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Items from 1 to 20 out of 94 results

article

Semi‐analytic time differencing methods for singularly perturbed initial value problems

Gung‐Min Gie, Chang‐Yeol Jung, Hoyeon Lee

Numerical Methods for Partial Differential Equations > 38 > 5 > 1367 - 1395

We implement our new semi‐analytic time differencing methods, applied to singularly perturbed non‐linear initial value problems. It is well‐known that, concerning the singularly perturbed initial problem, a very stiff layer, called initial layer, appears when the perturbation parameter is small, and this stiff initial layer causes significant difficulties to approximate the solution. To improve numerical...

article

Variation of parameters method for a three-dimensional problem of condensation film on an inclined rotating disk

Osman Güngör, Cihat Arslantürk

Journal of Applied Mathematics and Computational Mechanics > 2019 > Vol. 18, nr 1 > 15--28

In this paper, the steady three-dimensional problem of condensation film on an inclined rotating disk is considered. The governing nonlinear partial differential equations are reduced to the nonlinear ordinary differential equations system by a similarity transform. The equation system is solved by the variation of parameters method (VPM) which is rather used to solve nonhomogeneous linear differential...

article

The Gross–Pitaevskii equation: Bäcklund transformations and admitted solutions

Sandra Carillo, Federico Zullo

Ricerche di Matematica > 2019 > 68 > 2 > 503-512

Bäcklund transformations are applied to study the Gross–Pitaevskii equation. Supported by previous results, a class of Bäcklund transformations admitted by this equation are constructed. Schwarzian derivative as well as its invariance properties turn out to represent a key tool in the present investigation. Examples and explicit solutions of the Gross–Pitaevskii equation are obtained.

article

Approximation of enzyme kinetics for high enzyme concentration by a first order perturbation approach

Sebastian Kram, Maximilian Schäfer, Rudolf Rabenstein

Journal of Mathematical Chemistry > 2018 > 56 > 4 > 1153-1183

This contribution presents an approximate solution of the enzyme kinetics problem for the case of excess of an enzyme over the substrate. A first order perturbation approach is adopted where the perturbation parameter is the relation of the substrate concentration to the total amount of enzyme. As a generalization over existing solutions for the same problem, the presented approximation allows for...

article

Difference schemes for systems of second order nonlinear ODEs on a semi-infinite interval

M. Król, M.V. Kutniv, O.I. Pazdriy

Applied Numerical Mathematics > 2017 > 119 > C > 33-50

The three-point difference schemes of high order accuracy for the numerical solving boundary value problems on a semi-infinite interval for systems of second order nonlinear ordinary differential equations with a not self-conjugate operator are constructed and justified. We proved the existence and uniqueness of solutions of the three-point difference schemes and obtained the estimate of their accuracy...

article

Tangential velocity during impact with friction in three-dimensional rigid multibody systems

Hesham A. Elkaranshawy, Amr M. Abdelrazek, Hossam M. Ezzat

Nonlinear Dynamics > 2017 > 90 > 2 > 1443-1459

This paper introduces an analysis of tangential velocity of the contact point during a single-point impact with friction in three-dimensional rigid multibody systems. Considering the normal component of impulse at the contact point as the independent variable, a system of nonlinear differential equations describing the progress of the tangential velocity is derived. Quartic algebraic equations for...

article

Exploring inductive linearization for pharmacokinetic–pharmacodynamic systems of nonlinear ordinary differential equations

Chihiro Hasegawa, Stephen B. Duffull

Journal of Pharmacokinetics and Pharmacodynamics > 2018 > 45 > 1 > 35-47

Pharmacokinetic–pharmacodynamic systems are often expressed with nonlinear ordinary differential equations (ODEs). While there are numerous methods to solve such ODEs these methods generally rely on time-stepping solutions (e.g. Runge–Kutta) which need to be matched to the characteristics of the problem at hand. The primary aim of this study was to explore the performance of an inductive approximation...

article

Conditional linearization of the quintic nonlinear beam equation

Nuntawan Sripana, Waraporn Chatanin

Advances in Difference Equations > 2017 > 2017 > 1 > 1-6

In this paper, the linearization of third-order ordinary differential equations, which are the transformed equations of the quintic nonlinear beam equation, is presented. First of all, a third-order ordinary differential equation can be linearized if its coefficients satisfied the conditions of the linearization theorem. So, the conditions for linearization are investigated. After that, in each case,...

