Original Initial Value Problem Research Articles

Asymptotic stage of modulation instability for the nonlocal nonlinear Schrödinger equation

We study the initial value problem for the integrable nonlocal nonlinear Schrödinger (NNLS) equation iqt(x,t)+qxx(x,t)+2q2(x,t)q̄(−x,t)=0with symmetric boundary conditions: q(x,t)→Ae2iA2t as x→±∞, where A>0 is an arbitrary constant. We describe the asymptotic stage of modulation instability for the NNLS equation by computing the large-time asymptotics of the solution q(x,t) of this initial value problem. We show that it exhibits a non-universal, in a sense, behavior: the asymptotics of |q(x,t)| depends on details of the initial data q(x,0). This is in a sharp contrast with the local classical NLS equation, where the long-time asymptotics of the solution depends on the initial value through the phase parameters only. The main tool used in this work is the inverse scattering transform method applied in the form of the matrix Riemann–Hilbert problem. The Riemann–Hilbert problem associated with the original initial value problem is analyzed asymptotically by the nonlinear steepest decent method.

Time Vector Marching Method for Analyzing Complex Periodic Unsteady Flows Within Turbomachinery

The paper presents an efficient method for analyzing complex periodic unsteady flows within turbomachinery. The method transforms the original initial value problem of unsteady flows to a boundary value problem by taking advantage of temporal periodicity known a priori. This transform allows a pseudotime marching at all physical time instants to be performed simultaneously, for which the method is called the time vector marching (TVM) method. Time direction discretization can be achieved using finite difference schemes. A second-order backward scheme and a ninth-order mixed scheme are investigated. To demonstrate the stability, accuracy, and correctness of the TVM method, the one-dimensional (1-D) advection equation and a quasi-1-D Laval nozzle with time-varying inlet total pressures are used as the test cases. The unsteady flowfield within it was also analyzed using a time-domain dual time-stepping (DTS) method and the harmonic balance (HB) method to obtain the results as references. Further verification was carried out using a transonic compressor rotor with inlet absolute total pressure distortion. The phase-shift boundary condition was applied at geometric periodic boundaries to facilitate the use of a computational domain of one blade passage for a blade row. The time-averaged and instantaneous results of the DTS, HB, and TVM methods and associated time cost were compared and discussed.

Semigroups Generated by Volterra Integro-Differential Equations

We study abstract integro-differential equations that are operator models of problems in viscoelasticity. We present results based on an approach related to the study of one-parameter semigroups for linear evolution equations. The presented approach can also be used to study other integro-differential equations containing integral terms of the Volterra convolution form. A method is given for reducing the original initial value problem for a model integro-differential equation with operator coefficients in a Hilbert space to the Cauchy problem for a first-order differential equation. The existence of a contraction $$C_0 $$ -semigroup is proved under certain assumptions about the kernels of integral operators. Examples of exponential and fractional-exponential (Rabotnov functions) kernels of integral operators satisfying the above assumptions are given.

Numerical solution of coupled nonlinear Schrödinger equations on unbounded domains

In this paper, we investigate the numerical solution of general N-coupled nonlinear Schrödinger equations on unbounded domains, that describe the multiple solitary waves in physics. Due to the unboundedness of the physical domain and strong nonlinearity, how to design appropriate numerical approach to obtain the numerical solution and simulate the real physical phenomena is a major challenge. Based on the ideas of artificial boundary method and operator splitting approach, we develop local artificial boundary conditions for the N-coupled nonlinear Schrödinger equations on the introduced artificial boundaries. Then the original initial value problem was reduced into an initial boundary value problem on the truncated computational domain, which can be solved by the finite difference method efficiently. We rigorously prove the stability of the reduced problem by introducing a series of auxiliary variables, which are vital to overcome the difficulty of the mixed partial derivatives in the local artificial boundary conditions. Numerical simulations are also presented to validate accuracy and the stability of our proposed method.

A new approach to constructing of explicit one-step methods of high order for singular initial value problems for nonlinear ordinary differential equations

A new approach to construction of one-step numerical methods of high order for the initial value problems on the interval [0,a] with a singularity of the first kind in the point x=0 is proposed. Using the substitution of the independent variable x=et, we reduce the original initial value problem to the one on the interval (−∞,ln⁡a]. On some finite irregular grid {tn∈(−∞,ln⁡a], n=0,1,...,N,tN=ln⁡a} Taylor series and Runge-Kutta methods for this problem have been developed. For finding of an approximate solution at the grid node t0, new one-step methods have been constructed. For finding of the solution at other grid nodes, the standard one-step methods have been used. An algorithm for the automatic generation of a grid which guarantees the prescribed accuracy is presented. The effectiveness of presented approach is illustrated by a set of numerical examples. The applicability of the constructed method to systems of singular differential equations is shown.

