Solution Of Initial Value Problem Research Articles

Is Doom the Inescapable Solution of Initial Value Problems?

Some realistic applications of differential equations accessible to beginning calculus students appear to paint us a picture of imminent and inexorable catastrophe.Example 1. A model of Methanosarc...

Necessary and sufficient conditions for existence of solutions for initial value problem of fuzzy Bagley-Torvik equation and solution representation

In this paper, we investigate the necessary and sufficient conditions for existence of solutions for initial value problem of fuzzy Bagley-Torvik equation and the solution representation by using the multivariate Mittag-Leffler function. First we convert fuzzy initial value problem into the cut problem (system of fractional differential equations with inequality constraints) and obtain existence results for the solution of the cut problem under (1,1)- differentiability. Next we study the conditions for the solutions of the cut problem to constitute the solution of a fuzzy initial value problem and suggest a necessary and sufficient condition for the (1,1)-solution. Also, some examples are given to verify the effectiveness of our proposed method. The necessary and sufficient condition, solution representation for (1,2)-solution of initial value problem of fuzzy fractional Bagley-Torvik equation are shown in Appendix.

Parameter Identification of Lorenz System with Incomplete Information: Case of One Known and Two Unknown Functions

In the present paper, which is the continuation of the previous one, the problem of parameter identification of the Lorenz system is solved in assumption that only one of three functions is known at discrete time instants on finite time initial time interval. Two other functions are assumed to be unknown. The regular methods of guess values determination of the unknown parameters are developed. They are based on the Lagrange multiplier and auxiliary parameters approaches. A novel method of initial value problem solution is proposed in which the abovementioned guess values are used for more accurate estimation of the system parameters. It is demonstrated that the proposed IVP method simultaneously solves three different tasks: the problem of function interpolation from its discrete values on the initial time interval; the problem of unknown functions reconstruction on the same time interval, and the problem of extrapolation of all functions on limited time interval. It is also shown that the proposed method reconstructs the Lorenz attractor from limited data volume and data including random components.

On the boundedness of solution of the second order ordinary differential equation with damping term and involution

In the present paper the initial value problem for the second order ordinary differential equation with damping term and involution is investigated. We obtain equivalent initial value problem for the fourth order ordinary differential equations to the initial value problem for second order linear differential equations with damping term and involution. Theorem on stability estimates for the solution of the initial value problem for the second order ordinary linear differential equation with damping term and involution is proved. Theorem on existence and uniqueness of bounded solution of initial value problem for second order ordinary nonlinear differential equation with damping term and involution is established.

Analysis of the properties of a derived one-step numerical method of a transcendental function

The paper presents a new numerical method for the solution of Initial Value Problems (IVPs) of first order Ordinary Differential Equations (ODEs) via a transcendental basis function. FORTRAN 95 programing language was used for the implementation of the derived numerical scheme. The analysis of the properties of the newly derived method was investigated which includes the stability, accuracy and consistency through which the suitability of it was established. This scheme avoids the need for higher derivative of differential equation. This scheme was tested on some illustrative examples and the results compared favourably with exact solutions. We conclude that the derived numerical method is consistent, convergent, quite stable and more accurate for application to numerical problems and can be widely used in solving first order IVPs in ODEs.

Accuracy Analysis for the Solution of Initial Value Problem of ODEs Using Modified Euler Method

Accuracy Analysis for the Solution of Initial Value Problem of ODEs Using Modified Euler Method

Explicit Solutions of Initial Value Problems for Fractional Generalized Proportional Differential Equations with and without Impulses

The object of investigation in this paper is a scalar linear fractional differential equation with generalized proportional derivative of Riemann–Liouville type (LFDEGD). The main goal is the obtaining an explicit solution of the initial value problem of the studied equation. Note that the locally solvability, being the same as the existence of solutions to the initial value problem, is connected with the symmetry of a transformation of a system of differential equations. At the same time, several criteria for existence of the initial value problem for nonlinear fractional differential equations with generalized proportional derivative are connected with the linear ones. It leads to the necessity of obtaining an explicit solution of LFDEGD. In this paper two cases are studied: the case of no impulses in the differential equation are presented and the case when instantaneous impulses at initially given points are involved. All obtained formulas are based on the application of Mittag–Leffler function with two parameters. In the case of impulses, initially the appropriate impulsive conditions are set up and later the explicit solutions are obtained.

