Dynamical Approach Study of Spurious Steady-State Numerical Solutions of Nonlinear Differential Equations Part IV. Stability vs. Methods of Discretizing Nonlinear Source Terms in Reaction-Convection Equations

1996 ◽  
Vol 6 (2) ◽  
pp. 89-123 ◽  
Author(s):  
A. LAFON ◽  
H. C. YEE
Keyword(s):  
Steady State ◽  
Differential Equations ◽  
Numerical Solutions ◽  
Nonlinear Differential Equations ◽  
Source Terms ◽  
Nonlinear Source ◽  
Dynamical Approach

Finite Deflections of a Nonlinearly Elastic Bar

1970 ◽  
Vol 37 (1) ◽  
pp. 48-52 ◽  
Author(s):  
J. T. Oden ◽  
S. B. Childs
Keyword(s):  
Differential Equations ◽  
Axial Force ◽  
Classical Theory ◽  
Numerical Solutions ◽  
Nonlinear Differential Equations ◽  
Finite Deflections ◽  
Hyperbolic Tangent ◽  
Elastic Bar ◽  
Elastoplastic Materials ◽  
Nonlinearly Elastic

The problem of finite deflections of a nonlinearly elastic bar is investigated as an extension of the classical theory of the elastica to include material nonlinearities. A moment-curvature relation in the form of a hyperbolic tangent law is introduced to simulate that of a class of elastoplastic materials. The problem of finite deflections of a clamped-end bar subjected to an axial force is given special attention, and numerical solutions to the resulting system of nonlinear differential equations are obtained. Tables of results for various values of the parameters defining the material are provided and solutions are compared with those of the classical theory of the elastica.

scholarly journals Homotopy Perturbation Method

2017 ◽  
pp. 24-61
Keyword(s):  
Heat Transfer ◽  
Differential Equations ◽  
Perturbation Method ◽  
Exact Solutions ◽  
Homotopy Perturbation Method ◽  
Numerical Solutions ◽  
Nonlinear Differential Equations ◽  
Homotopy Perturbation ◽  
Wide Range ◽  
Transfer Problems

The homotopy perturbation method (HPM) is employed to compute an approximation to the solution of the system of nonlinear differential equations governing on the problem. It has been attempted to show the capabilities and wide-range applications of the homotopy perturbation method in comparison with the previous ones in solving heat transfer problems. The obtained solutions, in comparison with the exact solutions admit a remarkable accuracy. A clear conclusion can be drawn from the numerical results that the HPM provides highly accurate numerical solutions for nonlinear differential equations.

scholarly journals Modified Variational Iteration Algorithm-II: Convergence and Applications to Diffusion Models

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-14 ◽  
Author(s):  
Hijaz Ahmad ◽  
Tufail A. Khan ◽  
Predrag S. Stanimirović ◽  
Yu-Ming Chu ◽  
Imtiaz Ahmad
Keyword(s):  
Differential Equations ◽  
Numerical Solutions ◽  
Nonlinear Models ◽  
Nonlinear Differential Equations ◽  
Variational Iteration Method ◽  
Iteration Algorithm ◽  
Compact Finite Difference Method ◽  
Variational Iteration ◽  
Linear And Nonlinear ◽  
The Stability

Variational iteration method has been extensively employed to deal with linear and nonlinear differential equations of integer and fractional order. The key property of the technique is its ability and flexibility to investigate linear and nonlinear models conveniently and accurately. The current study presents an improved algorithm to the variational iteration algorithm-II (VIA-II) for the numerical treatment of diffusion as well as convection-diffusion equations. This newly introduced modification is termed as the modified variational iteration algorithm-II (MVIA-II). The convergence of the MVIA-II is studied in the case of solving nonlinear equations. The main advantage of the MVIA-II improvement is an auxiliary parameter which makes sure a fast convergence of the standard VIA-II iteration algorithm. In order to verify the stability, accuracy, and computational speed of the method, the obtained solutions are compared numerically and graphically with the exact ones as well as with the results obtained by the previously proposed compact finite difference method and second kind Chebyshev wavelets. The comparison revealed that the modified version yields accurate results, converges rapidly, and offers better robustness in comparison with other methods used in the literature. Moreover, the basic idea depicted in this study is relied upon the possibility of the MVIA-II being utilized to handle nonlinear differential equations that arise in different fields of physical and biological sciences. A strong motivation for such applications is the fact that any discretization, transformation, or any assumptions are not required for this proposed algorithm in finding appropriate numerical solutions.

SEMI-ANALYTICAL SOLUTIONS FOR THE BRUSSELATOR REACTION–DIFFUSION MODEL

2017 ◽  
Vol 59 (2) ◽  
pp. 167-182 ◽  
Author(s):  
H. Y. ALFIFI
Keyword(s):  
Steady State ◽  
Hopf Bifurcation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Analytical Solutions ◽  
Numerical Solutions ◽  
Partial Differential ◽  
Analytical Results ◽  
Transient Solutions ◽  
The Impact

Semi-analytical solutions are derived for the Brusselator system in one- and two-dimensional domains. The Galerkin method is processed to approximate the governing partial differential equations via a system of ordinary differential equations. Both steady-state concentrations and transient solutions are obtained. Semi-analytical results for the stability of the model are presented for the identified critical parameter value at which a Hopf bifurcation occurs. The impact of the diffusion coefficients on the system is also considered. The results show that diffusion acts to stabilize the systems better than the equivalent nondiffusive systems with the increasing critical value of the Hopf bifurcation. Comparison between the semi-analytical and numerical solutions shows an excellent agreement with the steady-state transient solutions and the parameter values at which the Hopf bifurcations occur. Examples of stable and unstable limit cycles are given, and Hopf bifurcation points are shown to confirm the results previously calculated in the Hopf bifurcation map. The usefulness and accuracy of the semi-analytical results are confirmed by comparison with the numerical solutions of partial differential equations.

