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1.
In this article, an iterative method for the approximate solution to one‐dimensional variable‐coefficient Burgers' equation is proposed in the reproducing kernel space W(3,2). It is proved that the approximation wn(x,t) converges to the exact solution u(x,t) for any initial function w0(x,t) ε W(3,2), and the approximate solution is the best approximation under a complete normal orthogonal system . Moreover the derivatives of wn(x,t) are also uniformly convergent to the derivatives of u(x,t).© 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009 相似文献
2.
Nguyen Dung Nguyen Vu Huy Pham Hoang Quan Dang Duc Trong 《Mathematische Nachrichten》2006,279(11):1147-1158
We consider the problem of finding u ∈ L 2(I ), I = (0, 1), satisfying ∫I u (x )x dx = μ k , where k = 0, 1, 2, …, (α k ) is a sequence of distinct real numbers greater than –1/2, and μ = (μ kl ) is a given bounded sequence of real numbers. This is an ill‐posed problem. We shall regularize the problem by finite moments and then, apply the result to reconstruct a function on (0, +∞) from a sequence of values of its Laplace transforms. Error estimates are given. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
3.
Rossitza I. Semerdjieva 《Mathematische Nachrichten》2002,237(1):89-104
Let k(y) > 0, 𝓁(y) > 0 for y > 0, k(0) = 𝓁(0) = 0 and limy → 0k(y)/𝓁(y) exists; then the equation L(u) ≔ k(y)uxx – ∂y(𝓁(y)uy) + a(x, y)ux = f(x, y, u) is strictly hyperbolic for y > 0 and its order degenerates on the line y = 0. Consider the boundary value problem Lu = f(x, y, u) in G, u|AC = 0, where G is a simply connected domain in ℝ2 with piecewise smooth boundary ∂G = AB∪AC∪BC; AB = {(x, 0) : 0 ≤ x ≤ 1}, AC : x = F(y) = ∫y0(k(t)/𝓁(t))1/2dt and BC : x = 1 – F(y) are characteristic curves. Existence of generalized solution is obtained by a finite element method, provided f(x, y, u) satisfies Carathéodory condition and |f(x, y, u)| ≤ Q(x, y) + b|u| with Q ∈ L2(G), b = const > 0. It is shown also that each generalized solution is a strong solution, and that fact is used to prove uniqueness under the additional assumption |f(x, y, u1) – f(x, y, u2| ≤ C|u1 – u2|, where C = const > 0. 相似文献
4.
A. I. Perov 《Functional Analysis and Its Applications》2010,44(1):69-72
Let M be a complete K-metric space with n-dimensional metric ρ(x, y): M × M → R
n
, where K is the cone of nonnegative vectors in R
n
. A mapping F: M → M is called a Q-contraction if ρ (Fx,Fy) ⩽ Qρ (x,y), where Q: K → K is a semi-additive absolutely stable mapping. A Q-contraction always has a unique fixed point x* in M, and ρ(x*,a) ⩽ (I - Q)-1 ρ(Fa, a) for every point a in M. The point x* can be obtained by the successive approximation method x
k
= Fx
k-1, k = 1, 2,..., starting from an arbitrary point x
0 in M, and the following error estimates hold: ρ (x*, x
k
) ⩽ Q
k
(I - Q)-1ρ(x
1, x
0) ⩽ (I - Q)-1
Q
k
ρ(x
1, x
0), k = 1, 2,.... Generally the mappings (I - Q)-1 and Q
k
do not commute. For n = 1, the result is close to M. A. Krasnosel’skii’s generalized contraction principle. 相似文献
5.
