General Initial Value Research Articles

Exact solutions of forced Burgers equations with time variable coefficients

In this paper, we consider a forced Burgers equation with time variable coefficients of the form Ut+(μ˙(t)/μ(t))U+UUx=(1/2μ(t))Uxx-ω2(t)x, and obtain an explicit solution of the general initial value problem in terms of a corresponding second order linear ordinary differential equation. Special exact solutions such as generalized shock and multi-shock waves, triangular wave, N-wave and rational type solutions are found and discussed. Then, we introduce forced Burgers equations with constant damping and an exponentially decaying diffusion coefficient as exactly solvable models. Different type of exact solutions are obtained for the critical, over and under damping cases, and their behavior is illustrated explicitly. In particular, the existence of inelastic type of collisions is observed by constructing multi-shock wave solutions, and for the rational type solutions the motion of the pole singularities is described.

Exact solutions and numerical approximations of mixed problems for the wave equation with delay

This work deals with the construction of exact and continuous numerical solutions of mixed problems for the wave equation with delayutt(t,x)=a2uxx(t,x)+bu(t-τ,x),t>τ,0⩽x⩽l.Using the method of separation of variables, and obtaining explicit expressions for the solution of a general initial value problem for a second order delay differential equation, an exact infinite series solution is derived. Bounds on the truncation errors, when this series is approximated by finite sums, are given, thus providing constructive continuous numerical solutions with prescribed accuracy in bounded domains.

Elastodynamics in micropolar fractal solids

This research explores elastodynamics and wave propagation in fractal micropolar solid media. Such media incorporate a fractal geometry while being modelled constitutively by the Cosserat elasticity. The formulation of the balance laws which govern the mechanics of fractal micropolar solid media is presented. Four eigenvalue-type elastodynamic problems admitting closed-form analytical solutions are introduced and discussed. A numerical procedure to solve general initial boundary value wave propagation problems in three-dimensional micropolar bodies exhibiting geometric fractality is then applied. Verification of the numerical procedure is discussed using the analytical solutions.

Continuous Block-Predictor Hybrid-Corrector Method

A method of collocation and interpolation of the power series approximate solution at some grid and off-grid points is considered to generate a continuous linear multistep method for the solution of general second order initial value problems at constant step size. We use continuous block method to generate independent solutions which serves as predictors at selected points within the interval of integration. The efficiency of the proposed method was tested and was found to compete favorably with the existing methods.

Stability of Traveling Front Solutions with Algebraic Spatial Decay for Some Autocatalytic Chemical Reaction Systems

This paper is concerned with the linear and nonlinear asymptotic stability of traveling front solutions with algebraic spatial decay for some autocatalytic chemical reaction systems, especially for the typical autocatalytic system with reaction rate $u^qv^p$ when $p>1$ and $q\ge 1$. First, for the autocatalytic systems with equal diffusion rates and with more general initial values, the wave fronts with noncritical speeds are proved to be nonlinearly asymptotically stable in some special polynomially weighted spaces for the case where $p>1$ and $q=1$, and the waves fronts with noncritical speeds are proved to be linearly asymptotically stable for the case where $p>1$ and $q>1$. Second, by applying special transformations and appropriate matrix perturbation theories, we establish some abstract results on the existence and analyticity of the Evans function for the more general linear ODE systems with slow algebraic decaying coefficients. Third, by detailed spectral estimates and by applying our abstract results on the Evans function to the autocatalytic systems for the case where $p>1$ and $q\ge 1$ and when the two diffusion rates are close, we prove that the wave fronts with noncritical speeds are linearly exponentially stable in some exponentially weighted spaces. Finally, for the autocatalytic systems with $p>1$ and $q=1$ it can be shown that if the initial perturbation of the wave in $C_{\rm unif}(R)$ space is small in both the unweighted norm and the exponentially weighted norm, then the perturbation stays small in the unweighted norm and decays exponentially in the exponentially weighted norm; further, we can prove that if the initial perturbation is, in addition, small in the $L^1$ norm, then the perturbation also stays small in the $L_1(R)$ norm and decays algebraically in the $C_{\rm unif}(R)$ norm.

