共查询到20条相似文献,搜索用时 62 毫秒
1.
We consider an inverse problem for the determination of a purely time-dependent source in a semilinear parabolic equation with a nonlocal boundary condition. An approximation scheme for the solution together with the well-posedness of the problem with the initial value u0 ∈ H1(Ω) is presented by means of the Rothe time-discretization method. Further approximation scheme via Rothe’s method is constructed for the problem when u0 ∈ L2(Ω) and the integral kernel in the nonlocal boundary condition is symmetric.
相似文献2.
Klaus Pflüger 《Mathematical Methods in the Applied Sciences》1996,19(5):363-374
Periodic solutions of arbitrary period to semilinear partial differential equations of Zabusky or Boussinesq type are obtained. More generally, for a linear differential operator A(y,∂), the equation A(y,∂)u = (−1)∣γ∣∂γf(y,∂γu), y = (t,x)∈ℝk×G is studied, where homogeneous boundary conditions on ∂G and periodicity conditions on t are imposed. The solutions are obtained by variational methods in anisotropic Sobolev spaces. 相似文献
3.
Guang‐Hua Gao Zhi‐Zhong Sun 《Numerical Methods for Partial Differential Equations》2013,29(5):1459-1486
This is the further work on compact finite difference schemes for heat equation with Neumann boundary conditions subsequent to the paper, [Sun, Numer Methods Partial Differential Equations (NMPDE) 25 (2009), 1320–1341]. A different compact difference scheme for the one‐dimensional linear heat equation is developed. Truncation errors of the proposed scheme are O(τ2 + h4) for interior mesh point approximation and O(τ2 + h3) for the boundary condition approximation with the uniform partition. The new obtained scheme is similar to the one given by Liao et al. (NMPDE 22 (2006), 600–616), while the major difference lies in no extension of source terms to outside the computational domain any longer. Compared with ones obtained by Zhao et al. (NMPDE 23 (2007), 949–959) and Dai (NMPDE 27 (2011), 436–446), numerical solutions at all mesh points including two boundary points are computed in our new scheme. The significant advantage of this work is to provide a rigorous analysis of convergence order for the obtained compact difference scheme using discrete energy method. The global accuracy is O(τ2 + h4) in discrete maximum norm, although the spatial approximation order at the Neumann boundary is one lower than that for interior mesh points. The analytical techniques are important and can be successfully used to solve the open problem presented by Sun (NMPDE 25 (2009), 1320–1341), where analyzed theoretical convergence order of the scheme by Liao et al. (NMPDE 22 (2006), 600–616) is only O(τ2 + h3.5) while the numerical accuracy is O(τ2 + h4), and convergence order of theoretical analysis for the scheme by Zhao et al. (NMPDE 23 (2007), 949–959) is O(τ2 + h2.5), while the actual numerical accuracy is O(τ2 + h3). Following the procedure used for the new obtained difference scheme in this work, convergence orders of these two schemes can be proved rigorously to be O(τ2 + h4) and O(τ2 + h3), respectively. Meanwhile, extension to the case involving the nonlinear reaction term is also discussed, and the global convergence order O(τ2 + h4) is proved. A compact ADI difference scheme for solving two‐dimensional case is derived. Finally, several examples are given to demonstrate the numerical accuracy of new obtained compact difference schemes. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013 相似文献
4.
Ujjwal Koley 《Central European Journal of Mathematics》2012,10(1):173-187
We are concerned with convergence of spectral method for the numerical solution of the initial-boundary value problem associated
to the Korteweg-de Vries-Kawahara equation (Kawahara equation, in short), which is a transport equation perturbed by dispersive
terms of the 3rd and 5th order. This equation appears in several fluid dynamics problems. It describes the evolution of small
but finite amplitude long waves in various problems in fluid dynamics. These equations are discretized in space by the standard
Fourier-Galerkin spectral method and in time by the explicit leap-frog scheme. For the resulting fully discrete, conditionally
stable scheme we prove an L
2-error bound of spectral accuracy in space and of second-order accuracy in time. 相似文献
5.
Theodore S. Papatheodorou Anastasia N. Kandili 《Journal of Computational and Applied Mathematics》2009
The unified transform method of A. S. Fokas has led to important new developments, regarding the analysis and solution of various types of linear and nonlinear PDE problems. In this work we use these developments and obtain the solution of time-dependent problems in a straightforward manner and with such high accuracy that cannot be reached within reasonable time by use of the existing numerical methods. More specifically, an integral representation of the solution is obtained by use of the A. S. Fokas approach, which provides the value of the solution at any point, without requiring the solution of linear systems or any other calculation at intermediate time levels and without raising any stability problems. For instance, the solution of the initial boundary value problem with the non-homogeneous heat equation is obtained with accuracy 10−15, while the well-established Crank–Nicholson scheme requires 2048 time steps in order to reach a 10−8 accuracy. 相似文献
6.
