Chaotic Vibrations of Beams: Numerical Solution of Partial Differential Equations

Output feedback vibration control of a string driven by a nonlinear actuator

The scope of this book is to present well known simple and advanced numerical methods for solving partial differential equations (PDEs) and how to implement these methods using the programming environment of the software package Diffpack. A basic background in PDEs and numerical methods is required by the potential reader. Further, a basic knowledge of the finite element method and its implementation in one and two space dimensions is required. The authors claim that no prior knowledge of the package Diffpack is required, which is true, but the reader should be at least familiar with an object oriented programming language like C++ in order to better comprehend the programming environment of Diffpack. Certainly, a prior knowledge or usage of Diffpack would be a great advantage to the reader.

Recently sufficiently deep knowledge has been accumulated about the numerical solution of special types of nonlinear partial differential equations (PDE) and also convergence analysis methods of discrete algorithms have been developed [1, 2]. An important problem is to work out general methods for the investigation of finite difference and variational algorithms for the solution of nonlinear PDE. Construction of such methodology is based on generalization of stability and convergence results for the solution of nonlinear ordinary differential equations (ODE) and the linear PDE. In the case of ODE we will mention the books [3, 4], in which a deep analysis of the basic numerical analysis concepts of approximation, stability, and convergence is given. Recent results of stability theory are comprehensively surveyed in [5]. A fairly complete theory has been already constructed for the numerical solution of the linear PDE. The theorems on necessary and sufficient stability conditions of finite-difference schemes are proved and efficient methods for verification these conditions are developed [6, 7]. Therefore attempts to relate investigation of the convergence of nonlinear finite difference schemes to the convergence analysis of some classes of linear problems seems to be fruitful. Usually we take the equation linearized around a given solution of the nonlinear PDE as an example of such a linear problem. Let us mention the most important works related to this trend. The results of [3] are generalized in [8, 9] and a quite general convergence theory for the nonlinear finite difference schemes is obtained. Similar results are also proved in [10], whose ideas are based on the analysis of nonlinear iterative methods developed in [11]. We point out also a rather general methodology for investigation of twoand threestep finite difference schemes developed in [12, 13]. The results given in [14] for the solution of the nonlinear operator equations are used substantially in this methodology. In this article we generalize the methodology for investigation of nonlinear difference schemes, which was proposed in our papers [15-19] for the solution of diffusion-reaction and nonlinear Schrdinger type problems. Our aim is to obtain sufficient stability conditions of finite difference schemes which will enable us to use the well-known investigation scheme "approXimation + stability = convergence" in the case of nonlinear difference schemes and to prove the existence and uniqueness of the finite difference scheme solution at the same time. First we consider thoroughly the case when the neighborhood of a solution of a differential problem is independent of discrete mesh parameters. The efficiency of the proposed methodology is demonstrated for a concrete finite difference scheme, which approximates the well-known Kuramoto-Tsuzuki problem. After this the given investigation scheme is generalized for the case when radius of the region, in which stability of nonlinear difference scheme is proved, depends on the discrete mesh parameters r, h, and also when, generally speaking,

<i>Self-Similarity and Beyond: Exact Solutions of Nonlinear Problems</i>

Local defect correction techniques : analysis and application to combustion

American Institute of Mining, Metallurgical, and Petroleum Engineers, Inc. Abstract Two dimensional spline functions were utilized in the solution of the linear diffusion equation both with and without a sink within the domain. A collocation method of solution was used and several types of spline representations were investigated. Results were compared with finite difference solutions and with the analytical solutions. B-splines with an extended region of definition were concluded to be superior. For problems with a sink the B-splines alone were not accurate, however, supplementing the splines with a log term produced very good results. Introduction In engineering design, ship builders and others often use a spline, an elastic ruler which can be bent so that it passes through a given set of points. Under certain assumptions the curve can be approximately described as being built up of different third degree polynomials (cubic polynomials) in such a way polynomials (cubic polynomials) in such a way that the function and its first two derivatives are everywhere continuous. The third derivative however, can have discontinuities at the given points. Such a function is called cubic spline. In recent years, spline functions have found wide applications in approximation theory, data fitting, interpolation and so on 4, 9, 15, 16, 17. A smaller amount of work has been devoted to the solution of ordinary and partial differential equations. partial differential equations. Application of spline functions in reservoir problems was initiated by Culham and Varga. problems was initiated by Culham and Varga. These authors used spline functions to approximate the space derivative in a one dimensional nonlinear parabolic partial differential equation which describes the real gas flow in a porous media. The time derivative, however, was approximated by means of various finite difference schemes. They employed Galarkin's and some non-Galarkin type methods to obtain the solutions. Cubic spline basis functions were recommended as the optimum choice for a single-phase flow. When a spline function is used to approximate the solution of a differential equation, two different methods may be used to eliminate the space dependency. These are namely, Galarkin's method (or Integral Methods) and collocation method. Comparisons of these two different solution methods were made by Panton and Salle on a one dimensional heat Panton and Salle on a one dimensional heat conduction equation. They reported that integral methods give higher accuracy in the solution but programming is quite laborous.

