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Initial Boundary Value Problem (initial + boundary_value_problem)
Selected AbstractsInitial boundary value problem for a class of non-linear strongly damped wave equationsMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 12 2003Yang Zhijian The paper studies the existence, asymptotic behaviour and stability of global solutions to the initial boundary value problem for a class of strongly damped non-linear wave equations. By a H00.5ptk-Galerkin approximation scheme, it proves that the above-mentioned problem admits a unique classical solution depending continuously on initial data and decaying to zero as t,+,as long as the non-linear terms are sufficiently smooth; they, as well as their derivatives or partial derivatives, are of polynomial growth order and the initial energy is properly small. Copyright © 2003 John Wiley & Sons, Ltd. [source] Initial boundary value problem for the evolution system of MHD type describing geophysical flow in three-dimensional domainsMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 9 2003Chunshan Zhao Abstract The initial boundary value problem for the evolution system describing geophysical flow in three-dimensional domains was considered. The existence and uniqueness of global strong solution to the evolution system were proved under assumption on smallness of data. Moreover, solvable compatibility conditions of initial data and boundary values which guarantee the existence and uniqueness of global strong solution were discussed. Copyright © 2003 John Wiley & Sons, Ltd. [source] On the stability and convergence of a Galerkin reduced order model (ROM) of compressible flow with solid wall and far-field boundary treatment,INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 10 2010I. Kalashnikova Abstract A 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. [source] Fatigue life prediction using 2-scale temporal asymptotic homogenizationINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 3 2004Caglar Oskay Abstract In this manuscript, fatigue of structures is modelled as a multiscale phenomenon in time domain. Multiple temporal scales are introduced due to the fact that the load period is orders of magnitude smaller than the useful life span of a structural component. The problem of fatigue life prediction is studied within the framework of mathematical homogenization with two temporal co-ordinates. By this approach the original initial boundary value problem is decomposed into coupled micro-chronological (fast time-scale) and macro-chronological (slow time-scale) problems. The life prediction methodology has been implemented in ABAQUS and validated against direct cycle-by-cycle simulations. Copyright © 2004 John Wiley & Sons, Ltd. [source] Shape reconstruction of an inverse boundary value problem of two-dimensional Navier,Stokes equationsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 6 2010Wenjing Yan Abstract This paper is concerned with the problem of the shape reconstruction of two-dimensional flows governed by the Navier,Stokes equations. Our objective is to derive a regularized Gauss,Newton method using the corresponding operator equation in which the unknown is the geometric domain. The theoretical foundation for the Gauss,Newton method is given by establishing the differentiability of the initial boundary value problem with respect to the boundary curve in the sense of a domain derivative. The numerical examples show that our theory is useful for practical purpose and the proposed algorithm is feasible. Copyright © 2009 John Wiley & Sons, Ltd. [source] Longtime behavior for a nonlinear wave equation arising in elasto-plastic flowMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 9 2009Yang Zhijian Abstract The paper studies the longtime behavior of solutions to the initial boundary value problem (IBVP) for a nonlinear wave equation arising in elasto-plastic flow utt,div{|,u|m,1,u},,,ut+,2u+g(u)=f(x). It proves that under rather mild conditions, the dynamical system associated with above-mentioned IBVP possesses a global attractor, which is connected and has finite Hausdorff and fractal dimension in the phase spaces X1=H(,) × L2(,) and X=(H3(,),H(,)) × H(,), respectively. Copyright © 2008 John Wiley & Sons, Ltd. [source] Resonance phenomena in compound cylindrical waveguidesMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 8 2006Günter Heinzelmann Abstract We study the large time asymptotics of the solutions u(x,t) of the Dirichlet and the Neumann initial boundary value problem for the wave equation with time-harmonic right-hand side in domains , which are composed of a finite number of disjoint half-cylinders ,1,,,,r with cross-sections ,,1,,,,,r and a bounded part (,compound cylindrical waveguides'). We show that resonances of orders t and t1/2 may occur at a finite or countable discrete set of frequencies ,, while u(x,t) is bounded as t,, for the remaining frequencies. A resonance of order t occurs at , if and only if ,2 is an eigenvalue of the Laplacian ,, in , with regard to the given boundary condition u=0 or ,u/,n=0, respectively. A resonance of order t1/2 occurs at , if and only if (i) ,2 is an eigenvalue of at least one of the Laplacians for the cross-sections ,,1,,,,r, with regard to the respective boundary condition and (ii) the respective homogeneous boundary value problem