This article presents a semigroup approach for the mathematical analysis of the inverse coefficient problems of identifying the unknown coefficient k(u(x,t)) in the quasi‐linear parabolic equation ut(x,t)=(k(u(x,t))ux(x,t))x, with Dirichlet boundary conditions u(0,t)=ψ0, u(1,t)=ψ1. The main purpose of this paper is to investigate the distinguishability of the input–output mappings Φ[?]:?? →C1[0,T], Ψ[?]:??→C1[0,T] via semigroup theory. In this paper, it is shown that if the null space of the semigroup T(t) consists of only zero function, then the input–output mappings Φ[?] and Ψ[?] have the distinguishability property. It is also shown that the types of the boundary conditions and the region on which the problem is defined play an important role in the distinguishability property of these mappings. Moreover, under the light of measured output data (boundary observations) f(t):=k(u(0,t))ux(0,t) or/and h(t):=k(u(1,t))ux(1,t), the values k(ψ0) and k(ψ1) of the unknown diffusion coefficient k(u(x,t)) at (x,t)=(0,0) and (x,t)=(1,0), respectively, can be determined explicitly. In addition to these, the values ku(ψ0) and ku(ψ1) of the unknown coefficient k(u(x,t)) at (x,t)=(0,0) and (x,t)=(1,0), respectively, are also determined via the input data. Furthermore, it is shown that measured output data f(t) and h(t) can be determined analytically by an integral representation. Hence the input–output mappings Φ[?]:??→ C1[0,T], Ψ[?]:??→C1[0,T] are given explicitly in terms of the semigroup. Copyright © 2007 John Wiley & Sons, Ltd. 相似文献
6.
In this paper, we investigate the existence of global weak solutions to the Cauchy problem of a modified two‐component Camassa‐Holm equation with the initial data satisfying limx → ±∞u0(x) = u±. By perturbing the Cauchy problem around a rarefaction wave, we obtain a global weak solution for the system under the assumption u? ≤ u+. The global weak solution is obtained as a limit of approximation solutions. The key elements in our analysis are the Helly theorem and the estimation of energy for approximation solutions in $H^1(\mathbb {R})\times H^1(\mathbb {R})In this paper, we investigate the existence of global weak solutions to the Cauchy problem of a modified two‐component Camassa‐Holm equation with the initial data satisfying limx → ±∞u0(x) = u±. By perturbing the Cauchy problem around a rarefaction wave, we obtain a global weak solution for the system under the assumption u? ≤ u+. The global weak solution is obtained as a limit of approximation solutions. The key elements in our analysis are the Helly theorem and the estimation of energy for approximation solutions in $H^1(\mathbb {R})\times H^1(\mathbb {R})$ and some a priori estimates on the first‐order derivatives of approximation solutions. 相似文献
7.
Susana Elena Trione 《Studies in Applied Mathematics》1988,79(2):127-141
Let t = (t1,…,tn) be a point of ?n. We shall write . We put, by the definition, Wα(u, m) = (m?2u)(α ? n)/4[π(n ? 2)/22(α + n ? 2)/2Г(α/2)]J(α ? n)/2(m2u)1/2; here α is a complex parameter, m a real nonnegative number, and n the dimension of the space. Wα(u, m), which is an ordinary function if Re α ≥ n, is an entire distributional function of α. First we evaluate {□ + m2}Wα + 2(u, m) = Wα(u, m), where {□ + m2} is the ultrahyperbolic operator. Then we express Wα(u, m) as a linear combination of Rα(u) of differntial orders; Rα(u) is Marcel Riesz's ultrahyperbolic kernel. We also obtain the following results: W?2k(u, m) = {□ + m2}kδ, k = 0, 1,…; W0(u, m) = δ; and {□ + m2}kW2k(u, m) = δ. Finally we prove that Wα(u, m = 0) = Rα(u). Several of these results, in the particular case µ = 1, were proved earlier by a completely different method. 相似文献
8.
Selina Yo-Ping Chang Justie Su-Tzu Juan Cheng-Kuan Lin Jimmy J. M. Tan Lih-Hsing Hsu 《Annals of Combinatorics》2009,13(1):27-52
A graph G is hamiltonian connected if there exists a hamiltonian path joining any two distinct nodes of G. Two hamiltonian paths and of G from u to v are independent if u = u
1 = v
1, v = u
v(G)
= v
v(G)
, and u
i
≠ v
i
for every 1 < i < v(G). A set of hamiltonian paths, {P
1, P
2, . . . , P
k
}, of G from u to v are mutually independent if any two different hamiltonian paths are independent from u to v. A graph is k mutually independent hamiltonian connected if for any two distinct nodes u and v, there are k mutually independent hamiltonian paths from u to v. The mutually independent hamiltonian connectivity of a graph G, IHP(G), is the maximum integer k such that G is k mutually independent hamiltonian connected. Let n and k be any two distinct positive integers with n–k ≥ 2. We use S
n,k
to denote the (n, k)-star graph. In this paper, we prove that IHP(S
n,k
) = n–2 except for S
4,2 such that IHP(S
4,2) = 1.
