Sensitivity of intrinsic mode functions of Lorenz system to initial values based on EMD method

Extreme sensitivity to initial values is an intrinsic character of chaotic systems. The evolution of a chaotic system has a spatiotemporal structure containing quasi-periodic changes of different spatiotemporal scales. This paper uses an empirical mode decomposition （EMD） method to decompose and compare the evolution of the time-dependent evolutions of the x-component of the Lorenz system. The results indicate that the sensitivity of intrinsic mode function （IMF） component is dependent on initial values, which provides some scientific evidence for the possibility of long-range climatic prediction.

THE QUASI-BOUNDARY VALUE METHOD FOR IDENTIFYING THE INITIAL VALUE OF THE SPACE-TIME FRACTIONAL DIFFUSION EQUATION

In this article,we consider to solve the inverse initial value problem for an inhomogeneous space-time fractional diffusion equation.This problem is ill-posed and the quasi-boundary value method is proposed to deal with this inverse problem and obtain the series expression of the regularized solution for the inverse initial value problem.We prove the error estimates between the regularization solution and the exact solution by using an a priori regularization parameter and an a posteriori regularization parameter choice rule.Some numerical results in one-dimensional case and two-dimensional case show that our method is efficient and stable.

Initial value problem for a class of fourth-order nonlinear wave equations

In this paper, existence and uniqueness of the generalized global solution and the classical global solution to the initial value problem for a class of fourth-order nonlinear wave equations are studied in the fractional order Sobolev space using the contraction mapping principle and the extension theorem. The sufficient conditions for the blow up of the solution to the initial value problem are given.

The Relationship between Nonconservative Schemes and Initial Values of Nonlinear Evolution Equations

For the nonconservative schemes of the nonlinear evolution equations, taking the one-dimensional shallow water wave equation as an example, the necessary conditions of computational stability are given. Based on numerical tests, the relationship between the nonlinear computational stability and the construction of difference schemes, as well as the form of initial values, is further discussed. It is proved through both theoretical analysis and numerical tests that if the construction of difference schemes is definite, the computational stability of nonconservative schemes is decided by the form of initial values.

Laguerre-Gauss collocation method for initial value problems of second order ODEs

This paper proposes a new collocation method for initial value problems of second order ODEs based on the Laguerre-Gauss interpolation. It provides the global numerical solutions and possesses the spectral accuracy. Numerical results demonstrate its high efficiency.

Numerical Treatment of Initial Value Problems of Nonlinear Ordinary Differential Equations by Duan-Rach-Wazwaz Modified Adomian Decomposition Method

We employ the Duan-Rach-Wazwaz modified Adomian decomposition method for solving initial value problems for the systems of nonlinear ordinary differential equations numerically. In order to confirm practicality, robustness and reliability of the method, we compare the results from the modified Adomian decomposition method with those from the MATHEMATICA solutions and also from the fourth-order Runge Kutta method solutions in some cases. Furthermore, we apply Padé approximants technique to improve the solutions of the modified decomposition method whenever the exact solutions exist.

THE STABILITY ON THE SOLUTION OF THE INITIAL VALUE PROBLEM

In this paper, using the differentiability of the solution with respect to the initial value and the parameter, we present a method which, different from Liapunov's direct method. will determine the stability oj the non-stationary solution of the initial value problem when the non-stationary solution remains unknown.

An Explicit Single-Step Nonlinear Numerical Method for First Order Initial Value Problems (IVPs)

Interest in the construction of efficient methods for solving initial value problems that have some peculiar properties with it or its solution is recently gaining wide popularity. Based on the assumption that the solution is representable by nonlinear trigonometric expressions, this work presents an explicit single-step nonlinear method for solving first order initial value problems whose solution possesses singularity. The stability and convergence properties of the constructed scheme are also presented. Implementation of the new method on some standard test problems compared with those discussed in the literature proved its accuracy and efficiency.

