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Correction factors in series solutions for one- and two-dimensional boundary value problems

JOURNAL OF ENGINEERING MECHANICS 2007年第11期133卷 1151-1161页

作者： GangaRao, Hota V. S. Prachasaree, Woraphot Constructed Facil Ctr Morgantown WV 26506 USA Prince Songkla Univ Fac Engn Dept Civil Engn Hat Yai 90010 Songkla Thailand W Virginia Univ Coll Engn & Mineral Resources Dept Civil & Environm Engn Morgantown WV 26506 USA

A Fourier cum polynomial series solution with correction factors is presented herein for differential equations with variable coefficients. The differential equations correspond to a wide range of boundary value problems. The correction factors included herein are: (1) modified Lanczos correction;(2) Bessel J;and (3) loading correction factor. These correction factors are introduced in terms of Fourier and polynomial series. The main purpose of using correction factors through a set of series is to improve convergence of the proposed solution, using the first two terms of the series. For the loading correction factor, a Fourier series expansion coupled with orthogonality conditions leads to evaluating undetermined Fourier coefficients of arbitrarily applied loads using concepts of summation equations. Representative boundary value problems are provided to demonstrate the efficiency and accuracy of the first two terms of the proposed solution with correction factors.

关键词： MECHANICAL engineering -- Research boundary value problems -- numerical solutions FINITE element method GIRDERS PLATES (Engineering) FOURIER series

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A modification of the method of continuous prolongation for finding the bifurcation solutions of steady-state self-adjoint boundary value problems

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PMM JOURNAL OF APPLIED MATHEMATICS AND MECHANICS 2012年第6期76卷 715-721页

作者： Lopanitsyn, Ye. A. Frolov, A. B. Moscow Russia

An algorithm for showing solution of systems of non-linear algebraic equations describing the steady-state behaviour of objects in the mechanics of a deformable solid is considered. The existence of limit points and simple bifurcation points on the trajectory of the solution of the system is admitted and, at these points, the Jacobian matrix of the system, assumed to be real, symmetric and continuous, degenerates. The basis of the algorithm is a transformation of the space of the arguments of the solution of systems of non-linear algebraic equations using a rotation matrix formed from the eigenvectors of the Jacobian matrix. (C) 2013 Elsevier Ltd. All rights reserved.

关键词： CONTINUOUS functions BIFURCATION theory boundary value problems -- numerical solutions ALGORITHMS LINEAR algebra ALGEBRAIC equations

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CLASSICAL solutions OF HYPERBOLIC INITIAL boundary-value problems WITH STATE-DEPENDENT DELAYS

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NONLINEAR OSCILLATIONS 2011年第4期13卷 595-612页

作者： Czernous, W. Univ Gdansk Inst Math PL-80952 Gdansk Poland

We consider the initial boundary-value problem for a system of quasilinear partial functional differential equations of the first order partial derivative(t)z(i)(t, x) + Sigma(n)(j=1) rho(ij)(t, x, V(z;t, x))partial derivative(xj)z(i)(t, x) = G(i)(t, x, V(z;t, x)), 1 <= i <= m, where V is a nonlinear operator of the Volterra type that maps bounded (with respect to seminorm) subsets of the space of Lipschitz continuously differentiable functions into bounded subsets of this space. Using the method of bicharacteristics and a fixed-point theorem, we prove the local existence, uniqueness, and continuous dependence on data of classical solutions of the problem. This approach covers systems of the form partial derivative(t)z(i)(t, x) + Sigma(n)(j=1) rho(ij)(t, x, z(psi(t, x, z(t, x))))partial derivative(xj)z(i)(t, x) = G(i)(t, x, z(psi(t, x, z(t, x)))), 1 <= i <= m, where (t, x) bar right arrow z((t, x)) is the Hale operator, and all components of psi may depend on. (t, x, z((t, x))). More specifically, problems with deviating arguments and integro-differential systems are included.