for the reduced wave equation ,U+,2U=0 in , has non-trivial solutions with suitable asymptotic properties as | x | ,, (,standing waves'). Copyright © 2006 John Wiley & Sons, Ltd. [source] Initial boundary value problem for a class of non-linear strongly damped wave equationsMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 12 2003Yang Zhijian The paper studies the existence, asymptotic behaviour and stability of global solutions to the initial boundary value problem for a class of strongly damped non-linear wave equations. By a H00.5ptk-Galerkin approximation scheme, it proves that the above-mentioned problem admits a unique classical solution depending continuously on initial data and decaying to zero as t,+,as long as the non-linear terms are sufficiently smooth; they, as well as their derivatives or partial derivatives, are of polynomial growth order and the initial energy is properly small. Copyright © 2003 John Wiley & Sons, Ltd. [source] Blowup of solutions for a class of non-linear evolution equations with non-linear damping and source termsMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 10 2002Yang Zhijian We consider the blowup of solutions of the initial boundary value problem for a class of non-linear evolution equations with non-linear damping and source terms. By using the energy compensation method, we prove that when p>max{m, ,}, where m, , and p are non-negative real numbers and m+1, ,+1, p+1 are, respectively, the growth orders of the non-linear strain terms, damping term and source term, under the appropriate conditions, any weak solution of the above-mentioned problem blows up in finite time. Comparison of the results with the previous ones shows that there exist some clear condition boundaries similar to thresholds among the growth orders of the non-linear terms, the states of the initial energy and the existence and non-existence of global weak solutions. Copyright © 2002 John Wiley & Sons, Ltd. [source] Non-linear initial boundary value problems of hyperbolic,parabolic type.MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 3 2002A general investigation of admissible couplings between systems of higher order. This is the second part of an article that is devoted to the theory of non-linear initial boundary value problems. We consider coupled systems where each system is of higher order and of hyperbolic or parabolic type. Our goal is to characterize systematically all admissible couplings between systems of higher order and different type. By an admissible coupling we mean a condition that guarantees the existence, uniqueness and regularity of solutions to the respective initial boundary value problem. In part 1, we develop the underlying theory of linear hyperbolic and parabolic initial boundary value problems. Testing the PDEs with suitable functions we obtain a priori estimates for the respective solutions. In particular, we make use of the regularity theory for linear elliptic boundary value problems that was previously developed by the author. In part 2 at hand, we prove the local in time existence, uniqueness and regularity of solutions to the quasilinear initial boundary value problem (3.4) using the so-called energy method. In the above sense the regularity assumptions (A6) and (A7) about the coefficients and right-hand sides define the admissible couplings. In part 3, we extend the results of part 2 to non-linear initial boundary value problems. In particular, the assumptions about the respective parameters correspond to the previous regularity assumptions and hence define the admissible couplings now. Moreover, we exploit the assumptions about the respective parameters for the case of two coupled systems. Copyright © 2002 John Wiley & Sons, Ltd. [source] Non-linear initial boundary value problems of hyperbolic,parabolic type.MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 3 2002A general investigation of admissible couplings between systems of higher order. This is the third part of an article that is devoted to the theory of non-linear initial boundary value problems. We consider coupled systems where each system is of higher order and of hyperbolic or parabolic type. Our goal is to characterize systematically all admissible couplings between systems of higher order and different type. By an admissible coupling we mean a condition that guarantees the existence, uniqueness and regularity of solutions to the respective initial boundary value problem. In part 1, we develop the underlying theory of linear hyperbolic and parabolic initial boundary value problems. Testing the PDEs with suitable functions we obtain a priori estimates for the respective solutions. In particular, we make use of the regularity theory for linear elliptic boundary value problems that was previously developed by the author. In part 2, we prove the local in time existence, uniqueness and regularity of solutions to quasilinear initial boundary value problems using the so-called energy method. In the above sense the regularity assumptions about the coefficients and right-hand sides define the admissible couplings. In part 3 at hand, we extend the results of part 2 to the nonlinear initial boundary value problem (4.2). In particular, assumptions (B8) and (B9) about the respective parameters correspond to the previous regularity assumptions and hence define the admissible couplings