A Study of Legendre Polynomials Approximation for Solving Initial Value Problems

Legendre polynomials (LPs) basis on the interval I = [-1;1] have been introduced and their properties are studied. Legendre polynomials with Gaussian integration method (GIM) are used to solve the initial value problems (IVPs) of third-order ordinary differential equations (ODEs). The aim of this work is to study the numerical solution of IVP which is widely applicable in the fields of engineering and science using Legendre base. The implementations have been discussed to demonstrate the validity and applicability of the technique of collections and variational methods which compared with the numerical results that obtained by classical RK and modern direct RKD numerical methods. Finally, the numerical results show the high accuracy and efficiency of the proposed method.

A New Method for Numerical Integration of Higher-Order Ordinary Differential Equations Without Losing the Periodic Responses

A new numerical method is presented for the solution of initial value problems described by systems of N linear ordinary differential equations (ODEs). Using the state-space representation, a differential equation of order n > 1 is transformed into a system of L = n×N first-order equations, thus the numerical method developed recently by Katsikadelis for first-order parabolic differential equations can be applied. The stability condition of the numerical scheme is derived and is investigated using several well-corroborated examples, which demonstrate also its convergence and accuracy. The method is simply implemented. It is accurate and has no numerical damping. The stability does not require symmetrical and positive definite coefficient matrices. This advantage is important because the scheme can find the solution of differential equations resulting from methods in which the space discretization does not result in symmetrical matrices, for example, the boundary element method. It captures the periodic behavior of the solution, where many of the standard numerical methods may fail or are highly inaccurate. The present method also solves equations having variable coefficients as well as non-linear ones. It performs well when motions of long duration are considered, and it can be employed for the integration of stiff differential equations as well as equations exhibiting softening where widely used methods may not be effective. The presented examples demonstrate the efficiency and accuracy of the method.

L 1-solutions of the initial value problems for implicit differential equations with Hadamard fractional derivative

Abstract In this paper, we study the existence of integrable solutions for initial value problems for fractional order implicit differential equations with Hadamard fractional derivative. Our results are based on Schauder’s fixed point theorem and the Banach contraction principle fixed point theorem.

Accuracy Analysis on Solution of Initial Value Problems of Ordinary Differential Equations for Some Numerical Methods with Different Step Sizes

In this article, three numerical methods namely Euler’s, Modified Euler, and Runge-Kutta method have been discussed, to solve the initial value problem of ordinary differential equations. The main goal of this research paper is to find out the accurate results of the initial value problem (IVP) of ordinary differential equations (ODE) by applying the proposed methods. To achieve this goal, solutions of some IVPs of ODEs have been done with the different step sizes by using the proposed three methods, and solutions for each step size are analyzed very sharply. To ensure the accuracy of the proposed methods and to determine the accurate results, numerical solutions are compared with the exact solutions. It is observed that numerical solutions are best fitted with exact solutions when the taken step size is very much small. Consequently, all the proposed three methods are quite efficient and accurate for solving the IVPs of ODEs. Error estimation plays a significant role in the establishment of a comparison among the proposed three methods. On the subject of accuracy and efficiency, comparison is successfully implemented among the proposed three methods.

Singular initial value problems for scalar quasi-linear ordinary differential equations

We discuss existence, non-uniqueness and regularity of one- and two-sided solutions of initial value problems for scalar quasi-linear ordinary differential equations where the initial condition corresponds to an impasse point of the equation. With a differential geometric approach, we reduce the problem to questions in dynamical systems theory. As an application, we discuss in detail second-order equations of the form g(x)u″=f(x,u,u′) with an initial condition imposed at a simple zero of g. This generalises results by Liang and also makes them more transparent via our geometric approach.