关键词： EXPONENTIAL functions boundary value problems -- numerical solutions NONLINEAR operators INTEGRO-differential equations FUNCTIONAL differential equations MATHEMATICAL analysis MAPPINGS (Mathematics)

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NON-UNIQUE solutions TO boundary value problems FOR NON-SYMMETRIC DIVERGENCE FORM EQUATIONS

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TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY 2010年第2期362卷 661-672页

作者： Axelsson, Andreas Stockholms Univ Inst Matemat S-10691 Stockholm Sweden

We calculate explicitly solutions to the Dirichlet and Neumann boundary value problems in the upper half plane, for a family of divergence form equations having non-symmetric coefficients with a jump discontinuity. It is shown that the boundary equation method and the Lax-Milgram method for constructing solutions may give two different solutions when the coefficients are sufficiently non-symmetric.

关键词： boundary value problems -- numerical solutions SYMMETRY (Mathematics) NEUMANN problem numerical calculations MATHEMATICAL analysis DIRICHLET problem

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Bifurcation of solutions of singular Fredholm boundary value problems

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DIFFERENTIAL EQUATIONS 2011年第4期47卷 453-461页

作者： Boichuk, A. A. Shegda, L. M. Natl Acad Sci Inst Math Kiev Ukraine Ivano Frankivsk Natl Tech Univ Oil & Gas Ivano Frankivsk Ukraine

We obtain conditions for the bifurcation of solutions of linear singular Fredholm boundary value problems with a small parameter under the assumption that the unperturbed singular differential system can be reduced to central canonical form. Using the Vishik-Lyusternik method and the technique of Moore-Penrose pseudoinverse matrices, we suggest an algorithm for finding a family of linearly independent solutions of such boundary value problems for the general case in which the number of boundary conditions specified by a linear vector functional does not coincide with the number of unknowns in the singular differential system.

关键词： BIFURCATION theory FREDHOLM equations boundary value problems -- numerical solutions PSEUDOINVERSES VECTOR spaces MATHEMATICAL optimization

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Existence of the solutions for a class of nonlinear fractional order three-point boundary value problems with resonance

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boundary value problems 2012年第1期2012卷 1-13页

作者： Ouyang, Zigen Li, Gangzhao Univ South China Sch Nucl Sci & Technol Sch Math & Phys Hengyang 421001 Peoples R China

A class of nonlinear fractional order differential equation D(0+)(alpha)u(t) + f (t, u(t)) = 0, 0 < t < 1, u(0) = 0, u(1) = 1/eta(alpha-1) u(eta) is investigated in this paper, where D-0+(alpha) is the standard Riemann-Liouville fractional derivative of order 1 < alpha <= 2, 0 < eta < 1, f is an element of C([0, 1] x R, R). Using intermediate value theorem, we obtain a sufficient condition for the existence of the solutions for the above fractional order differential equations.

关键词： EXISTENCE theorems NONLINEAR theories -- numerical solutions SET theory FRACTIONAL differential equations boundary value problems -- numerical solutions FRACTIONAL calculus

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Block elements and analytical solutions of boundary-value problems for sets of differential equations

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DOKLADY PHYSICS 2014年第1期59卷 30-34页

作者： Babeshko, V. A. Evdokimova, O. V. Babeshko, O. M. Kuban State Univ Krasnodar 350640 Russia Russian Acad Sci Southern Res Ctr Rostov Na Donu 341004 Russia Southern Fed Univ Rostov Na Donu 344049 Russia

The article discusses the solutions of boundary-value problems for sets of differential equations using block elements and analytical form. Topics discussed include the general boundary-value problem described by the set of partial differential equations, the exact solution of boundary-value problem based on the block-element method, and the accomplishment of quotient topology to construct the exact solution for the boundary-value problem through the implementation of the boundaries of layers.