NUMERICAL SOLUTION FOR SOLVING SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS USING BLOCK METHOD

The purpose of this paper is to present a four point direct block one-step method for solving directly the general second order nonstiff initial value problems (IVPs) of ordinary differential equations (ODEs). The mathematical problems in real world can be written in the form of differential equations and arise in the fields of science and engineering such as fluid dynamic, electric circuit, motion of rocket or satellite and other area of application. The proposed method will estimate the approximation solutions at four points simultaneously by using variable step size. Numerical results are given to show the efficiency of the proposed method.

Finding attractors of continuous-time systems by parameter switching

The review presents a parameter switching algorithm and his applications which allows numerical approximation of any attractor of a class of continuous-time dynamical systems depending linearly on a real parameter. The considered classes of systems are modeled by a general initial value problem which embeds dynamical systems continuous and discontinuous with respect to the state variable, of integer, and fractional order. The numerous results, presented in several papers, are systematized here on four representative known examples representing the four classes. The analytical proof of the algorithm convergence for the systems belonging to the continuous class is presented briefly, while for the other categories of systems, the convergence is numerically verified via computational tools. The utilized numerical tools necessary to apply the algorithm are contained in Appendices A, B, C, D and E.

An Alternate Approach to Optimal L 2-Error Analysis of Semidiscrete Galerkin Methods for Linear Parabolic Problems with Nonsmooth Initial Data

In this article, we propose and analyze an alternate proof of a priori error estimates for semidiscrete Galerkin approximations to a general second order linear parabolic initial and boundary value problem with rough initial data. Our analysis is based on energy arguments without using parabolic duality. Further, it follows the spirit of the proof technique used for deriving optimal error estimates for finite element approximations to parabolic problems with smooth initial data and hence, it unifies both theories, that is, one for smooth initial data and other for nonsmooth data. Moreover, the proposed technique is also extended to a semidiscrete mixed method for linear parabolic problems. In both cases, optimal L 2-error estimates are derived, when the initial data is in L 2. A superconvergence phenomenon is also observed, which is then used to prove L ∞-estimates for linear parabolic problems defined on two-dimensional spatial domain again with rough initial data.

On the stability and convergence of a Galerkin reduced order model (ROM) of compressible flow with solid wall and far‐field boundary treatment

AbstractA reduced order model (ROM) based on the proper orthogonal decomposition (POD)/Galerkin projection method is proposed as an alternative discretization of the linearized compressible Euler equations. It is shown that the numerical stability of the ROM is intimately tied to the choice of inner product used to define the Galerkin projection. For the linearized compressible Euler equations, a symmetry transformation motivates the construction of a weighted L2 inner product that guarantees certain stability bounds satisfied by the ROM. Sufficient conditions for well‐posedness and stability of the present Galerkin projection method applied to a general linear hyperbolic initial boundary value problem (IBVP) are stated and proven. Well‐posed and stable far‐field and solid wall boundary conditions are formulated for the linearized compressible Euler ROM using these more general results. A convergence analysis employing a stable penalty‐like formulation of the boundary conditions reveals that the ROM solution converges to the exact solution with refinement of both the numerical solution used to generate the ROM and of the POD basis. An a priori error estimate for the computed ROM solution is derived, and examined using a numerical test case. Published in 2010 by John Wiley & Sons, Ltd.

A direct variable step block multistep method for solving general third-order ODEs

This paper discusses a direct three-point implicit block multistep method for direct solution of the general third-order initial value problems of ordinary differential equations using variable step size. The method is based on a pair of explicit and implicit of Adams type formulas which are implemented in PE(CE) t mode and in order to avoid calculating divided difference and integration coefficients all the coefficients are stored in the code. The method approximates the numerical solution at three equally spaced points simultaneously. The Gauss Seidel approach is used for the implementation of the proposed method. The local truncation error of the proposed scheme is studied. Numerical examples are given to illustrate the efficiency of the method.