相似文献
9.
Claire Chainais‐Hillairet 《Mathematical Methods in the Applied Sciences》2000,23(5):467-490
We study second‐order finite‐volume schemes for the non‐linear hyperbolic equation ut(x, t) + div F(x, t, u(x, t)) = 0 with initial condition u0. The main result is the error estimate between the approximate solution given by the scheme and the entropy solution. It is based on some stability properties verified by the scheme and on a discrete entropy inequality. If u0 ∈ L∞ ∩ BVloc(ℝN), we get an error estimate of order h1/4, where h defines the size of the mesh. Copyright © 2000 John Wiley & Sons, Ltd. 相似文献
10.
G. I. Shishkin L. P. Shishkina 《Computational Mathematics and Mathematical Physics》2010,50(4):633-645
A boundary value problem for a singularly perturbed elliptic reaction-diffusion equation in a vertical strip is considered.
The derivatives are written in divergent form. The derivatives in the differential equation are multiplied by a perturbation
parameter ɛ2, where ɛ takes arbitrary values in the interval (0, 1]. As ɛ → 0, a boundary layer appears in the solution of this problem.
Using the integrointerpolational method and the condensing grid technique, conservative finite difference schemes on flux
grids are constructed that converge ɛ-uniformly at a rate of O(N
1−2ln2
N
1 + N
2−2), where N
1 + 1 and N
2 + 1 are the number of mesh points on the x
1-axis and the minimal number of mesh points on a unit interval of the x
2-axis respectively. The normalized difference derivatives ɛ
k
(∂
k
/∂x
1
k
)u(x) (k = 1, 2), which are ɛ-uniformly bounded and approximate the normalized derivatives in the direction across the boundary layer,
and the derivatives along the boundary layer (∂
k
/∂
x
2
k
)u(x) (k = 1, 2) converge ɛ-uniformly at the same rate. 相似文献
11.
Dirk Segers 《Mathematische Annalen》2006,336(3):659-669
Let K be a p-adic field, R the valuation ring of K, P the maximal ideal of R and q the cardinality of the residue field R/P. Let f be a polynomial over R in n >1 variables and let χ be a character of . Let M
i
(u) be the number of solutions of f = u in (R/P
i
)
n
for and. These numbers are related with Igusa’s p-adic zeta function Z
f,χ(s) of f. We explain the connection between the M
i
(u) and the smallest real part of a pole of Z
f,χ(s). We also prove that M
i
(u) is divisible by , where the corners indicate that we have to round up. This will imply our main result: Z
f,χ(s) has no poles with real part less than − n/2. We will also consider arbitrary K-analytic functions f. 相似文献
12.
Alemdar Hasanov 《Applied mathematics and computation》2003,140(2-3):501-515
An inverse polynomial method of determining the unknown leading coefficient k=k(x) of the linear Sturm–Liouville operator Au=−(k(x)u′(x))′+q(x)u(x), x(0,1), is presented. As an additional condition only two measured data at the boundary (x=0,x=1) are used. In absence of a singular point (u′(x)≠0,u″(x)≠0,x[0,1]) the inverse problem is classified as a well-conditioned . If there exists at least one singular point, then the inverse problem is classified as moderately ill-conditioned (u′(x0)=0,x0(0,1);u′(x)≠0,x≠x0;u″(x)≠0,x[0,1]) and severely ill-conditioned (u′(x0)=u″(x0)=0,x0(0,1);u′(x)≠0,u″(x)≠0,x≠x0). For each of the cases direct problem solution is approximated by corresponding polynomials and the inverse problem is reformulated as a Cauchy problem for to the first order differential equation with respect the unknown function k=k(x). An approximate analytical solution of the each Cauchy problems are derived in explicit form. Numerical simulations all the above cases are given for noise free and noisy data. An accuracy of the presented approach is demonstrated on numerical test solutions. 相似文献
13.
