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1.
M. L. Gorbachuk 《Ukrainian Mathematical Journal》2000,52(5):680-693
We find conditions on a closed operator A in a Banach space that are necessary and sufficient for the existence of solutions
of a differential equation y′(t) = Ay(t), t ∈[0,∞),in the classes of entire vector functions with given order of growth and type. We present criteria for the denseness
of classes of this sort in the set of all solutions. These criteria enable one to prove the existence of a solution of the
Cauchy problem for the equation under consideration in the class of analytic vector functions and to justify the convergence
of the approximate method of power series. In the special case where A is a differential operator, the problem of applicability
of this method was first formulated by Weierstrass. Conditions under which this method is applicable were found by Kovalevskaya. 相似文献
2.
Polina Vinogradova Anatoli Zarubin 《Numerical Functional Analysis & Optimization》2013,34(1-2):148-167
In the current paper, we study a projection method for a Cauchy problem for an operator-differential equation with a leading self-adjoint operator A(t) and a subordinate linear operator K(t) in a Hilbert space. The projection subspaces are linear spans of eigenvectors of an operator similar to A(t). It is assumed that the operators A(t) and K(t) are sufficiently smooth. Error estimates for the approximate solutions and their derivatives are obtained. The application of the developed method for solving the initial boundary value problems is given. 相似文献
3.
Analytical solutions for the Cahn-Hilliard initial value problem are obtained through an application of the homotopy analysis method. While there exist numerical results in the literature for the Cahn-Hilliard equation, a nonlinear partial differential equation, the present results are completely analytical. In order to obtain accurate approximate analytical solutions, we consider multiple auxiliary linear operators, in order to find the best operator which permits accuracy after relatively few terms are calculated. We also select the convergence control parameter optimally, through the construction of an optimal control problem for the minimization of the accumulated L 2-norm of the residual errors. In this way, we obtain optimal homotopy analysis solutions for this complicated nonlinear initial value problem. A variety of initial conditions are selected, in order to fully demonstrate the range of solutions possible. 相似文献
4.
To prove the existence of a solution of a two-point boundary value problem for an nth-order operator equation by the a priori estimate method, we study extremal solutions of auxiliary boundary value problems
for an nth-order differential equation with simplest right-hand side, which have a unique solution under certain restrictions on the
boundary conditions. 相似文献
5.
We study a projection-difference method of solving the Cauchy problem for an operatordifferential equation with a selfadjoint leading operator A(t) and a nonlinear monotone subordinate operator K(·) in a Hilbert space. This method leads to a solution of a system of linear algebraic equations at each time level. Error estimates are derived for approximate solutions as well as for fractional powers of the operator A(t). The method is applied to a model parabolic problem. 相似文献
6.
Explicit and approximate solutions of second‐order evolution differential equations in Hilbert space
Ivan P. Gavrilyuk Vladimir L. Makarov 《Numerical Methods for Partial Differential Equations》1999,15(1):111-131
The explicit closed‐form solutions for a second‐order differential equation with a constant self‐adjoint positive definite operator coefficient A (the hyperbolic case) and for the abstract Euler–Poisson–Darboux equation in a Hilbert space are presented. On the basis of these representations, we propose approximate solutions and give error estimates. The accuracy of the approximation automatically depends on the smoothness of the initial data. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 111–131, 1999 相似文献
7.
Time harmonic Maxwell equations in lossless media lead to a second order differential equation for the electric field involving a differential operator that is neither elliptic nor definite. A Galerkin method using Nedelec spaces can be employed to get approximate solutions numerically. The problem of preconditioning the indefinite matrix arising from this method is discussed here. Specifically, two overlapping Schwarz methods will be shown to yield uniform preconditioners.
8.
Petr Harasim 《Applications of Mathematics》2011,56(5):459-480
We apply a theoretical framework for solving a class of worst scenario problems to a problem with a nonlinear partial differential
equation. In contrast to the one-dimensional problem investigated by P. Harasim in Appl. Math. 53 (2008), No. 6, 583–598, the two-dimensional problem requires stronger assumptions restricting the admissible set to ensure
the monotonicity of the nonlinear operator in the examined state problem, and, as a result, to show the existence and uniqueness
of the state solution. The existence of the worst scenario is proved through the convergence of a sequence of approximate
worst scenarios. Furthermore, it is shown that the Galerkin approximation of the state solution can be calculated by means
of the Kachanov method as the limit of a sequence of solutions to linearized problems. 相似文献
9.