A fast-converging recursive approach for Lane–Emden type initial value problems arising in astrophysics

In this paper, we describe a new technique, called the least square recursive approach (LSRA), to approximate the solution of Lane–Emden type initial value problems, which describe many phenomena in mathematical physics and astrophysics. The proposed method reduces the solution of the original initial value problem to the solution of an integral equation and provides us with a simple way to improve the convergence of the solution. Convergence analysis of the method is carried out. The obtained results are compared with exiting methods (Chowdhary and Hashim, 2007; Wazwaz, 2001; Singh et al., 2009) to show the advantage of the proposed method. The method is shown to be very reliable, accurate and computationally efficient.

Approximations of stable manifolds in the vicinity of hyperbolic equilibrium points for fractional differential equations

This paper is devoted to the numerical analysis of the abstract semilinear fractional problem $$D^\alpha u(t) = Au(t) + f(u(t)), u(0)=u^0,$$ in a Banach space E. We are developing a general approach to establish a semidiscrete approximation of stable manifolds. The phase space in the neighborhood of the hyperbolic equilibrium can be split in such a way that the original initial value problem is reduced to systems of initial value problems in the invariant subspaces corresponding to positive and negative real parts of the spectrum. We show that such a decomposition of the equation keeps the same structure under general approximation schemes. The main assumption of our results are naturally satisfied, in particular, for operators with compact resolvents and can be verified for finite element as well as finite difference methods.

On uniqueness of weak solutions for the thin-film equation

In any number of space variables, we study the Cauchy problem related to the fourth-order degenerate diffusion equation ∂sh+∇⋅(h∇Δh)=0. This equation, derived from a lubrication approximation, also models the surface tension dominated flow of a thin viscous film in the Hele-Shaw cell. Our focus is on uniqueness of weak solutions in the “complete wetting” regime, when a zero contact angle between liquid and solid is imposed. In this case, we transform the problem by zooming into the free boundary and look at small Lipschitz perturbations of a quadratic stationary solution. In the perturbational setting, the main difficulty is to construct scale invariant function spaces in such a way that they are compatible with the structure of the nonlinear equation. Here we rely on the theory of singular integrals in spaces of homogeneous type to obtain linear estimates in these functions spaces which provide “optimal” conditions on the initial data under which a unique solution exists. In fact, this solution can be used to define a class of functions in which the original initial value problem has a unique (weak) solution. Moreover, we show that the interface between empty and wetted regions is an analytic hypersurface in time and space.

Approximations of Parabolic Equations at the Vicinity of Hyperbolic Equilibrium Point

This article is devoted to the numerical analysis of the abstract semilinear parabolic problem u′(t) = Au(t) + f(u(t)), u(0) = u 0, in a Banach space E. We are developing a general approach to establish a discrete dichotomy in a very general setting and prove shadowing theorems that compare solutions of the continuous problem with those of discrete approximations in space and time. In [3] the discretization in space was constructed under the assumption of compactness of the resolvent. It is a well-known fact (see [10, 11]) that the phase space in the neighborhood of the hyperbolic equilibrium can be split in a such way that the original initial value problem is reduced to initial value problems with exponential bounded solutions on the corresponding subspaces. We show that such a decomposition of the flow persists under rather general approximation schemes, utilizing a uniform condensing property. The main assumption of our results are naturally satisfied, in particular, for operators with compact resolvents and condensing semigroups and can be verified for finite elements as well as finite differences methods.

Application of radial basis functions to evolution equations arising in image segmentation

In this paper, radial basis functions are used to obtain the solution of evolution equations which appear in variational level set method based image segmentation. In this method, radial basis functions are used to interpolate the implicit level set function of the evolution equation with a high level of accuracy and smoothness. Then, the original initial value problem is discretized into an interpolation problem. Accordingly, the evolution equation is converted into a set of coupled ordinary differential equations, and a smooth evolution can be retained. Compared with finite difference scheme based level set approaches, the complex and costly re-initialization procedure is unnecessary. Numerical examples are also given to show the efficiency of the method.

The exponential dichotomy on general approximation scheme

This paper is devoted to the numerical analysis of abstract parabolic problems u′(t) = Au(t), u(0) = u 0 with hyperbolic generator A. We develop a general approach to establish a discrete dichotomy in a very general setting in the case of discrete approximation in space and time. It is a well-known fact that the phase space in the neighborhood of the hyperbolic equilibrium can be split in such a way that the original initial value problem is reduced to initial value problems with exponentially decaying solutions in opposite time directions. We use the theory of compact approximation principle and collectively condensing approximation to show that such a decomposition of the flow persists under rather general approximation schemes. The main assumption of our results are naturally satisfied, in particular, for operators with compact resolvents and condensing semigroups and can be verified for the finite element method as well as finite difference methods.

