A perturbation method for nonlinear dispersive wave problems

Stochastic differential equation models play an increasingly prominent role in a wide variety of application areas, particularly those characterized by complexity and random behavior. Examples of such areas include biology, epidemiology, mechanics, economics, and finance. This issue's Education section contains a paper, "An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations," by Desmond Higham, that makes this important material accessible to a wide range of readers. Higham's paper is written from the perspective that the best way to gain insight quickly into the topics of stochastic differential equations (SDEs) is to see examples of numerical methods in use. The author feels that it is possible to explain and understand how simple numerical methods in this area work, without requiring the reader to be familiar with SDE theory. Indeed, the paper takes the approach that simulations of the dynamic behavior of SDEs and experimentation with numerical examples can help develop an intuitive understanding of SDEs that lays the groundwork for subsequent theoretical treatments. Thus Higham's paper expects only that the reader's background includes Euler's method for deterministic ordinary differential equations and an intuitive understanding of random variables. It provides a beautifully written treatment that is aimed at upper-division undergraduates and beginning graduate students. The topics covered in Higham's paper include investigation of stochastic integration, linear stability, and strong and weak convergence. The paper is built around ten complete downloadable MATLAB programs that are used as examples and contain ample comment lines. The programs are linked to relevant discussion points in the body of the paper. This paper is suitable as a module in a numerical methods course, such as a course on numerical methods for ordinary differential equations. It could also find application in courses in the areas where SDEs arise, such as finance, epidemiology, or general mathematical modeling. And it can serve excellently as the starting point for a self-study project in the area of SDEs for advanced undergraduates or beginning graduate students. Finally, many professionals in the SIAM community will find this article useful as an introduction to an area of numerical computation that has grown in prominence since many of us were students!

Separation of variables is a well‐known technique for solving differential equations. However, it is seldom used in practical applications since it is impossible to carry out a separation of variables in most cases. In this paper, we propose the amplitude–shape approximation (ASA) which may be considered as an extension of the separation of variables method for ordinary differential equations. The main idea of the ASA is to write the solution as a product of an amplitude function and a shape function, both depending on time, and may be viewed as an incomplete separation of variables. In fact, it will be seen that such a separation exists naturally when the method of lines is used to solve certain classes of coupled partial differential equations. We derive new conditions which may be used to solve the shape equations directly and present a numerical algorithm for solving the resulting system of ordinary differential equations for the amplitude functions. Alternatively, we propose a numerical method, similar to the well‐established exponential time differencing method, for solving the shape equations. We consider stability conditions for the specific case corresponding to the explicit Euler method. We also consider a generalization of the method for solving systems of coupled partial differential equations. Finally, we consider the simple reaction diffusion equation and a numerical example from chemical kinetics to demonstrate the effectiveness of the method. The ASA results in far superior numerical results when the relative errors are compared to the separation of variables method. Furthermore, the method leads to a reduction in CPU time as compared to using the Rosenbrock semi‐implicit method for solving a stiff system of ordinary differential equations resulting from a method of lines solution of a coupled pair of partial differential equations. The present amplitude–shape method is a simplified version of previous ones due to the use of a linear approximation to the time dependence of the shape function. Copyright © 2007 John Wiley & Sons, Ltd.

In order to provide simple and accurate wave theory in design of offshore structure, an analytical approximation is introduced in this paper. The solution is limited to flat bottom having a constant water depth. Water is considered as inviscid, incompressible and irrotational. The solution satisfies the continuity equation, bottom boundary condition and non-linear kinematic free surface boundary condition exactly. Error for dynamic condition is quite small. The solution is suitable in description of breaking waves. The solution is presented with closed form and dispersion relation is also presented with closed form. In the last century, there have been two main approaches to the nonlinear problems. One of these is perturbation method. Stokes wave and Cnoidal wave are based on the method. The other is numerical method. Dean’s stream function theory is based on the method. In this paper, power series method was considered. The power series method can be applied to certain nonlinear differential equations (initial value problems). The series coefficients are specified by a nonlinear recurrence inherited from the differential equation. Because the non-linear wave problem is a boundary value problem, the power series method cannot be applied to the problem in general. But finite number of coefficients is necessary to describe the wave profile, truncated power series is enough. Therefore the power series method can be applied to the problem. In this case, the series coefficients are specified by a set of equations instead of recurrence. By using the set of equations, the nonlinear wave problem has been solved in this paper.

On stability and convergence of semi-Lagrangian methods for the first-order time-dependent nonlinear partial differential equations in 1D

A new analytical solution method is proposed for ordinary differential equations. Starting with an initial perturbation solution, a first iteration solution is obtained by directly integrating the highest order term while substituting the assumed perturbed solution to the remaining terms. The iterations can be repeated to obtain a satisfactory converged solution in the domain of interest. The method has been applied to four sample first and second order linear and nonlinear differential equations. The iteration solutions are contrasted with the available exact solutions. The method is simple, straightforward and converges to the real solution if the initial perturbation solution is a good starting assumption.