The solutions of a class of matrix optimization problems (including
the Nehari problem and its multidisk generalization) can be identified with
the solutions of an abstract operator equation of the form T(., ., .) = 0. This
equation can be solved numerically by Newton's method if the differential
T' of T is invertible at the points of interest. This is typically too difficult
to verify. However, it turns out that under reasonably broad conditions we
can identify T' as the sum of a block Toeplitz operator and a compact block
Hankel operator. Moreover, we can show that the block Toeplitz operator is
a Fredholm operator and and in some cases can calculate its Fredholm index.
Thus, T' will also be a Fredholm operator of the same index. In a number of
cases that have been checked todate, numerical methods perform well when
the Fredholm index is equal to zero and poorly otherwise. The main focus
of this paper is on the multidisk problem alluded to above. However, a number
of analogies with existing work on matrix optimization have been worked out
and incorporated.
Submitted: April 23, 2002. 相似文献
10.
Pham Loi Vu 《Acta Appl Math》2010,109(3):789-787
We derive the continual system of nonlinear interaction waves from the compatibility of the transport equation on the whole
line and the equation governing the time-evolution of the eigenfunctions of the transport operator. The transport equation
represents the continual generalization from the n-component system of first-order ordinary differential equations. The continual system describes a nonlinear interaction of
waves. We prove that the continual system can be integrated by the inverse scattering method. The method is based on applying
the results of the inverse scattering problem for the transport equation to finding the solution of the Cauchy initial-value
problem for the continual system. Indeed, the transition operator for the scattering problem admits right and left Volterra
factorizations. The intermediate operator for this problem determines the one-to-one correspondence between the preimages
of a solution of the transport equation. This operator is related to the transition operator and admits not only right and
left Volterra factorizations but also the analytic factorization. By virtue of this fact the potential in the transport equation
is uniquely reconstructed in terms of the solutions of the fundamental equations in inverse problem. 相似文献
11.
The contacts problem of the theory of elasticity and bending theory of plates for finite or infinite plates with an elastic
inclusion of variable rigidity are considered. The problems are reduced to integral differential equation or to the system
of integral differential equations with variable coefficient of singular operator. If such coefficient varies with power law
we can manage to investigate the obtained equations, to get exact or approximate solutions and to establish behavior of unknown
contact stresses at the ends of elastic inclusion.
相似文献
12.
We study the Galerkin method for a third-order differential-operator equation with self-adjoint leading operator A and subordinate linear operator K(t) in a separable Hilbert space. We prove a theorem on the existence and uniqueness of a strong solution of the original problem. We derive estimates for the accuracy of the approximate solutions constructed by the Galerkin method. An application of the suggested method to the solution of a model problem is described. 相似文献
13.
This paper determines the solvability of multipoint boundary value problems for p-Laplacian generalized fractional differential systems with Riesz–Caputo derivative, which exhibits two-sided nonlocal memory effects. An equivalent integral form for the generalized fractional differential system is deduced by transformation. First, we obtain the existence of solutions on the basis of the upper–lower solutions method, in which an explicit iterative approach for approximating the solution is established. Second, we deal with a special case of our fractional differential system; in order to obtain novel results, an abstract sum-type operator equation A(x,x)+Bx+e=x on ordered Banach space is discussed. Without requiring the existence of upper–lower solutions or compactness conditions, we get several unique results of solutions for this operator equation, which provide new inspiration for the study of boundary value problems. Then, we apply these abstract results to get the uniqueness of solutions for our differential system. 相似文献
14.