A class of collocation methods for general second order ordinary differential equations

In this paper, self starting hybrid block method of order is proposed for the solution of general second order initial value problem of the form directly without reducing it to first systems of odes. The continuous formation of the integrator enable us to differentiate and evaluate at some grids and off-grid points to take care of y

A self-starting linear multistep method for a direct solution of the general second-order initial value problem

In this paper, we propose a linear multistep method of order 5 that is self-starting for the direct solution of the general second-order initial value problem (IVP). The method is derived by the interpolation and collocation of the assumed approximate solution and its second derivative at x=x n+j , j=1, 2, …, r−1, and x=x n+j , j=1, 2, …, s−1, respectively, where r and s are the number of interpolation and collocation points, respectively. The interpolation and collocation procedures lead to a system of (r+s) equations involving (r+s) unknown coefficients, which are determined by the matrix inversion approach. The resulting coefficients are used to construct the approximate solution from which multiple finite difference methods (MFDMs) are obtained and simultaneously applied to provide a direct solution to IVPs. In particular, the method is implemented without the need for either predictors or starting values from other methods. Numerical examples are given to illustrate the efficiency of the method.

Method for solving nonlinear initial value problems by combining homotopy perturbation and reproducing kernel Hilbert space methods

The nonlinear singular initial value problems including generalized Lane–Emden-type equations are investigated by combining homotopy perturbation method (HPM) and reproducing kernel Hilbert space method (RKHSM). He’s HPM is based on the use of traditional perturbation method and homotopy technique and can reduce a nonlinear problem to some linear problems and generate a rapid convergent series solution in most cases. RKHSM is also an analytical technique, which can overcome the difficulty at the singular point of non-homogeneous, linear singular initial value problems; especially when the singularity appears on the right-hand side of this type of equations, so it can solve powerfully linear singular initial value problems. Therefore, using advantages of these two methods, more general nonlinear singular initial value problems can be solved powerfully. Some numerical examples are presented to illustrate the strength of the method.

An interpretation of variational principles for gauge theories: a cyclic coordinate alternative to ADM split

I show that there is an ambiguity in how one treats auxiliary variables in gauge theories including general relativity cast as 3 + 1 geometrodynamics. Auxiliary variables may be treated pre-variationally as multiplier coordinates or as the velocities corresponding to cyclic coordinates. The latter treatment works through the physical meaninglessness of auxiliary variables' values applying also to the end points (or end spatial hypersurfaces) of the variation, so that these are free rather than fixed. (This is also known as variation with natural boundary conditions.) Further principles of dynamics workings such as Routhian reduction and the Dirac procedure are shown to have parallel counterparts for this new formalism. One advantage of the new scheme is that the corresponding actions are more manifestly relational. While the electric potential is usually regarded as a multiplier coordinate and Arnowitt, Deser and Misner have regarded the lapse and shift likewise, this paper's scheme considers new flux, instant and grid variables whose corresponding velocities are, respectively, the above-mentioned previously used variables. This paper's way of thinking about gauge theory furthermore admits interesting generalizations as regards variational principles for the conformal form of general relativity's initial value problem.

Characterization of Lyapunov pairs in the nonlinear case and applications

The paper presents a characterization of the Lyapunov pairs for a general initial value problem with a possibly multivalued m -accretive operator on an arbitrary Banach space by means of the contingent derivative related to the operator. The proof is based on tangency and flow-invariance arguments combined with a priori estimates and approximation. The abstract results are applied to obtain precise a priori estimates for the mild solutions. They readily lead to the existence of global solutions and various controllability properties. Important Lyapunov pairs are pointed out in connection with specific problems.

A new analytical technique to find periodic solutions of non-linear systems

Based on the classical harmonic balance method a new technique is presented to determine higher approximate periodic solutions of the non-linear differential equations. The new method is systematic and simple. The solution covers the general initial value problem (i.e., for [ x ( 0 ) = a 0 , x ˙ ( 0 ) = b 0 ] ) while the existing solution is determined for a particular case, especially for [ x ( 0 ) = a 0 , x ˙ ( 0 ) = 0 ] . The solution is easily transformed to perturbation solution. The method is used in various non-linear problems possessing second and more than second derivatives.