An analytic method for the initial value problem of the electric field system in vertical inhomogeneous anisotropic media

The time-dependent system of partial differential equations of the second order describing the electric wave propagation in vertically inhomogeneous electrically and magnetically biaxial anisotropic media is considered. A new analytical method for solving an initial value problem for this system is the main object of the paper. This method consists in the following: the initial value problem is written in terms of Fourier images with respect to lateral space variables, then the resulting problem is reduced to an operator integral equation. After that the operator integral equation is solved by the method of successive approximations. Finally, a solution of the original initial value problem is found by the inverse Fourier transform.

The Exponential Dichotomy under Discretization on General Approximation Scheme

This paper is devoted to the numerical analysis of abstract parabolic problem ; , with hyperbolic generator . We are developing a general approach to establish a discrete dichotomy in a very general setting in case of discrete approximation in space and time. It is a well-known fact that the phase space in the neighborhood of the hyperbolic equilibrium can be split in a such way that the original initial value problem is reduced to initial value problems with exponential decaying solutions in opposite time direction. We use the theory of compact approximation principle and collectively condensing approximation to show that such a decomposition of the flow persists under rather general approximation schemes. The main assumption of our results is naturally satisfied, in particular, for operators with compact resolvents and condensing semigroups and can be verified for finite element as well as finite difference methods.

Solving an initial value problem in inhomogeneous electrically and magnetically anisotropic uniaxial media

An initial value problem for a system of the second order partial differential equations, describing the electric wave propagation in vertically inhomogeneous electrically and magnetically anisotropic uniaxial media, is the main object of the study. The present paper suggests and justifies a new algorithm for solving this problem. This algorithm has several steps. On the first step the original initial value problem is written in terms of the Fourier images with respect to lateral variables. After that the obtained problem is transformed into an equivalent second kind vector integral equation of the Volterra type. A solution of this integral equation is constructed by successive approximations. At last, using the real Paley–Wiener theorem, a solution of the original initial value problem is found.

HOPF SOLUTIONS TO A FLUID-ELASTIC INTERACTION MODEL

In this paper, we give a global existence theorem of weak solutions to model equations governing interaction fluid structure in a two-dimensional layer, cf. Refs. 8 and 14. To our knowledge this is the first existence theorem of global in time solutions for such model. The interest of our result is double because, first, we change the original initial value problem by deleting one initial condition, second, we construct a solution through the classical Galerkin method for which several computing codes have been constructed.

A level set‐based parameterization method for structural shape and topology optimization

AbstractThis paper presents an effective parametric approach by extending the conventional level set method to structural shape and topology optimization using the compactly supported radial basis functions (RBFs) and the optimality criteria (OC) method. The structural design boundary is first represented implicitly by embedding into a higher‐dimensional level set function as its zero level set, and the RBFs of a favorable smoothness are then applied to interpolate the level set function. The original initial value problem is thus converted to a parametric optimization, with the expansion coefficients of the interplant posed as the design variables.The OC method is then applied to advance the structure boundary in terms of the velocity field derived from the parametric optimization. Hence, the structural shape and topology optimization is now transformed into a process of iteratively finding coefficients to update the level set function to achieve an optimal configuration. The numerical considerations of the conventional level set method, including upwind schemes, velocity extension, and reinitialization, are eliminated. The proposed scheme is capable of addressing structural shape fidelity and topology change simultaneously and of keeping the boundary smooth during the optimization process. Furthermore, numerical convergence is expected to be improved. A widely investigated example, in the framework of structural stiffness designs, is applied to demonstrate the efficiency and accuracy of the proposed approach. Copyright © 2007 John Wiley & Sons, Ltd.

An extended level set method for shape and topology optimization

In this paper, the conventional level set methods are extended as an effective approach for shape and topology optimization by the introduction of the radial basis functions (RBFs). The RBF multiquadric splines are used to construct the implicit level set function with a high level of accuracy and smoothness and to discretize the original initial value problem into an interpolation problem. The motion of the dynamic interfaces is thus governed by a system of coupled ordinary differential equations (ODEs) and a relatively smooth evolution can be maintained without reinitialization. A practical implementation of this method is further developed for solving a class of energy-based optimization problems, in which approximate solution to the original Hamilton–Jacobi equation may be justified and nucleation of new holes inside the material domain is allowed for. Furthermore, the severe constraints on the temporal and spatial discretizations can be relaxed, leading to a rapid convergence to the final design insensitive to initial guesses. The normal velocities are chosen to perform steepest gradient-based optimization by using shape sensitivity analysis and a bi-sectioning algorithm. A physically meaningful and efficient extension velocity method is also presented. The proposed method is implemented in the framework of minimum compliance design and its efficiency over the existing methods is highlighted. Numerical examples show its accuracy, convergence speed and insensitivity to initial designs in shape and topology optimization of two-dimensional (2D) problems that have been extensively investigated in the literature.

