共查询到18条相似文献,搜索用时 171 毫秒
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对来源于平面弹性问题的Hamilton算子的本征值问题进行了研究.在矩形域内含位移和应力的混合边界条件下,首先求解了相应算子的本征函数.接着,证明了本征函数系的完备性,这为施行分离变量法求解相应问题提供了可行性.最后,利用文中的辛本征展开定理获得了问题的一般解. 相似文献
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本文研究一边简支对边滑支边界条件的矩形板方程的无穷维Hamilton算子本征函数系,证明该无穷维Hamilton算子广义本征函数系在Cauchy主值意义下是完备的,为应用辛本征函数展开法求解该平面弹性问题提供理论基础.进而推导出原方程的通解,并对该平面弹性问题指出什么样的边界条件可按此方法求解.最后应用具体的算例说明所得结论的合理性. 相似文献
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本文研究对边滑支边界条件的矩形板方程的无穷维Hamilton算子本征函数系,证明该无穷维Hamilton算子广义本征函数系在Cauchy主值意义下的完备性.进而推导出原矩形板方程的一般解,并对该平面弹性问题指出什么样的边界条件可按此方法求解.最后应用具体的算例说明所得结论的合理性. 相似文献
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研究了Sturm-Liouvile偏微分方程导出的无穷维Hamilton算子的本征值问题.证明了导出的无穷维Hamilton算子族本征函数系的完备性,为对此类方程应用基于Hamilton体系的分离变量法提供了理论基础.最后举例说明了结果的有效性. 相似文献
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In the theory of mechanics and/or mathematical physics problems in a prismatic domain, the method of separation of variables ususally leads to the Sturm–Liouville-type eigenproblems of self-adjoint operators, and then the eigenfunction expansion method can be used in equation solving. However, a number of important application problems cannot lead to self-adjoint operator for the transverse coordinate. From the minimum potential energy variational principle, by selection of the state and its dual variables, the generalized variational principle is deduced. Then, based on the analogy between the theory of structural mechanics and optimal control, the present article leads the problem to the Hamiltonian system. The finite-dimensional theory for the Hamiltonian system is extended to the corresponding theory of the Hamiltonian operator matrix and adjoint symplectic spaces. The adjoint symplectic orthonormality relation is proved for the whole state eigenfunction vectors, and then the expansion of an arbitrary whole state function vector by the eigenfunction vectors is established. Thus the range of classical method of separation of variables is considerably extended. The eigenproblem derived from a plate bending problem in a strip domain is used for illustration. © 1993 John Wiley & Sons, Inc. 相似文献
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The Complete Biorthogonal Expansion Theorem
and Its Application to a Class of Rectangular
Plate Equations 下载免费PDF全文
In this paper, we first establish the separable $Hamiltonian$ system of rectangular cantilever thin plate bending problems by choosing proper dual vectors. Then using the characteristics of off-diagonal infinite-dimensional $Hamiltonian$ operator matrix, we derive the biorthogonal relationships of the eigenfunction systems and based on it we further obtain the complete biorthogonal expansion theorem. Finally, applying this theorem we obtain the general solutions of rectangular cantilever thin plate bending problems with two opposite edges slidingly supported. 相似文献
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P. N. Vabishchevich 《Computational Mathematics and Mathematical Physics》2014,54(6):953-962
In this paper, the schemes of the alternating triangular method are set out in the class of splitting methods used for the approximate solution of Cauchy problems for evolutionary problems. These schemes are based on splitting the problem operator into two operators that are conjugate transposes of each other. Economical schemes for the numerical solution of boundary value problems for parabolic equations are designed on the basis of an explicit-implicit splitting of the problem operator. The alternating triangular method is also of interest for the construction of numerical algorithms that solve boundary value problems for systems of partial differential equations and vector systems. The conventional schemes of the alternating triangular method used for first-order evolutionary equations are two-level ones. The approximation properties of such splitting methods can be improved by transiting to three-level schemes. Their construction is based on a general principle for improving the properties of difference schemes, namely, on the regularization principle of A.A. Samarskii. The analysis conducted in this paper is based on the general stability (or correctness) theory of operator-difference schemes. 相似文献
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Farshid Mirzaee Sahar Alipour 《Numerical Methods for Partial Differential Equations》2019,35(3):1134-1151
In this paper, an effective numerical approach based on a new two‐dimensional hybrid of parabolic and block‐pulse functions (2D‐PBPFs) is presented for solving nonlinear partial quadratic integro‐differential equations of fractional order. Our approach is based on 2D‐PBPFs operational matrix method together with the fractional integral operator, described in the Riemann–Liouville sense. The main characteristic behind this approach is to reduce such problems to those of solving systems of algebraic equations, which greatly simplifies the problem. By using Newton's iterative method, this system is solved, and the solution of fractional nonlinear partial quadratic integro‐differential equations is achieved. Convergence analysis and an error estimate associated with the proposed method is obtained, and it is proved that the numerical convergence order of the suggested numerical method is O(h3) . The validity and applicability of the method are demonstrated by solving three numerical examples. Numerical examples are presented in the form of tables and graphs to make comparisons with the exact solutions much easier. 相似文献
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Regular and well-posed formulation of the boundary integral method for a singular biharmonic problem
By adequate choice of a fundamental solution, the singular point of the solution is excluded from the integral equations. The use of a special differential operator yields a well-posed formulation of the system of two integral equations. Moreover, the application of the symmetry principle for biharmonic functions improves the efficiency of the method. Finally, the results are used to compute the coefficients of the William's series (stress intensity factors) which is the eigenfunction expansion of the solution around the singular point.The research was supported in part by the Technion VPR Fund-M. R. Saulson Research Fund. 相似文献
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In this paper we apply matrix splitting techniques and a conjugate gradient algorithm to the problem of minimizing a convex quadratic form subject to upper and lower bounds on the variables. This method exploits sparsity structure in the matrix of the quadratic form. Choices of the splitting operator are discussed, and convergence results are established. We present the results of numerical experiments showing the effectiveness of the algorithm on free boundary problems for elliptic partial differential equations, and we give comparisons with other algorithms. 相似文献
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Robert I Jewett 《Advances in Mathematics》1975,18(1):1-101
The self-adjoint subspace extensions of a possibly nondensely defined symmetric ordinary differential operator in a Hilbert space are described. The operator part of these extensions involve not only the differential operator but boundary-integral terms, and the side conditions which determine the domains of the extensions also involve boundary-integral terms. Corresponding to each self-adjoint subspace extension in a possibly larger Hilbert space an eigenfunction expansion result is obtained. Analogous results for first-order systems of ordinary differential operators are shown to be valid. 相似文献