Development of a new numerical scheme for the solution of exponential growth and decay models

This paper presents the development of a new numerical scheme for the solution of exponential growth and decay models emanated from biological sciences. The scheme has been derived via the combination of two interpolants namely, polynomial and exponential functions. The analysis of the local truncation error of the derived scheme is investigated by means of the Taylor’s series expansion. In order to test the performance of the scheme in terms of accuracy in the context of the exact solution, four biological models were solved numerically. The absolute error has been computed successfully at each mesh point of the integration interval under consideration. The numerical results generated via the scheme agree with the exact solution and with the fifth order convergence based upon the analysis carried out. Hence, the scheme is found to be of order five, accurate and is a good approach to be included in the class of linear explicit numerical methods for the solution of initial value problems in ordinary differential equations.

The eigenvalue problem of the general Einstein–Weyl metric equation and exact self-similar and multi-traveling waves solutions

The integrability of the general Einstein–Weyl Metric equation is studied . Here, the approach proposed is based on using the extended and generalized unified methods. These methods inspect the integrability conditions, whenever exist. The exact solutions are thus obtained. They provide the existence and uniqueness of solution for initial value problem where initial value is taken from the exact solutions obtained. Similarity variables are introduced, and the extended unified method is applied to find a class of explicit self-similar wave solutions to the GEWM equation. Numerical evaluations of the solutions are done via symbolic computation. These results show periodic solitons and multi-lumps. It is found that the generalized heavenly equation admits a class of infinite solutions, among them the chaotic ones. The generalized unified method is used to find multi-traveling solitary waves solutions. We think that the presented methods generalize the known ones in the literature. This will be illustrated later on.

Existence and Uniqueness Results for Two-Term Nonlinear Fractional Differential Equations via a Fixed Point Technique

The work addressed in this paper is to ensure the existence and uniqueness of positive solutions for initial value problems for nonlinear fractional differential equations with two terms of fractional orders. By virtue of recent fixed point theorems on mixed monotone operators, we get some new straightforward results with a wide range of applications.

A new approach to integrable evolution equations on the circle.

We propose a new approach for the solution of initial value problems for integrable evolution equations in the periodic setting based on the unified transform. Using the nonlinear Schrödinger equation as a model example, we show that the solution of the initial value problem on the circle can be expressed in terms of the solution of a Riemann–Hilbert problem whose formulation involves quantities which are defined in terms of the initial data alone. Our approach provides an effective solution of the problem on the circle which is conceptually analogous to the solution of the problem on the line via the inverse scattering transform.

On definition of solution of initial value problem for fractional differential equation of variable order

We propose a new definition of continuous approximate solution to initial value problem for differential equations involving variable order Caputo fractional derivative based on the classical definition of solution of integer order (or constant fractional order) differential equation. Some examples are presented to illustrate these theoretical results.

Revised version of exponentially fitted pseudo-Runge-Kutta method

In this paper, we have proposed the revised version of exponentially-fitted (ef) pseudo-Runge-Kutta method (ef-PRKM). The motivation behind the revision is to fill the leakage of error in the internal stages during the fitting process. Generally, the internal stage operator, in an ef-PRKM or ef-Runge-Kutta method (ef-RKM) integrates two exponential functions exp(± ωx), with unknown frequency ω e R (for trigonometric-fitting exp(± ωx), ω e iR). However, these internal stage operators produce some amount of errors in integrating other functions like xk exp(± ωx), k ≥ 0 . Here, we first measure the error expression and taking into account this error, we redefine the external stage (solution) operators and compute the revised weights of the ef-PRKM. The revised ef-PRKM is tested on two initial value problems (IVPs). The results are reported in tables and figure. The proposed method would be a good option to find the numerical solution of IVPs efficiently with less cost consumption in the form of slopes/function evaluations than the standard Runge-Kutta methods.

On the Boundedness of Solution of the Second Order Ordinary Differential Equation with Involution

In the present paper, the initial value problem for the second order ordinary differential equation with involution is investigated. We obtain equivalent initial value problem for the fourth order ordinary differential equations to the initial value problem for second order linear differential equations with involution. Theorem on stability estimates for the solution of the initial value problem for the second order ordinary linear differential equation with involution is proved. Theorem on existence and uniqueness of bounded solution of initial value problem for second order ordinary nonlinear differential equation with involution is established.

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.