Leonid Knizhnerman Vladimir Druskin Qing-Huo Liu Fikri J. Kuchuk 《Numerical Methods for Partial Differential Equations》1994,10(5):569-580
A three-dimensional well model (r ? θ ? z) for the simulation of single-phase fluid flow in porous media is developed. Rather than directly solving the 3-D parabolic PDE (partial differential equation) for fluid flow, the PDE is transformed to a linear operator problem that is defined as u = f( A ) σ , where A is a real symmetric square matrix and σ is a vector. The linear operator problem is solved by using the spectral Lanczos decomposition method. This formulation gives continuous solutions in time. A 7-point finite difference scheme is used for the spatial discretization. The model is useful for well testing problems as well as for the simulation of the wireline formation tester tool behavior in heterogeneous reservoirs. The linear operator formulation also permits us to obtain solutions in the Laplace domain, where the wellbore storage and skin can be incorporated analytically. The infinite-conductivity (uniform pressure) wellbore condition is preserved when mixed boundary conditions, such as partial penetration, occur. The numerical solutions are compared with the analytical solutions for fully and partially penetrated wells in a homogeneous reservoir. © 1994 John Wiley & Sons, Inc. 相似文献
15.
A 2 + 1-dimensional nonlinear differential equation integrable by the inverse-spectral-transform method with the quartet operator representation is proposed. This GL(2, C)-valued chiral-field-type equation is the generating (prototype) equation for the Davey-Stewartson and Ishimori equations. It coincides with the nonlinear equation for the Davey-Stewartson eigenfunction ψDS. The initial-value problem for this equation is solved by the techniques for the and the nonlocal Riemann-Hilbert problem. The classes of exact solutions with the functional parameters and exponential-rational solutions are constructed by the method. The static lump solution in the case α = i and the exponentially localized solution at α = i are found. Other similar examples of nonlinear integrable equations in 2 + 1 and 1 + 1 dimensions are discussed. 相似文献
16.
We consider an approximate method for the solution of the Cauchy problem for an operator differential equation. This method
is based on the expansion of an exponential in orthogonal Laguerre polynomials. We prove that the fact that an initial value
belongs to a certain space of smooth elements of the operator A is equivalent to the convergence of a certain weighted sum of integral residuals. As a corollary, we obtain direct and inverse
theorems of the theory of approximation in the mean.
__________
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 4, pp. 557–563, April, 2008. 相似文献
17.
In this paper, we study a final value problem for first order abstract differential equation with positive self-adjoint unbounded operator coefficient. This problem is ill-posed. Perturbing the final condition we obtain an approximate nonlocal problem depending on a small parameter. We show that the approximate problems are well posed and that their solutions converge if and only if the original problem has a classical solution. We also obtain estimates of the solutions of the approximate problems and a convergence result of these solutions. Finally, we give explicit convergence rates. 相似文献
18.
In this article we consider differential equations which generate oscillating solutions. These oscillations are due to the presence of a small parameter l>0 ; however, they are not present in the coefficients but instead they are caused by a penalty term involving an antisymmetric operator. Our aims are twofold. In the first part we study asymptotics at all orders, for lM 0 , construct approximate solutions, and derive estimates of the error between the exact solution and the approximate ones. One of the motivations of this part is the study to high orders of the geostrophic asymptotics in atmospheric science, but there are many other possible applications involving in particular the wave equation. The actual applications of our results to atmospheric science will be discussed elsewhere [STW], as well as, on the mathematical side, the application to partial differential equations [TW1]. In the second part of this article we study a control problem involving such an equation and study the behavior of the state equation, of the optimal control, and of the optimality equation as lM 0 . For the control part we restrict ourselves to a linear equation and to the first order in the asymptotics lM 0 , leaving nonlinear problems and higher orders to a future work. 相似文献
19.
A general approximation scheme for minimization of functionals in a Banach space is considered. Inequalities are proved which supply bounds on the rate of convergence of the approximate solutions to the exact solution. These bounds are applied to an optimal control problem for an abstract operator equation in a Hilbert space with control in the right-hand side.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 59, pp. 117–121, 1986. 相似文献
20.
In this article we use the C 1 wavelet bases on Powell-Sabin triangulations to approximate the solution of the Neumann problem for partial differential equations. The C 1 wavelet bases are stable and have explicit expressions on a three-direction mesh. Consequently, we can approximate the solution of the Neumann problem accurately and stably. The convergence and error estimates of the numerical solutions are given. The computational results of a numerical example show that our wavelet method is well suitable to the Neumann boundary problem. 相似文献