We consider the scalar linear second-order differential-difference equation with delay {fx159-01}. This equation is investigated
by the method of polynomial quasisolutions based on the representation of an unknown function in the form of a polynomial
{ie159-01}. Upon the substitution of this polynomial in the original equation, the residual Δ(t) = O(t
N−1) appears. An exact analytic representation of this residual is obtained. We show the close connection between a linear differential-difference
equation with variable coefficients and a model equation with constant coefficients, the structure of whose solution is determined
by the roots of the characteristic quasipolynomial.
__________
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 1, pp. 140–152, January, 2008. 相似文献
7.
A. R. Bechelova 《Ukrainian Mathematical Journal》1998,50(7):1131-1134
For the diffusion equation of fractional order, we construct an approximation difference scheme of order 0(h
2 + τ). We present an algorithm for the solution of boundary-value problems for a generalized transfer equation of fractional
order.
Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 7, pp. 994–996, July, 1998. 相似文献
8.
G. I. Shishkin L. P. Shishkina 《Computational Mathematics and Mathematical Physics》2010,50(3):437-456
The Dirichlet problem on a vertical strip is examined for a singularly perturbed semilinear elliptic convection-diffusion
equation. For this problem, the basic nonlinear difference scheme based on the classical approximations on piecewise uniform
grids condensing in the vicinity of boundary layers converges ɛ-uniformly with an order at most almost one. The Richardson
technique is used to construct a nonlinear scheme that converges ɛ-uniformly with an improved order, namely, at the rate O(N
1−2ln2
N
1 + N
2−2), where N
1 + 1 and N
2 + 1 are the number of grid nodes along the x
1-axis and per unit interval of the x
2-axis, respectively. This nonlinear basic scheme underlies the linearized iterative scheme, in which the nonlinear term is
calculated using the values of the sought function found at the preceding iteration step. The latter scheme is used to construct
a linearized iterative Richardson scheme converging ɛ-uniformly with an improved order. Both the basic and improved iterative
schemes converge ɛ-uniformly at the rate of a geometric progression as the number of iteration steps grows. The upper and
lower solutions to the iterative Richardson schemes are used as indicators, which makes it possible to determine the iteration
step at which the same ɛ-uniform accuracy is attained as that of the non-iterative nonlinear Richardson scheme. It is shown
that no Richardson schemes exist for the convection-diffusion boundary value problem converging ɛ-uniformly with an order
greater than two. Principles are discussed on which the construction of schemes of order greater than two can be based. 相似文献
9.
《偏微分方程通讯》2013,38(11-12):2267-2303
We prove a weighted L ∞ estimate for the solution to the linear wave equation with a smooth positive time independent potential. The proof is based on application of generalized Fourier transform for the perturbed Laplace operator and a finite dependence domain argument. We apply this estimate to prove the existence of global small data solution to supercritical semilinear wave equations with potential. 相似文献
10.
Qifeng
Zhang Lingling Liu Jiyuan Zhang 《Numerical Methods for Partial Differential Equations》2020,36(6):1790-1810
In the article, two linearized finite difference schemes are proposed and analyzed for the Benjamin–Bona–Mahony–Burgers (BBMB) equation. For the construction of the two-level scheme, the nonlinear term is linearized via averaging k and k + 1 floor, we prove unique solvability and convergence of numerical solutions in detail with the convergence order O(τ2 + h2) . For the three-level linearized scheme, the extrapolation technique is utilized to linearize the nonlinear term based on ψ function. We obtain the conservation, boundedness, unique solvability and convergence of numerical solutions with the convergence order O(τ2 + h2) at length. Furthermore, extending our work to the BBMB equation with the nonlinear source term is considered and a Newton linearized method is inserted to deal with it. The applicability and accuracy of both schemes are demonstrated by numerical experiments. 相似文献
11.
Kenji Nishihara 《Mathematische Zeitschrift》2003,244(3):631-649
It has been asserted that the damped wave equation has the diffusive structure as t→∞. In this paper we consider the Cauchy problem in 3-dimensional space for the linear damped wave equation and the corresponding
parabolic equation, and obtain the L
p
−L
q
estimates of the difference of each solution, which represent the assertion precisely. Explicit formulas of the solutions
are analyzed for the proof. The second aim is to apply the L
p
−L
q
estimates to the semilinear damped wave equation with power nonlinearity. If the power is larger than the Fujita exponent,
then the time global existence of small weak solution is proved and its optimal decay order is obtained.
Received: 8 June 2001; in final form: 12 August 2002 /
Published online: 1 April 2003
Mathematical Subject Classification (2000): 35L15. 相似文献
12.