Perturbation by Decomposition:A New Approach to Singular Initial Value Problems with Mamadu-Njoseh Polynomials as Basis Functions

This paper focuses on the application of Mamadu-Njoseh polynomials(MNPs)as basis functions for the solution of singular initial value problems in the second-order ordinary differential equations in a perturbation by decomposition approach.Here,the proposed method is an hybrid of the perturbation theory and decomposition method.In this approach,the approximate solution is slihtly perturbed with the MNPs to ensure absolute convergence.Nonlinear cases are first treated by decomposition.The method is,easy to execute with well-posed mathematical formulae.The existence and convergence of the method is also presented explicitly.Resulting numerical evidences show that the proposed method,in comparison with the Adomian Decomposition Method(ADM),Homotpy Pertubation Method and the exact solution is reliable,efficient and accuarate.

Influence of Initial Value of Transient Time Step on Numerical Simulation of Blow Molding Balloon

The correlation between the initial time value of transient iterative parameters and the blowing pressure in the numerical simulation process of blowing balloon is investigated by POLYFLOW. The results show that: 1) As the blow molding pressure increases, the boundary value of the iterative time step decreases rapidly at first and then slowly. At the end of the first step of iterative calculation for each boundary value, the balloon parison is in the mold core cavity. 2) If the initial time value of transient iterative parameters is smaller than the boundary value of the iterative time step, the balloon parison is still in the mold core cavity at the end of the first iteration. However, if the iterative calculation continues, the calculation process may be interrupted when the time step is smaller than the initial time value of the transient iterative parameters, which makes the blow molding simulation of balloon unable to continue. 3) It is suggested that the initial time value of transient iterative parameters is one order of magnitude smaller than the boundary value of the iterative time step to complete smoothly the simulation of blow molding balloon.

WELL-POSEDNESS OF INITIAL VALUE PROBLEM FOR EULER EQUATIONS OF INVISCID COMPRESSIBLE ADIABATIC FLUID

The well-posedness of the initial value problem of the Euler equations was mainly discussed based on the stratification theory, and the necessary and sufficient conditions of well-posedness are presented for some representative initial or boundary value problem, thus the structure of solution space for local （exact） solution of the Euler equations is determined. Moreover the computation formulas of the analytical solution of the well-posed problem are also given.

LEGENDRE-GAUSS-RADAU SPECTRAL COLLOCATION METHOD FOR NONLINEAR SECOND-ORDER INITIAL VALUE PROBLEMS WITH APPLICATIONS TO WAVE EQUATIONS

We propose and analyze a single-interval Legendre-Gauss-Radau(LGR)spectral collocation method for nonlinear second-order initial value problems of ordinary differential equations.We design an efficient iterative algorithm and prove spectral convergence for the single-interval LGR collocation method.For more effective implementation,we propose a multi-interval LGR spectral collocation scheme,which provides us great flexibility with respect to the local time steps and local approximation degrees.Moreover,we combine the multi-interval LGR collocation method in time with the Legendre-Gauss-Lobatto collocation method in space to obtain a space-time spectral collocation approximation for nonlinear second-order evolution equations.Numerical results show that the proposed methods have high accuracy and excellent long-time stability.Numerical comparison between our methods and several commonly used methods are also provided.

Representation of measures of noncompactness and its applications related to an initial value problem in Banach spaces

This paper is devoted to studying the representation of measures of non-generalized compactness,in particular,measures of noncompactness,of non-weak compactness and of non-super weak compactness,defined on Banach spaces and its applications.With the aid of a three-time order-preserving embedding theorem,we show that for every Banach space X,there exist a Banach function space C(K)for some compact Hausdorff space K and an order-preserving affine mapping T from the super space B of all the nonempty bounded subsets of X endowed with the Hausdorff metric to the positive cone C(K)^(+) of C(K),such that for every convex measure,in particular,the regular measure,the homogeneous measure and the sublinear measure of non-generalized compactnessμon X,there is a convex function F on the cone V=T(B)which is Lipschitzian on each bounded set of V such that F(T(B))=μ(B),■B∈B.As its applications,we show a class of basic integral inequalities related to an initial value problem in Banach spaces,and prove a solvability result of the initial value problem,which is an extension of some classical results due to Bana′s and Goebel(1980),Goebel and Rzymowski(1970)and Rzymowski(1971).