article

Time Optimal Control for Some Ordinary Differential Equations with Multiple Solutions

Ping Lin, Shu Luan

Journal of Optimization Theory and Applications > 2017 > 173 > 1 > 78-90

This paper studies a time optimal control problem for a class of ordinary differential equations. The control systems may have multiple solutions. Based on the properties fulfilled by the solutions of the concerned equations, we get both the existence and the Pontryagin maximum principle for optimal controls.

article

Colombeau algebra as a mathematical tool for investigating step load and step deformation of systems of nonlinear springs and dashpots

Vít Průša, Martin Řehoř, Karel Tůma

Zeitschrift für angewandte Mathematik und Physik > 2017 > 68 > 1 > 1-13

The response of mechanical systems composed of springs and dashpots to a step input is of eminent interest in the applications. If the system is formed by linear elements, then its response is governed by a system of linear ordinary differential equations. In the linear case, the mathematical method of choice for the analysis of the response is the classical theory of distributions. However, if the...

article

Solutions to a Periodic Nonlinear Differential Equation: Nonlinear Floquet Theorem

M. Fernández-Guasti

International Journal of Applied and Computational Mathematics > 2017 > 3 > 4 > 3379-3388

Floquet’s theorem applied to Hill’s equation, is translated to its Ermakov pair, namely, the nonlinear amplitude differential equation with periodic parameter. The nonlinear version states that if $$\rho \left( z\right) $$ ρ z is a solution within one period d, to the nonlinear differential equation $$d^{2}\rho /dz^{2}+\rho \varOmega ^{2}-Q^{2}\rho ^{-3}=0$$ d 2 ρ / d z 2 +...

article

Solution for the Nonlinear Relativistic Harmonic Oscillator via Laplace-Adomian Decomposition Method

O. González-Gaxiola, J. A. Santiago, J. Ruiz Chávez

International Journal of Applied and Computational Mathematics > 2017 > 3 > 3 > 2627-2638

In this paper, the Adomian decomposition method in combination with the Laplace transform is used to solve the nonlinear differential equation that governs the oscillation of the relativistic oscillator. The solution obtained is a power series of functions that have never been reported and which show a very good match when compared with other approximate solutions, obtained by different methods. The...

article

Boundary integral solution for a class of fourth-order two-point boundary value problems

H. J. Al-Gahtani

Journal of Applied Mathematics and Computational Mechanics > 2016 > Vol. 15, nr 3 > 5--13

In this paper, a boundary integral method is proposed for the solution of a class of fourth-order two-boundary value problems described by the equation y^iv+P(x, y, y^’, y^’’, y^’’’) = 0, x ∈ ( 0,L), where P is a polynomial function of its arguments. The differential equation is cast in an integral form and the weighted residual technique is used to generate the corresponding boundary integral equations...

article

Analysis of a posteriori error estimates of the discontinuous Galerkin method for nonlinear ordinary differential equations

Mahboub Baccouch

Applied Numerical Mathematics > 2016 > 106 > C > 129-153

We develop and analyze a new residual-based a posteriori error estimator for the discontinuous Galerkin (DG) method for nonlinear ordinary differential equations (ODEs). The a posteriori DG error estimator under investigation is computationally simple, efficient, and asymptotically exact. It is obtained by solving a local residual problem with no boundary condition on each element. We first prove...

article

Inference in dynamic systems using B‐splines and quasilinearized ODE penalties

Gianluca Frasso, Jonathan Jaeger, Philippe Lambert

Biometrical Journal > 58 > 3 > 691 - 714

Nonlinear (systems of) ordinary differential equations (ODEs) are common tools in the analysis of complex one‐dimensional dynamic systems. We propose a smoothing approach regularized by a quasilinearized ODE‐based penalty. Within the quasilinearized spline‐based framework, the estimation reduces to a conditionally linear problem for the optimization of the spline coefficients. Furthermore, standard...

