enhanced mixing. Although the laminar flow was developed, it can be assumed to be a well-mixed medium. The kinetic data were evaluated through a mechanistic model, the modified bi-bi ping-pong model, which is adequate to represent the enzymatic degradation using peroxidases. The model was composed of an initial value problem for ordinary differential equations that were solved using MATLAB. Some kinetic constants were estimated using the least square function. The results of the model fit well the experimental data.

A New Two-step Hybrid Singularly P-stable Method for the Numerical Solution of Second-order IVPs with Oscillating Solutions

In this paper, a new two-step hybrid method of twelfth algebraic order is constructed and analyzed for the numerical solution of initial value problems of second-order ordinary differential equations. The proposed methods are symmetric and belongs to the family of multiderivative methods. Each methods of the new family appears to be hybrid, but after implementing the hybrid terms, it will continue as a multiderivative method. Therefore, the name semi-hybrid is used. The consistency, convergence, stability and periodicity of the methods are investigated and analyzed. The numerical results for some chemical (e.g. undamped Dufng's equation) as well as quantum chemistry problems (i.e. orbit problems of Stiefel and Bettis) indicated that the new method is superior, efcient, accurate and stable.

ON THE ALGORITHM FOR SOLVING OF A LINEAR BOUNDARY VALUE PROBLEM FOR ORDINARY DIFFERENTIAL EQUATION WITH PARAMETER

A linear boundary value problem for a system of ordinary differential equations containing a parameter is considered on a bounded segment. For a fixed parameter value, the Cauchy problem for an ordinary differential equation is solved. Using the fundamental matrix of differential part and assuming uniqueness solvability of the Cauchy problem an origin boundary value problem is reduced to the system of linear algebraic equation with respect to unknown parameter. The existence of a solution to this system ensures the existence of a solution to the boundary value problem under study. The algorithm of finding of solution for initial problem is offered based on a construction and solving of the linear algebraic equation. The basic auxiliary problem of algorithm is: the Cauchy problem for ordinary differential equations. The numerical implementation of algorithm offered in the article uses the method of Runge-Kutta of fourth order to solve the Cauchy problem for ordinary differential equations.

REQUIRED OPTIMALITY CONDITIONS IN ONE OPTIMAL CONTROL PROBLEM WITH MULTIPOINT FUNCTIONAL

One specific optimal control problem with distributed parameters of the Moskalenko type with a multipoint quality functional is considered. To date, the theory of necessary first-order optimality conditions such as the Pontryagin maximum principle or its consequences has been sufficiently developed for various optimal control problems described by ordinary differential equations, i.e. for optimal control problems with lumped parameters. Many controlled processes are described by various partial differential equations (processes with distributed parameters). Some features are inherent in optimal control problems with distributed parameters, and therefore, when studying the optimal control problem with distributed parameters, in particular, when deriving various necessary optimality conditions, non-trivial difficulties arise. In particular, in the study of cases of degeneracy of the established necessary optimality conditions, fundamental difficulties arise. In the present work, we study one optimal control problem described by a system of first-order partial differential equations with a controlled initial condition under the assumption that the initial function is a solution to the Cauchy problem for ordinary differential equations. The objective function (quality criterion) is multi-point. Therefore, it becomes necessary to introduce an unconventional conjugate equation, not in differential (classical), but in integral form. In the work, using one version of the increment method, using the explicit linearization method of the original system, the necessary optimality condition is proved in the form of an analog of the maximum principle of L.S. Pontryagin. It is known that the maximum principle of L.S. Pontryagin for various optimal control problems is the strongest necessary condition for optimality. But the principle of a maximum of L.S. Pontryagin, being a necessary condition of the first order, often degenerates. Such cases are called special, and the corresponding management, special management. Based on these considerations, in the considered problem, we study the case of degeneration of the maximum principle of L.S. Pontryagin for the problem under consideration. For this purpose, a formula for incrementing the quality functional of the second order is constructed. By introducing auxiliary matrix functions, it was possible to obtain a second-order increment formula that is constructive in nature. The necessary optimality condition for special controls in the sense of the maximum principle of L.S. Pontryagin is proved. The proved necessary optimality conditions are explicit.