In this paper, we study the existence of infinitely many solutions to p‐Kirchhoff‐type equation (0.1) where f(x,u) = λh1(x)|u|m ? 2u + h2(x)|u|q ? 2u,a≥0,μ > 0,τ > 0,λ≥0 and . The potential function verifies , and h1(x),h2(x) satisfy suitable conditions. Using variational methods and some special techniques, we prove that there exists λ0>0 such that problem 0.1 admits infinitely many nonnegative high‐energy solutions provided that λ∈[0,λ0) and . Also, we prove that problem 0.1 has at least a nontrivial solution under the assumption f(x,u) = h2|u|q ? 2u,p < q< min{p*,p(τ + 1)} and has infinitely many nonnegative solutions for f(x,u) = h1|u|m ? 2u,1 < m < p. Copyright © 2015 John Wiley & Sons, Ltd. 相似文献
14.
Alaa E. Hamza 《Journal of Difference Equations and Applications》2013,19(3):233-253
We suppose that M is a closed subspace of l ∞(J, X), the space of all bounded sequences {x(n)} n?J ? X, where J ? {Z+,Z} and X is a complex Banach space. We define the M-spectrum σM (u) of a sequence u ? l ∞(J,X). Certain conditions will be supposed on both M and σM (u) to insure the existence of u ? M. We prove that if u is ergodic, such that σM (u,) is at most countable and, for every λ ? σM (u), the sequence e?iλnu(n) is ergodic, then u ? M. We apply this result to the operator difference equationu(n + 1) = Au(n) + ψ(n), n ? J,and to the infinite order difference equation Σ r k=1 ak (u(n + k) ? u(n)) + Σ s ? Z?(n ? s)u(s) = h(n), n?J, where ψ?l ∞(Z,X) such that ψ| J ? M, A is the generator of a C 0-semigroup of linear bounded operators {T(t)} t>0 on X, h ? M, ? ? l 1(Z) and ak ?C. Certain conditions will be imposed to guarantee the existence of solutions in the class M. 相似文献
15.
In this paper, we obtain a sequence of approximate solution converging uniformly to the exact solution of a class of fourth‐order nonlinear boundary value problems. Its exact solution is represented in the form of series in the reproducing kernel space. The n‐term approximation un(x) is proved to converge to the exact solution u(x). Moreover, the derivatives of un(x) are also convergent to the derivatives of u(x). Copyright © 2008 John Wiley & Sons, Ltd. 相似文献
16.
We study the variable‐bottom, generalized Korteweg—de Vries (bKdV) equation ?tu = ??x(?u + f(u) ? b(t,x)u), where f is a nonlinearity and b is a small, bounded, and slowly varying function related to the varying depth of a channel of water. Many variable‐coefficient KdV‐type equations, including the variable‐coefficient, variable‐bottom KdV equation, can be rescaled into the bKdV. We study the long‐time behavior of solutions with initial conditions close to a stable, b = 0 solitary wave. We prove that for long time intervals, such solutions have the form of the solitary wave whose center and scale evolve according to a certain dynamical law involving the function b(t,x) plus an H1(?)‐small fluctuation. © 2005 Wiley Periodicals, Inc. 相似文献
17.
The paper studies differential equations of the form u′(x) = f(x, u(x), λ(x)), u(x0) = u0, where the right‐hand side is merely measurable in x. In particular sufficient conditions for the continuous and the differentiable dependence of solution u on the data and on the parameter λ are stated. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim 相似文献
18.
19.
S. Yu. Antonov 《Russian Mathematics (Iz VUZ)》2012,56(11):1-16
We determine the least degree of identities in the subspace M 1 (m,k) (F) of the matrix superalgebra M (m,k)(F) over a field F for arbitrary m and k. For the subspace M 1 (m,k) (F) (k > 1) we obtain concrete minimal identities and generalize some results by Chang and Domokos. 相似文献
20.
V. V. Karachik 《Siberian Advances in Mathematics》2008,18(2):103-117
Let u(x) be a function analytic in some neighborhood D about the origin, $ \mathcal{D} Let u(x) be a function analytic in some neighborhood D about the origin, ⊂ ℝ
n
. We study the representation of this function in the form of a series u(x) = u
0(x) + |x|2
u
1(x) + |x|4
u
2(x) + …, where u
k
(x) are functions harmonic in . This representation is a generalization of the well-known Almansi formula.
Original Russian Text ? V. V. Karachik, 2007, published in Matematicheskie Trudy, 2007, Vol. 10, No. 2, pp. 142–162. 相似文献