chapter

An Algorithm for Construction of Normal Forms

Victor F. Edneral

Lecture Notes in Computer Science > Computer Algebra in Scientific Computing > 134-142

The normal form method is widely used in the theory of nonlinear ordinary differential equations (ODEs). But in practice it is impossible to evaluate the corresponding transformations without computer algebra packages. Here we describe an algorithm for normalization of nonlinear autonomous ODEs. Some implementations of these algorithms are also discussed.

article

Solution of Lane–Emden type equations using rational Bernoulli functions

Velinda Calvert, Somayeh Mashayekhi, Mohsen Razzaghi

Mathematical Methods in the Applied Sciences > 39 > 5 > 1268 - 1284

In this paper, a numerical method for solving Lane‐Emden type equations, which are nonlinear ordinary differential equations on the semi‐infinite domain, is presented. The method is based upon the modified rational Bernoulli functions; these functions are first introduced. Operational matrices of derivative and product of modified rational Bernoulli functions are then given and are utilized to reduce...

article

Exponential function method for solving nonlinear ordinary differential equations with constant coefficients on a semi-infinite domain

Edmund Chadwick, Ali Hatam, Saeed Kazem

Proceedings - Mathematical Sciences > 2016 > 126 > 1 > 79-97

A new approach, named the exponential function method (EFM) is used to obtain solutions to nonlinear ordinary differential equations with constant coefficients in a semi-infinite domain. The form of the solutions of these problems is considered to be an expansion of exponential functions with unknown coefficients. The derivative and product operational matrices arising from substituting in the proposed...

article

On the nondimensionalization of coupled, nonlinear ordinary differential equations

M. Conesa, J. F. Sánchez Pérez, I. Alhama, F. Alhama

Nonlinear Dynamics > 2016 > 84 > 1 > 91-105

The first step to simplify the analysis of a mathematical model is to search the dimensionless groups that control its solution patterns since the solution of any equation or system of equations that define the laws that rule a physical or engineering problem can be represented as a relation between these groups. One of the current techniques used to attain this aim is the nondimensionalization of...

article

On the behavior of Kneser solutions of nonlinear ordinary differential equations

Andrej A. Kon’kov

Annali di Matematica Pura ed Applicata ( 01923 -) > 2016 > 195 > 3 > 977-994

We obtain sufficient conditions for Kneser solutions of ordinary differential equations to be identically zero in a neighborhood of infinity.

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NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS (94)
NONLINEAR DIFFERENTIAL EQUATIONS (21)
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DELAY (2)
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ENERGY EQUATION (2)
HEAT TRANSFER (2)
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Semi‐analytic time differencing methods for singularly perturbed initial value problems

Variation of parameters method for a three-dimensional problem of condensation film on an inclined rotating disk

The Gross–Pitaevskii equation: Bäcklund transformations and admitted solutions

Approximation of enzyme kinetics for high enzyme concentration by a first order perturbation approach

Difference schemes for systems of second order nonlinear ODEs on a semi-infinite interval

Tangential velocity during impact with friction in three-dimensional rigid multibody systems

Exploring inductive linearization for pharmacokinetic–pharmacodynamic systems of nonlinear ordinary differential equations

Conditional linearization of the quintic nonlinear beam equation

Time Optimal Control for Some Ordinary Differential Equations with Multiple Solutions

Colombeau algebra as a mathematical tool for investigating step load and step deformation of systems of nonlinear springs and dashpots

Solutions to a Periodic Nonlinear Differential Equation: Nonlinear Floquet Theorem

Solution for the Nonlinear Relativistic Harmonic Oscillator via Laplace-Adomian Decomposition Method

Boundary integral solution for a class of fourth-order two-point boundary value problems

Analysis of a posteriori error estimates of the discontinuous Galerkin method for nonlinear ordinary differential equations

Inference in dynamic systems using B‐splines and quasilinearized ODE penalties

An Algorithm for Construction of Normal Forms

Solution of Lane–Emden type equations using rational Bernoulli functions

Exponential function method for solving nonlinear ordinary differential equations with constant coefficients on a semi-infinite domain

On the nondimensionalization of coupled, nonlinear ordinary differential equations

On the behavior of Kneser solutions of nonlinear ordinary differential equations

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