Classical considerations of stability in ODE initial and boundary problems are mirrored by corresponding properties (stiff stability, di- stability) in problem discretizations. However, computational cate- gories are not precise, and qualitative descriptors such as of moderate size cannot be avoided where size varies with the sensitivity of the Newton iteration in nonlinear problems, for example. Sensitive New- ton iterations require close enough initial estimates. The main tool for providing this in boundary value problems is continuation with respect to a parameter. If stable discretizations are not available, then adaptive meshing is needed to follow rapidly changing solutions. Use of such tools can be necessary in stable nonlinear situations not covered by classical considerations. Implications for the estimation problem are sketched. It is shown how to choose appropriate bound- ary conditions for the embedding method. The simultaneous method is formulated as a constrained optimization problem. It avoids explicit ODE characterization and appears distinctly promising. However, its properties are not yet completely understood.

Solutions to Axon Equations

This comprehensive introduction to global spectral methods for fractional differential equations from leaders of this emerging field is designed to be accessible to graduate students and researchers across math, science, and engineering. The book begins by covering the foundational fractional calculus concepts needed to understand and model anomalous transport phenomena. The book proceeds to introduce a series of new spectral theories and new families of orthogonal and log-orthogonal functions, then presents corresponding spectral and spectral element methods for fractional differential equations. The book also covers the fractional Laplacian in unbounded and bounded domains and major developments in time-integration of fractional models. The book ends by sampling the wide variety of real-world applications of fractional modeling, including: concentration transport in surface/subsurface dynamics; complex rheology and material damage; and fluid turbulence and geostrophic transport.

In this paper, we present a modified and new version of spectral method which is based on minimization of obtained residual term in norm, where is a weight function with respect to Jacobi polynomials. Using this approach is efficient and effective rather than Tau and collocation methods. It reduces the nonlinear ordinary differential equations to the nonlinear programming problems which is an easy problem to solve. Hence, easy implementation of the method is the importance of our approach and some numerical test experiments show the accuracy and efficiency of this method. Key words: A new spectral method, nonlinear ordinary differential equations.

Continuous numerical methods for ODEs with defect control

Decentralized Predictor Output Feedback for Large-scale Systems with Large Delays

The purpose of this study is to develop a numerical method for bifurcation analysis of nonlinear partial differential equations, based on the reduction of partial differential equations to ordinary ones, using the Kolmogorov-Arnold theorem. Methods. This paper describes a method for reducing partial differential equations to ordinary ones using the Kolmogorov-Arnold theorem, as well as methods for the bifurcation analysis of nonlinear boundary value problems for ordinary differential equations. Results. The paper presents a new method for solving and bifurcation analysis of nonlinear boundary value problems for partial differential equations, which allow variational formulation. The method was applied to a nonlinear two-dimensional Bratu problem with Dirichlettype boundary conditions. Conclusion. A new method of bifurcation analysis for nonlinear partial differential equations has been developed. Specifically, a method has been proposed for reducing partial different equations to ordinary equations, which allows the use of the developed apparatus of bifurcation analysis for boundary value problems of ordinary differential equations. This method allows conducting bifurcation analysis for arbitrary nonlinear partial differential equations.

Homotopy perturbation method (HPM) is a semi-analytical technique for solving linear as well as nonlinear ordinary/partial differential equations. The method may also be used to solve a system of coupled linear and nonlinear differential equations. This chapter first illustrates the basic idea of the HPM. It uses the present method to solve a nonlinear ordinary differential equation, a nonlinear partial differential equation, and a coupled nonlinear system of partial differential equations. It is worth mentioning that the HPM can also be used for handling linear ordinary and partial differential equations. The test problems demonstrated in the chapter confirm that the HPM is an efficient method for solving linear/nonlinear ordinary, partial, and coupled differential equations. The series usually converges with an increase in the number of terms, but one may not expect the compact form solution always.

: Complex physical phenomena involving chemically reacting systems or the transport of heat or fluids are often modeled by coupled systems of time-dependent, nonlinear partial differential equations. The difficulties in understanding the stability of the differential equation systems and in designing efficient, accurate numerical methods for their solution are widely recognized and were the focus of this research. We have worked on four general aspects of the analysis and numerical approximation of systems of partial differential equations. These areas of research are: (1) modeling aspects and stability analysis for nonlinear time-dependent partial differential equations; (2) use and analysis of finite element or finite difference methods to discretize coupled systems of nonlinear differential equations; (3) development of adaptive or local grid refinement capabilities to resolve local phenomena in large-scale applications; and (4) development of data structures, preconditioners, and efficient solution algorithms for large-scale problems on new computer architectures. Emphasis has been placed upon multiphase or multicomponent, transport-dominated flow processes with dynamic local phenomena. The research also involved a mix of analysis, algorithm development, and large-scale computation using newer computer architectures.

We consider an enzyme-substrate reaction-diffusion problem. Unsteady and steady state models are recalled. For the unsteady state case, the model is in the form of a second order partial differential equation. We solve the unsteady state model using the explicit numerical finite difference method, which is forward difference in time and centered difference in space. For the general steady state case, the model is in the form of a second order ordinary differential equation. We solve the general steady state model using the explicit first order Euler's numerical method. For the particular steady state case of the unsaturated catalytic kinetics, we derive the exact analytical solution using the characteristic method of ordinary differential equations. For the particular steady state case of the saturated catalytic kinetics, we derive the exact analytical solution using the direct-integration method. The obtained exact analytical solutions are identical with the existing exact analytical solutions derived using the variational iteration method. With the aid of computer, the enzyme-substrate reaction-diffusion problem can be solved and simulated successfully for both unsteady and steady state cases.