Radial basis functions and level set method for structural topology optimization

AbstractLevel set methods have become an attractive design tool in shape and topology optimization for obtaining lighter and more efficient structures. In this paper, the popular radial basis functions (RBFs) in scattered data fitting and function approximation are incorporated into the conventional level set methods to construct a more efficient approach for structural topology optimization. RBF implicit modelling with multiquadric (MQ) splines is developed to define the implicit level set function with a high level of accuracy and smoothness. A RBF–level set optimization method is proposed to transform the Hamilton–Jacobi partial differential equation (PDE) into a system of ordinary differential equations (ODEs) over the entire design domain by using a collocation formulation of the method of lines. With the mathematical convenience, the original time dependent initial value problem is changed to an interpolation problem for the initial values of the generalized expansion coefficients. A physically meaningful and efficient extension velocity method is presented to avoid possible problems without reinitialization in the level set methods. The proposed method is implemented in the framework of minimum compliance design that has been extensively studied in topology optimization and its efficiency and accuracy over the conventional level set methods are highlighted. Numerical examples show the success of the present RBF–level set method in the accuracy, convergence speed and insensitivity to initial designs in topology optimization of two‐dimensional (2D) structures. It is suggested that the introduction of the radial basis functions to the level set methods can be promising in structural topology optimization. Copyright © 2005 John Wiley & Sons, Ltd.

Existence results for non-autonomous multiple-fragmentation models

We investigate an initial-value problem modelling fragmentation processes where particles split into two or more pieces at a rate, γ, that not only depends on the sizes of the particles involved but also on time. The existence of non-negative, mass-conserving solutions is established by considering a truncated version of an associated non-autonomous abstract Cauchy problem. The latter has solutions of the form u(t)=Un(t,t0)f, t⩾t0, where f is the known data at some fixed time t0⩾0 and {Un(t,s)} is a uniformly continuous evolution system. A limit evolution system {U(t,s)} is shown to exist. Depending on the form of the known data f at time t0, the scalar-valued function u, obtained from the limit evolution system via u(x, t)=[U(t, t0)f](x) for a.e. x>0, t⩾t0, is a solution of either the original initial-value problem or an integral version of this problem. © 1997 B. G. Teubner Stuttgart–John Wiley & Sons Ltd.

The Drifting Confluence Zone

Alongshore migration of a western boundary current separation is investigated with a nonlinear inviscid reduced gravity model on an f plane. Separation due to a collision with an opposing current is considered. It is known that for such a case stationary collision and separation is possible only for boundary currents with “balanced” transports, that is, equal near-wall depths. The authors perturb this stationary solution with a small steplike variation of the opposing current transport and focus on the resulting time-dependent flow. Two different analytical methods to compute the migration rate are used. The first method involves integrated balances, and the second involves the path equation for the separated flow. Using the fist approach, it is found analytically that the flow consists of one current intruding into the area occupied by the other. After an initial adjustment period the intrusion becomes steadily propagating. The width of intrusion is much greater than the width of boundary currents, whereas the migration speed is much smaller than the speed of the currents. The speed of the opposing current intrusion into the main western boundary current is given approximately by the formula C ≈ (g′H)1/2(D21 − D22)2/4D3/22, where H is the undisturbed depth of the main poleward flowing current, D1 is its depth near the wall (nondimensionalized by H), and D2 is the opposing current near-wall (nondimensional) depth. Due to the nonlinearity of the process, the expression for the opposite speed (i.e., the main western boundary current intrusion into the opposing current) has a similar looking but different form. Using the second approach, the original initial value problem is reduced to a time-dependent path equation for the separated current. It is shown analytically that, as should be the case, in the limit t → ∞ the path equation solution is identical to the earlier solution for the steadily propagating intrusion. The migration process exhibits an hysteresis; that is, the progression of the separated currents differs from its corresponding regression. Application of the theory to the collision and separation of the Brazil and the Malvinas Currents is discussed. It is suggested that the observed migrations of the separation latitude may be caused by seasonal changes of the Malvinas Current transport.