In this manuscript, by using undetermined parameter method, an efficient iterative method with eighth-order is designed to solve nonlinear systems. The new method requires one matrix inversion per iteration, which means that computational cost of our method is low. The theoretical efficiency of the proposed method is analyzed, which is superior to other methods. Numerical results show that the proposed method can reduce the computational time, remarkably. New method is applied to solve the numerical solution of nonlinear ordinary differential equations (ODEs) and partial differential equations (PDEs). The nonlinear ODEs and PDEs are discretized by finite difference method. The validity of the new method is verified by comparison with analytic solutions.

The geometrical optical concept of rays is generalized for fields generated by any system of coupled non-linear partial differential equations of arbitrary order like the vector wave equations for (non)linear media, the Maxwell equations, the equations of magneto-hydrodynamics, the system of continuity (transport) equations for an inhomogeneous semiconductor, etc. It is shown that the pertinent solutions of such a system of partial differential equations can be obtained exactly from a set of ordinary non-linear differential equations, the so-called characteristic or ray equations. These characteristic (ray) equations are connected with the key equation of the theory, the generalized Hamilton–Jacobi equation. The generalized Hamilton–Jacobi equation is a first-order non-linear partial differential equation, a special case of which is known to be of great importance for classical mechanics. This equivalence between partial and ordinary differential equations implies an enormous simplification for the numerical evaluation of the original problem, as solving partial differential equations is much more difficult than solving ordinary differential equations. From the theoretical point of view this equivalence between partial and ordinary differential equations is also interesting for various reasons. We mention, for example, that interesting properties like stability, soliton behaviour, decay, growth, etc of the solutions of the field equations of physics can be directly deduced from the relatively easily obtained behaviour of the solutions of the ordinary non-linear differential equations for the ‘rays’. As an example of the potential powers of this new method we analyse the non-linear Schrödinger equation and derive its soliton solutions.

The Laplace transform provides a technique for the solution of ordinary and partial differential equations for initial-value problems in which the number of independent variables is reduced by one. Ordinary differential equations become algebraic equations and equations such as the one-dimensional wave or diffusion equation become ordinary differential equations. The difficulty associated with the method manifests itself in the inversion which is required after the algebraic or ordinary differential equations have been solved. If the equations have suitable analytic solutions then the inversion may be effected either directly from tables or by using the complex inversion formula. If, however, such solutions are not suitable or if numerical solutions are obtained then inversion can cause serious problems. A numerical procedure developed some 25 years ago provides the opportunity for a straightforward numerical inversion. The method works well for ordinary and partial differential equations. In the latter case the resulting boundary-value problem may be solved by either the finite difference method, the finite element method or the boundary element method. The major advantage is that the method does not suffer from possible stability problems that may occur with the usual finite difference procedures in the time variable. The method is ideally suited to implementation on a spreadsheet.

Quadrature solution of ordinary and partial differential equations

In this paper, we propose a method of solution for both nonlinear Ordinary Differential Equations (ODEs) and Partial Differential Equations (PDEs). Shehu Transform is combined with Homotopy Analysis Method (HAM) to handle both homogeneous and nonhomogeneous problems in the family of differential equations considered. The properties of homotopy derivatives are exploited while handling the nonlinear terms encountered. Several examples are solved using HAM and the Shehu Transform Homotopy Analysis Method (STHAM) proposed in this work, and the effectiveness of the proposed method is obvious in terms of reduction in the volume of computations and time. All computations are carried out with the aid of Mathematica 12.0.

Chapter 3 - Finite difference methods and interpolation

Efficient computational system for transient neutron diffusion model via finite difference and theta methods

This paper develops and validates through a series of presentable examples, a comprehensive high-precision, and ultrafast computing concept for solving nonlinear ordinary differential equations (ODEs) and partial differential equations (PDEs) with cellular neural networks (CNN). The core of this concept is a straightforward scheme that we call "nonlinear adaptive optimization (NAOP),” which is used for a precise template calculation for solving nonlinear ODEs and PDEs through CNN processors. One of the key contributions of this work is to demonstrate the possibility of transforming different types of nonlinearities displayed by various classical and well-known nonlinear equations (e.g., van der Pol-, Rayleigh-, Duffing-, Rössler-, Lorenz-, and Jerk-equations, just to name a few) unto first-order CNN elementary cells, and thereby enabling the easy derivation of corresponding CNN templates. Furthermore, in the case of PDE solving, the same concept also allows a mapping unto first-order CNN cells while considering one or even more nonlinear terms of the Taylor's series expansion generally used in the transformation of a PDE in a set of coupled nonlinear ODEs. Therefore, the concept of this paper does significantly contribute to the consolidation of CNN as a universal and ultrafast solver of nonlinear ODEs and/or PDEs. This clearly enables a CNN-based, real-time, ultraprecise, and low-cost computational engineering. As proof of concept, two examples of well-known ODEs are considered namely a second-order linear ODE and a second order nonlinear ODE of the van der Pol type. For each of these ODEs, the corresponding precise CNN templates are derived and are used to deduce the expected solutions. An implementation of the concept developed is possible even on embedded digital platforms (e.g., field programmable gate array (FPGA), digital signal processor (DSP), graphics processing unit (GPU), etc.). This opens a broad range of applications. Ongoing works (as outlook) are using NAOP for deriving precise templates for a selected set of practically interesting ODEs and PDEs equation models such as Lorenz-, Rössler-, Navier Stokes-, Schrödinger-, Maxwell-, etc.

Finite Difference Method: A Brief Study