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Complete symmetry analysis is presented for non-linear Klein Gordon equations utt=uxx+f(u). A group classification is carried out by finding f(u) that give larger symmetry algebra. One-dimensional optimal system is determined for symmetry algebras obtained through group classification. The subalgebras in one-dimensional optimal system and their conjugacy classes in the corresponding normalizers are employed to obtain, up to conjugacy, all reductions of equation by two-dimensional subalgebras. This is a new idea which improves the computational complexity involved in finding all possible reductions of a PDE of the form F(x,t,u,ux,ut,uxx,utt,uxt)=0 to a first order ODE. Some exact solutions are also found.  相似文献   

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In this paper, we study the existence of solutions for the boundary value problem of fractional hybrid differential equationsD0+αx(t)f(t,x(t))+g(t,x(t))=0,0<t<1,x(0)=x(1)=0,where 1<α?2 is a real number, D0+α is the Riemann–Liouville fractional derivative. By a fixed point theorem in Banach algebra due to Dhage, an existence theorem for fractional hybrid differential equations is proved under mixed Lipschitz and Carathéodory conditions. As an application, examples are presented to illustrate the main results.  相似文献   

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We investigate a forced Korteweg–de Vries (fKdV) equation, u,t+cu,x+αuu,x+βu,xxx=βF(t), which arises in the modelling of tsunami generation by submarine landslides. Approximate symmetries are found for the fKdV equation using the method as proposed by Fushchich and Shtelen [6]. Symmetries are found to second order in the perturbation parameter using the MAPLE symmetry package ASP [11], an add-on to the symmetry package DESOLVII [18]. ASP also allows particular forms of the arbitrary function F(t) to be found which extend the symmetry algebra and hence a full approximate symmetry classification of the fKdV equation is obtained. Optimal systems of one-dimensional subalgebras are also determined. Corresponding approximate invariant solutions to the fKdV equation are then constructed for particular F(t) using DESOLVII routines.  相似文献   

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ASYMPTOTIC STABILITY OF RAREFACTION WAVE FOR GENERALIZED BURGERS EQUATION   总被引:3,自引:2,他引:1  
This paper is concerned with the stability of the rarefaction wave for the Burgers equationwhere 0 ≤ a < 1/4p (q is determined by (2.2)). Roughly speaking, under the assumption that u_ < u , the authors prove the existence of the global smooth solution to the Cauchy problem (I), also find the solution u(x, t) to the Cauchy problem (I) satisfying sup |u(x, t) -uR(x/t)| → 0 as t → ∞, where uR(x/t) is the rarefaction wave of the non-viscous Burgersequation ut f(u)x = 0 with Riemann initial data u(x, 0) =  相似文献   

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We apply a fixed point theorem to obtain sufficient conditions for existence of triple positive solutions of the symmetric nonlinear fourth-order boundary value problemu(4)(t)=a(t)f(t,u(t),|u(t)|,u(t),|u(t)|),t(-1,1),u(-1)=u(-1)=0,u(-t)=u(t),t[-1,1],where f satisfies certain growth conditions. In the process, we obtain nontrivial a priori bounds on the derivatives of the solution.  相似文献   

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Let Clt(A) denote the t-class group of an integral domain A. P. Samuel has established that if A is a Krull domain then the mapping Clt(A)Clt(A?X?), is injective and if A is a regular UFD, then Clt(A)Clt(A?X?), is bijective. Later, L. Claborn extended this result in case A is a regular Noetherian domain. In the first part of this paper we prove that the mapping Clt(A)Clt(A?X?); [I]?[(I.A?X?)t] is an injective homomorphism and in case of an integral domain A such that each υ-invertible υ-ideal of A has υ-finite type, we give an equivalent condition for Clt(A)Clt(A?X?), to be bijective, thus generalizing the result of Claborn. In the second part of this paper, we define the S-class group of an integral domain A: let S be a (not necessarily saturated) multiplicative subset of an integral domain A. Following [11], a nonzero fractional ideal I of A is S-principal if there exist an sS and aI such that sI?aA?I. The S-class group of A, S-Clt(A) is the group of fractional t-invertible t-ideals of A under t-multiplication modulo its subgroup of S-principal t-invertible t-ideals of A. We generalize some known results developed for the classic contexts of Krull and PυMD domain and we investigate the case of isomorphism S-Clt(A)?S-Clt(A?X?).  相似文献   

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In this paper we discuss the asymptotic stability as well as the well-posedness of the damped wave equation posed on a bounded domain Ω of Rn,n2,
ρ(x)utt?Δu+0g(s)div[a(x)?u(?,t?s)]ds+b(x)ut=0,
subject to a locally distributed viscoelastic effect driven by a nonnegative function a(x) and supplemented with a frictional damping b(x)0 acting on a region A of Ω, where a=0 in A. Assuming that ρ(x) is constant, considering that the well-known geometric control condition (ω,T0) holds and supposing that the relaxation function g is bounded by a function that decays exponentially to zero, we prove that the solutions to the corresponding partial viscoelastic model decay exponentially to zero, even in the absence of the frictional dissipative effect. In addition, in some suitable cases where the material density ρ(x) is not constant, it is also possible to remove the frictional damping term b(x)ut, that is, the localized viscoelastic damping is strong enough to assure that the system is exponentially stable. The semi-linear case is also considered.  相似文献   

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In this paper, we use the coincidence degree theory to establish new results on the existence of T-periodic solutions for the Liénard equation with two deviating arguments of the formx(t)+f(x(t))x(t)+g1(t,x(t-τ1(t)))+g2(t,x(t-τ2(t)))=p(t).  相似文献   

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