Chein-Shan Liu 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2004,15(3):606-625
In this paper we present a linear representation of the
Landau-Lifshitz-Gilbert equation for describing the magnetization
of ferromagnetic materials. According to Lie
P \Bbb M3+1,P {\Bbb M}^{3+1},
of which the
projective proper orthochronous Lorentz group
PSO
o(3,1)
left acts. By the Lie symmetry a group preserving scheme is developed,
which improves the computational accuracy and efficiency. 相似文献
13.
The purpose of this paper is to study the finite element method for second order semilinear elliptic interface problems in
two dimensional convex polygonal domains. Due to low global regularity of the solution, it seems difficult to achieve optimal
order of convergence with straight interface triangles [Numer. Math., 79 (1998), pp. 175–202]. For a finite element discretization based on a mesh which involve the approximation of the interface,
optimal order error estimates in L
2 and H
1-norms are proved for linear elliptic interface problem under practical regularity assumptions of the true solution. Then
an extension to the semilinear problem is also considered and optimal error estimate in H
1 norm is achieved. 相似文献
14.
Phase separation of an initially homogeneous mixture into two different phases can be modeled on a mesoscopic scale by the Cahn-Hilliard equation. The interface thickness between the pure phases enters as a small parameter γ into this mass conserving fourth order semilinear parabolic equation. Numerical analysis is well established for a fixed parameter size, but error estimates depend exponentially on γ–1 and thus become useless if γ → 0. We consider the case, that elastic stresses due to a lattice misfit become important and the equation has to be coupled to a system of linear elasticity. Applications include e. g. the simulation of Sn-Cu alloys for the production of lead free solder or Ni-Al alloys used for rotor blade surfaces. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
15.
L. A. Mamunya 《Journal of Mathematical Sciences》1994,71(4):2592-2595
The nonlinear equation of gas seepage is reduced by a change of variables to a so-called semilinear form, which is then solved by a numerical method using simple iteration.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 72, pp. 68–71, 1990. 相似文献
16.
Talha Achouri 《Numerical Methods for Partial Differential Equations》2019,35(1):200-221
In this article, two finite difference schemes for solving the semilinear wave equation are proposed. The unique solvability and the stability are discussed. The second‐order accuracy convergence in both time and space in the discrete H1‐norm for the two proposed difference schemes is proved. Numerical experiments are performed to support our theoretical results. 相似文献
17.
In this paper, we consider the existence and multiplicity of nodal solutions of semilinear elliptic equations. We prove that a semilinear elliptic equation in large domains does not admit any least energy nodal (sign-changing) solution and in an upper half strip with m-holes has at least m2 2-nodal solutions. 相似文献
18.
This paper deals with semilinear evolution equations with unbounded observation operators. Sufficient conditions are given
guaranteeing that the output function of a semilinear system is in L2loc([0, ∞); Y). We prove that the Lebesgue extension of the observation operators are invariant under nonlinear globally Lipschitz continuous
perturbations. Further, relations between the corresponding -extensions are studied. We show that exact observability of linear autonomous system is conserved under small Lipschitz perturbations.
The obtained results are illustrated by several examples.
相似文献
19.
Zhonghua Qiao Zhi‐zhong Sun Zhengru Zhang 《Numerical Methods for Partial Differential Equations》2012,28(6):1893-1915
The numerical simulation of the dynamics of the molecular beam epitaxy (MBE) growth is considered in this article. The governing equation is a nonlinear evolutionary equation that is of linear fourth order derivative term and nonlinear second order derivative term in space. The main purpose of this work is to construct and analyze two linearized finite difference schemes for solving the MBE model. The linearized backward Euler difference scheme and the linearized Crank‐Nicolson difference scheme are derived. The unique solvability, unconditional stability and convergence are proved. The linearized Euler scheme is convergent with the convergence order of O(τ + h2) and linearized Crank‐Nicolson scheme is convergent with the convergence order of O(τ2 + h2) in discrete L2‐norm, respectively. Numerical stability with respect to the initial conditions is also obtained for both schemes. Numerical experiments are carried out to demonstrate the theoretical analysis. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011 相似文献
20.
To recover the full accuracy of discretized fractional derivatives, nonuniform mesh technique is a natural and simple approach to efficiently resolve the initial singularities that always appear in the solutions of time-fractional linear and nonlinear differential equations. We first construct a nonuniform L2 approximation for the fractional Caputo's derivative of order 1 < α < 2 and present a global consistency analysis under some reasonable regularity assumptions. The temporal nonuniform L2 formula is then utilized to develop a linearized difference scheme for a time-fractional Benjamin–Bona–Mahony-type equation. The unconditional convergence of our scheme on both uniform and nonuniform (graded) time meshes are proven with respect to the discrete H1-norm. Numerical examples are provided to justify the accuracy. 相似文献