now. Moreover, we exploit assumptions (B8) and (B9) for the case of two coupled systems. Copyright © 2002 John Wiley & Sons, Ltd. [source] Determining the temperature from incomplete boundary dataMATHEMATISCHE NACHRICHTEN, Issue 16 2007B. Tomas Johansson Abstract An iterative procedure for determining temperature fields from Cauchy data given on a part of the boundary is presented. At each iteration step, a series of mixed well-posed boundary value problems are solved for the heat operator and its adjoint. A convergence proof of this method in a weighted L2 -space is included, as well as a stopping criteria for the case of noisy data. Moreover, a solvability result in a weighted Sobolev space for a parabolic initial boundary value problem of second order with mixed boundary conditions is presented. Regularity of the solution is proved. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] Modification of upwind finite difference fractional step methods by the transient state of the semiconductor deviceNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 2 2008Yirang Yuan Abstract The mathematical model of the three-dimensional semiconductor devices of heat conduction is described by a system of four quasi-linear partial differential equations for initial boundary value problem. One equation of elliptic form is for the electric potential; two equations of convection-dominated diffusion type are for the electron and hole concentration; and one heat conduction equation is for temperature. Upwind finite difference fractional step methods are put forward. Some techniques, such as calculus of variations, energy method multiplicative commutation rule of difference operators, decomposition of high order difference operators, and the theory of prior estimates and techniques are adopted. Optimal order estimates in L2 norm are derived to determine the error in the approximate solution.© 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008 [source] Homogenization in the Theory of ViscoplasticityPROCEEDINGS IN APPLIED MATHEMATICS & MECHANICS, Issue 1 2005Sergiy Nesenenko We study the homogenization of the quasistatic initial boundary value problem with internal variables which models the deformation behavior of viscoplastic bodies with a periodic microstructure. This problem is represented through a system of linear partial differential equations coupled with a nonlinear system of differential equations or inclusions. Recently it was shown by Alber [2] that the formally derived homogenized initial boundary value problem has a solution. From this solution we construct an asymptotic solution for the original problem and prove that the difference of the exact solution and the asymptotic solution tends to zero if the lengthscale of the microstructure goes to zero. The work is based on monotonicity properties of the differential equations or inclusions. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] The generalized Dirichlet-to-Neumann map for certain nonlinear evolution PDEsCOMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 5 2005A. S. Fokas Let q(x,t) satisfy a nonlinear integrable evolution PDE whose highest spatial derivative is of order n. An initial boundary value problem on the half-line for such a PDE is at least linearly well-posed if one prescribes initial conditions, as well as N boundary conditions at x = 0, where for n even N equals n/2 and for n odd, depending on the sign of the highest derivative, N equals either n,1/2 or n+1/2. For example, for the nonlinear Schrödinger (NLS) and the sine-Gordon (sG), N = 1, while for the modified Korteweg-deVries (mKdV) N = 1 or N = 2 depending on the sign of the third derivative. Constructing the generalized Dirichlet-to-Neumann map means determining those boundary values at x = 0 that are not prescribed as boundary conditions in terms of the given initial and boundary conditions. A general methodology is presented that constructs this map in terms of the solution of a system of two nonlinear ODEs. This formulation implies that for the focusing NLS, for the sG, and for the two focusing versions of the mKdV, this map is global in time. It appears that this is the first time in the literature that such a characterization for nonlinear PDEs is explicitly described. It is also shown here that for particular choices of the boundary conditions the above map can be linearized. © 2005 Wiley Periodicals, Inc. [source] Non-linear initial boundary value problems of hyperbolic,parabolic type.MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 3 2002A general investigation of admissible couplings between systems of higher order. This is the second part of an article that is devoted to the theory of non-linear initial boundary value problems. We consider coupled systems where each system is of higher order and of hyperbolic or parabolic type. Our goal is to characterize systematically all admissible couplings between systems of higher order and different type. By an admissible coupling we mean a condition that guarantees the existence, uniqueness and regularity of solutions to the respective initial boundary value problem. In part 1, we develop the underlying theory of linear hyperbolic and parabolic initial boundary value problems. Testing the PDEs with suitable functions we obtain a priori estimates for the respective solutions. In particular, we make use of the regularity