Self-similar solutions in the problem of generalized vortex diffusion

The analysis of the group properties and the search for self-similar solutions in problems of mathematical physics and continuum mechanics have always been of interest, both theoretical and applied [1–3]. Self-similar solutions of parabolic problems that depend only on a variable of the type η = x/√t are classical fundamental solutions of the one-dimensional linear and nonlinear heat conduction equations describing numerous physical phenomena with initial discontinuities on the boundary [4]. In this study, the term “generalized vortex diffusion” is introduced in order to unify the different processes in mechanics modeled by these problems. Here, vortex layer diffusion and vortex filament diffusion in a Newtonian fluid [5] can serve as classical hydrodynamic examples. The cases of self-similarity with respect to the variable η are classified for fairly general kinematics of the processes, physical nonlinearities of the medium, and types of boundary conditions at the discontinuity points. The general initial and boundary value problem thus formulated is analyzed in detail for Newtonian and non-Newtonian power-law fluids and a medium similar in behavior to a rigid-ideally plastic body. New self-similar solutions for the shear stress are derived.

A concise functional neural network computing the largest modulus eigenvalues and their corresponding eigenvectors of a real skew matrix

Quick extraction of the largest modulus eigenvalues of a real antisymmetric matrix is important for some engineering applications. As neural network runs in concurrent and asynchronous manner in essence, using it to complete this calculation can achieve high speed. This paper introduces a concise functional neural network (FNN), which can be equivalently transformed into a complex differential equation, to do this work. After obtaining the analytic solution of the equation, the convergence behaviors of this FNN are discussed. Simulation result indicates that with general initial complex values, the network will converge to the complex eigenvector which corresponds to the eigenvalue whose imaginary part is positive, and modulus is the largest of all eigenvalues. Comparing with other neural networks designed for the like aim, this network is applicable to real skew matrices.

A class of hybrid collocation methods for third-order ordinary differential equations

A class of numerical methods is proposed for solving general third-order ordinary differential equations directly by collocation at the grid points x = x n+j , i = 0(1)k and at an off grid point x = x n+u , where k is the step number of the method and u is an arbitrary rational number in (x n , x n+k ). A predictor of order 2k − 1 is also proposed to cater for y n+k in the main method. Taylor series expansion is employed for the calculation of y n+1, y n+2, y n+u and their higher derivatives. Evaluation of the resulting method at x = x n+k for any value of u in the specified open interval yields a particular discrete scheme as a special case of the method. The efficiency of the method is tested on some general initial value problems of third-order ordinary differential equations.

Relating optimal growth to counterpropagating Rossby waves in shear instability

The optimal dynamics of conservative disturbances to plane parallel shear flows is interpreted in terms of the propagation and mutual interaction of components called counterpropagating Rossby waves (CRWs). Pairs of CRWs were originally used by Bretherton to provide a mechanistic explanation for unstable normal modes in the barotropic Rayleigh model and baroclinic two-layer model. One CRW has large amplitude in regions of positive mean cross-stream potential vorticity (PV) gradient, while the second CRW has large amplitude in regions of negative PV gradient. Each CRW propagates to the left of the mean PV gradient vector, parallel to the mean flow. If the mean flow is more positive where the PV gradient is positive, the intrinsic phase speeds of the two CRWs will be similar. The CRWs interact because the PV anomalies of one CRW induce cross-stream velocity at the location of the other CRW, thus advecting the mean PV. Although a single Rossby wave is neutral, their interaction can result in phase locking and mutual growth. Here the general initial value problem for disturbances to shear flow is analyzed in terms of CRWs. For the discrete spectrum (which could alternatively be described using normal modes), the singular value decomposition of the dynamical propagator can be obtained analytically in terms of the CRW interaction coefficient and the intrinsic CRW phase speeds. Using this formalism, optimal perturbations, the disturbances which grow fastest in a given norm over a specified time interval, can readily be found. The most natural norm for CRWs is related to air parcel displacements or enstrophy. However, if an energy norm is taken, it is shown to grow due to both mutual amplification of air parcel displacements and the untilting of PV structures (the Orr mechanism) associated with decreasing phase difference between the CRWs. A generalization of the CRW description to the optimal dynamics of the complete spectrum solution is outlined. Although the dynamics then involves the interaction between an infinite number of “CRW kernels,” the form of the simple interaction between any two CRW kernels is the same as in the discrete case.