scholarly journals Effect of Material Nonlinearity on Large Deflection of Variable-Arc-Length Beams Subjected to Uniform Self-Weight

2012 ◽  
Vol 2012 ◽  
pp. 1-9 ◽  
Author(s):  
Chainarong Athisakul ◽  
Boonchai Phungpaingam ◽  
Gissanachai Juntarakong ◽  
Somchai Chucheepsakul
Keyword(s):  
Differential Equations ◽  
Large Deflection ◽  
Numerical Solutions ◽  
Nonlinear Differential Equations ◽  
Optimization Technique ◽  
Material Constant ◽  
Constitutive Law ◽  
Arc Length ◽  
Stress Strain Relation ◽  
Equilibrium Configurations

This paper presents a large deflection of variable-arc-length beams, which are made from nonlinear elastic materials, subjected to its uniform self-weight. The stress-strain relation of materials obeys the Ludwick constitutive law. The governing equations of this problem, which are the nonlinear differential equations, are derived by considering the equilibrium of a differential beam element and geometric relations of a beam segment. The model formulation presented herein can be applied to several types of nonlinear elastica problems. With presence of geometric and material nonlinearities, the system of nonlinear differential equations becomes complicated. Consequently, the numerical method plays an important role in finding solutions of the presented problem. In this study, the shooting optimization technique is employed to compute the numerical solutions. From the results, it is found that there is a critical self-weight of the beam for each value of a material constantn. Two possible equilibrium configurations (i.e., stable and unstable configurations) can be found when the uniform self-weight is less than its critical value. The relationship between the material constantnand the critical self-weight of the beam is also presented.

A Comparative Assessment of Simplified Models for Simulating Parametric Roll

2017 ◽  
Vol 139 (2) ◽  
Author(s):  
Abhilash S. Somayajula ◽  
Jeffrey Falzarano
Keyword(s):  
Differential Equations ◽  
Analytical Solutions ◽  
Time Domain ◽  
Numerical Solutions ◽  
Nonlinear Differential Equations ◽  
Nonlinear Phenomenon ◽  
Concept Design ◽  
Parametric Roll ◽  
Statistical Confidence ◽  
Restoring Forces

The motion of a ship/offshore platform at sea is governed by a coupled set of nonlinear differential equations. In general, analytical solutions for such systems do not exist and recourse is taken to time-domain simulations to obtain numerical solutions. Each simulation is not only time consuming but also captures only a single realization of the many possible responses. In a design spiral when the concept design of a ship/platform is being iteratively changed, simulating multiple realizations for each interim design is impractical. An analytical approach is preferable as it provides the answer almost instantaneously and does not suffer from the drawback of requiring multiple realizations for statistical confidence. Analytical solutions only exist for simple systems, and hence, there is a need to simplify the nonlinear coupled differential equations into a simplified one degree-of-freedom (DOF) system. While simplified methods make the problem tenable, it is important to check that the system still reflects the dynamics of the complicated system. This paper systematically describes two of the popular simplified parametric roll models in the literature: Volterra GM and improved Grim effective wave (IGEW) roll models. A correction to the existing Volterra GM model described in current literature is proposed to more accurately capture the restoring forces. The simulated roll motion from each model is compared against a corresponding simulation from a nonlinear coupled time-domain simulation tool to check its veracity. Finally, the extent to which each of the models captures the nonlinear phenomenon accurately is discussed in detail.

An implicit differential equation governing lumped capacitance, radiation dominated, unsteady, heat transfer

2012 ◽  
Vol 22 (7) ◽  
pp. 896-906
Author(s):  
Lawrence J. De Chant
Keyword(s):  
Heat Transfer ◽  
Differential Equation ◽  
Simple Model ◽  
Differential Equations ◽  
Large Scale ◽  
Numerical Solutions ◽  
Nonlinear Differential Equations ◽  
Implicit Differential Equation ◽  
Content Type ◽  
Integration Schemes

PurposeAlthough most physical problems in fluid mechanics and heat transfer are governed by nonlinear differential equations, it is less common to be confronted with a “so – called” implicit differential equation, i.e. a differential equation where the highest order derivative cannot be isolated. The purpose of this paper is to derive and analyze an implicit differential equation that arises from a simple model for radiation dominated heat transfer based upon an unsteady lumped capacitance approach.Design/methodology/approachHere we discuss an implicit differential equation that arises from a simple model for radiation dominated heat transfer based upon an unsteady lumped capacitance approach. Due to the implicit nature of this problem, standard integration schemes, e.g. Runge‐Kutta, are not conveniently applied to this problem. Moreover, numerical solutions do not provide the insight afforded by an analytical solution.FindingsA predictor predictor‐corrector scheme with secant iteration is presented which readily integrates differential equations where the derivative cannot be explicitly obtained. These solutions are compared to numerical integration of the equations and show good agreement.Originality/valueThe paper emphasizes that although large‐scale, multi‐dimensional time‐dependent heat transfer simulation tools are routinely available, there are instances where unsteady, engineering models such as the one discussed here are both adequate and appropriate.

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