关键词： boundary value problems -- numerical solutions PARTIAL differential equations SET theory ANALYTICAL mechanics TOPOLOGY

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Existence and Uniqueness of Positive solutions of boundary-value problems for Fractional Differential Equations with p-Laplacian Operator and Identities on the Some Special Polynomials

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JOURNAL OF FUNCTION SPACES AND APPLICATIONS 2013年第1期2013卷 1-11页

作者： Sen, Erdogan Acikgoz, Mehmet Seo, Jong Jin Araci, Serkan Orucoglu, Kamil Namik Kemal Univ Fac Sci & Letters Dept Math TR-59030 Tekirdag Turkey Istanbul Tech Univ Dept Engn Math TR-34469 Istanbul Turkey Gaziantep Univ Fac Sci & Arts Dept Math TR-27310 Gaziantep Turkey Pukyong Natl Univ Dept Appl Math Pusan 608737 South Korea Kwangwoon Univ Dept Math Coll Nat Sci Seoul 139701 South Korea

We consider the following boundary- value problem of nonlinear fractional differential equation with p-Laplacian operator D-0+(beta)(phi(p)(D(0+)(alpha)u(t))) + a(t)f(u) = 0, 0 < t < 1, u(0) = gamma u(h) + lambda, u'(0) = mu, phi(p)(D(0+)(alpha)u(0)) = (phi(p)(D-0+(alpha) u(0)) = (phi(p)(D(0+)(alpha)u(1)))' = (phi(p)(D(0+)(alpha)u(0)))'' = (phi(p)(D(0+)(alpha)u(0)))''' = 0, where 1 < alpha <= 2, 3 < beta <= 4 are real numbers are real numbers, D-0+(alpha), D-0+(beta) are the standard Caputo fractional derivatives, phi(p)(s) = vertical bar s vertical bar(p-2)s. p > 1, phi(-1)(p) = phi(q), 1/p + 1/q = 1, 0 <= gamma < 1, 0 <= h <= 1, lambda, mu > 0 are parameters a : (0, 1) -> [0, +infinity), and f : [ 0, +infinity) -> [0, +infinity) are continuous. By the properties of Green function and Schauder fixed point theorem, several existence and nonexistence results for positive solutions, in terms of the parameters.. and.. are obtained. The uniqueness of positive solution on the parameters lambda and mu is also studied. In the final section of this paper, we derive not only new but also interesting identities related special polynomials by which Caputo fractional derivative.

关键词： boundary value problems -- numerical solutions EXISTENCE theorems UNIQUENESS (Mathematics) FRACTIONAL calculus DIFFERENTIAL equations -- numerical solutions LAPLACIAN operator IDENTITIES (Mathematics) POLYNOMIALS

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On solutions of Fractional Order boundary value problems with Integral boundary Conditions in Banach Spaces

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JOURNAL OF FUNCTION SPACES AND APPLICATIONS 2013年第1期2013卷 1-13页

作者： Salem, Hussein A. H. Cichon, Mieczyslaw Univ Alexandria Fac Sci Dept Math Alexandria Egypt Taibah Univ Fac Sci Yanbu Saudi Arabia Adam Mickiewcz Univ Fac Math & Comp Sci PL-61614 Poznan Poland

The object of this paper is to investigate the existence of a class of solutions for some boundary value problems of fractional order with integral boundary conditions. The considered problems are very interesting and important from an application point of view. They include two, three, multipoint, and nonlocal boundary value problems as special cases. We stress on single and multivalued problems for which the nonlinear term is assumed only to be Pettis integrable and depends on the fractional derivative of an unknown function. Some investigations on fractional Pettis integrability for functions and multifunctions are also presented. An example illustrating the main result is given.

关键词： boundary value problems -- numerical solutions FRACTIONAL calculus boundary element methods BANACH spaces PETTIS integral NONLINEAR theories

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Asymptotic estimates for the solutions of boundary-value problems with initial jump for linear differential equations with small parameter in the coefficients of derivatives

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UKRAINIAN MATHEMATICAL JOURNAL 2013年第5期65卷 694-708页

作者： Kasymov, K. A. Nurgabyl, D. N. Uaisov, A. B. Kazakhstan Natl Univ Alma Ata Inst Math & Mech Alma Ata Kazakhstan Zhetysu State Univ Taldykorgan Kazakhstan

We establish asymptotic estimates for the solutions of singularly perturbed boundary-value problems with initial jumps.

关键词： ASYMPTOTIC expansions boundary value problems -- numerical solutions LINEAR differential equations PARAMETER estimation COEFFICIENTS (Statistics) DERIVATIVES (Mathematics) SINGULARITIES (Mathematics)

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