theory for linear elliptic boundary value problems that was previously developed by the author. In part 2 at hand, we prove the local in time existence, uniqueness and regularity of solutions to the quasilinear initial boundary value problem (3.4) using the so-called energy method. In the above sense the regularity assumptions (A6) and (A7) about the coefficients and right-hand sides define the admissible couplings. In part 3, we extend the results of part 2 to non-linear initial boundary value problems. In particular, the assumptions about the respective parameters correspond to the previous regularity assumptions and hence define the admissible couplings now. Moreover, we exploit the assumptions about the respective parameters for the case of two coupled systems. Copyright © 2002 John Wiley & Sons, Ltd. [source] Non-linear initial boundary value problems of hyperbolic,parabolic type.MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 3 2002A general investigation of admissible couplings between systems of higher order. This is the third part of an article that is devoted to the theory of non-linear initial boundary value problems. We consider coupled systems where each system is of higher order and of hyperbolic or parabolic type. Our goal is to characterize systematically all admissible couplings between systems of higher order and different type. By an admissible coupling we mean a condition that guarantees the existence, uniqueness and regularity of solutions to the respective initial boundary value problem. In part 1, we develop the underlying theory of linear hyperbolic and parabolic initial boundary value problems. Testing the PDEs with suitable functions we obtain a priori estimates for the respective solutions. In particular, we make use of the regularity theory for linear elliptic boundary value problems that was previously developed by the author. In part 2, we prove the local in time existence, uniqueness and regularity of solutions to quasilinear initial boundary value problems using the so-called energy method. In the above sense the regularity assumptions about the coefficients and right-hand sides define the admissible couplings. In part 3 at hand, we extend the results of part 2 to the nonlinear initial boundary value problem (4.2). In particular, assumptions (B8) and (B9) about the respective parameters correspond to the previous regularity assumptions and hence define the admissible couplings now. Moreover, we exploit assumptions (B8) and (B9) for the case of two coupled systems. Copyright © 2002 John Wiley & Sons, Ltd. [source] The homotopy analysis method for solving higher dimensional initial boundary value problems of variable coefficientsNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 5 2010H. Jafari Abstract In this article, higher dimensional initial boundary value problems of variable coefficients are solved by means of an analytic technique, namely the Homotopy analysis method (HAM). Comparisons are made between the Adomian decomposition method (ADM), the exact solution and the homotopy analysis method. The results reveal that the proposed method is very effective and simple. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010 [source] A numerical study of the accuracy and stability of symmetric and asymmetric RBF collocation methods for hyperbolic PDEsNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 2 2008Scott A. Sarra Abstract Differentiation matrices associated with radial basis function (RBF) collocation methods often have eigenvalues with positive real parts of significant magnitude. This prevents the use of the methods for time-dependent problems, particulary if explicit time integration schemes are employed. In this work, accuracy and eigenvalue stability of symmetric and asymmetric RBF collocation methods are numerically explored for some model hyperbolic initial boundary value problems in one and two dimensions. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008 [source] A Crank-Nicolson and ADI Galerkin method with quadrature for hyperbolic problemsNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 1 2005M. Ganesh Abstract We propose, analyze, and implement fully discrete two-time level Crank-Nicolson methods with quadrature for solving second-order hyperbolic initial boundary value problems. Our algorithms include a practical version of the ADI scheme of Fernandes and Fairweather [SIAM J Numer Anal 28 (1991), 1265,1281] and also generalize the methods and analyzes of Baker [SIAM J Numer Anal 13 (1976), 564,576] and Baker and Dougalis [SIAM J Numer Anal 13 (1976), 577,598]. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005 [source] Analysis of lattice Boltzmann boundary conditionsPROCEEDINGS IN APPLIED MATHEMATICS & MECHANICS, Issue 1 2003M. Junk Dr. The correct implementation of Navier-Stokes boundary conditions in the framework of lattice Boltzmann schemes is complicated by the non-availability of analytical methods to assess the consistency of such discretizations. To close this gap, we propose a simple direct asymptotic analysis which is readily applicable to finite difference discretizations of initial boundary value problems in general and to lattice Boltzmann methods in particular. Results of the analysis applied to the classical lattice Boltzmann scheme with bounce back boundary condition are reported. [source] | |||||||||||||