Local and Global Existence of Solutions to Initial Value Problems of Modified Nonlinear Kawahara Equations

This paper is devoted to studying the initial value problem of the modified nonlinear Kawahara equation the first partial dervative of u to t ,the second the third ＋α the second partial dervative of u to x ,the second the third ＋β the third partial dervative of u to x ,the second the thire ＋γ the fifth partial dervative of u to x = 0,（x,t）∈R^2.We first establish several Strichartz type estimates for the fundamental solution of the corresponding linear problem. Then we apply such estimates to prove local and global existence of solutions for the initial value problem of the modified nonlinear Karahara equation. The results show that a local solution exists if the initial function uo（x） ∈ H^s（R） with s ≥ 1/4, and a global solution exists if s ≥ 2.

Local and Global Existence of Solutions to Initial Value Problems of Nonlinear Kaup-Kupershmidt Equations

This paper is devoted to studying the initial value problems of the nonlinear Kaup Kupershmidt equations δu/δt ＋ α1 uδ^2u/δx^2 ＋ βδ^3u/δx^3 ＋ γδ^5u/δx^5 = 0, （x,t）∈ E R^2, and δu/δt ＋ α2 δu/δx δ^2u/δx^2 ＋ βδ^3u/δx^3 ＋ γδ^5u/δx^5 = 0, （x, t） ∈R^2. Several important Strichartz type estimates for the fundamental solution of the corresponding linear problem are established. Then we apply such estimates to prove the local and global existence of solutions for the initial value problems of the nonlinear Kaup- Kupershmidt equations. The results show that a local solution exists if the initial function u0（x） ∈ H^s（R）, and s ≥ 5/4 for the first equation and s≥301/108 for the second equation.

Initial and Boundary Value Problems for Two-Dimensional Non-hydrostatic Boussi nesq Equations

Based on the theory of stratification, the well-posedness of the init ial and boundary value problems for the system of two-dimensional non-hydrosta ti c Boussinesq equations was discussed. The sufficient and necessary conditions of the existence and uniqueness for the solution of the equations were given for s ome representative initial and boundary value problems. Several special cases we re discussed.

INITIAL VALUE PROBLEMS AND FIRST BOUNDARY PROBLEMS FOR A CLASS OF QUASILINEAR WAVE EQUATIONS

The initial value problems and the first boundary problems for the quasilinear wave equation u_(tt)-[a_0+na_1(u_x)^(n-1)]u_(xx)-a_2u_(xxtt)=0 are considered,where a_0,a_2>0 are constants,a_1 is an arbitrary real number,n is a natural number.The existence and uniqueness of the classical solutions for the initial value problems and the first boundary problems of the equation (1) are proved by the Galerkin method.

A Note on Initial Value Problem for the Generalized Tricomi Equation in a Mixed-type Domain

In this paper, we study the well-posedness of initial value problem for n-dimensional gener-alized Tricomi equation in the mixed-type domain {（t,x）：t∈[1,＋∞）,x∈Rn} with the initial data given on the line t=1 in Hadamard＇s sense. By taking partial Fourier transformation, we obtain the explicit expression of the solution in terms of two integral operators and further establish the global estimate of such a solution for a class of initial data and source term. Finally, we establish the global solution in time direction for a semilinear problem used the estimate.

A Collocation Method for Initial Value Problems of Second-Order ODEs by Using Laguerre Functions

We propose a collocation method for solving initial value problems of secondorder ODEs by using modified Laguerre functions.This new process provides global numerical solutions.Numerical results demonstrate the efficiency of the proposed algorithm.

INITIAL VALUE TECHNIQUES FOR THE HELMHOLTZ AND MAXWELL EQUATIONS

We study the initial value problem of the Helmholtz equation with spatially variable wave number. We show that it can be stabilized by suppressing the evanescent waves. The stabilized Helmholtz equation can be solved numerically by a marching scheme combined with FFT. The resulting algorithm has complexity n^2 log n on a n x n grid. We demonstrate the efficacy of the method by numerical examples with caustics. For the Maxwell equation the same treatment is possible after reducing it to a second order system. We show how the method can be used for inverse problems arising in acoustic tomography and microwave imaging.