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1
Content available An intial-value technique for self-adjoint singularly perturbed two-point boundary value problems
Padmaja P., Aparna P., Gorla R. S. R.
International Journal of Applied Mechanics and Engineering
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2020
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Vol. 25, no. 1
106--126
EN
In this paper, we present an initial value technique for solving self-adjoint singularly perturbed linearvboundary value problems. The original problem is reduced to its normal form and the reduced problem is converted to first order initial value problems. This replacement is significant from the computational point of view. The classical fourth order Runge-Kutta method is used to solve these initial value problems. This approach to solve singularly perturbed boundary-value problems is numerically very appealing. To demonstrate the applicability of this method, we have applied it on several linear examples with left-end boundary layer and rightend layer. From the numerical results, the method seems accurate and solutions to problems with extremely thin boundary layers are obtained.
2
Content available remote Theory of residual stresses with application to an arterial geometry
Klarbring A., Olsson T., Stalhand J.
Archives of Mechanics
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2007
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Vol. 59, nr 4-5
341-364
EN
This paper presents a theory of residual stresses, with applications to biomechanics, especially to arteries. For a hyperelastic material, we use an initial local deformation tensor K as a descriptor of residual strain. This tensor, in general, is not the gradient of a global deformation, and a stress-free reference configuration, denoted ..., therefore, becomes incompatible. Any compatible reference configuration ... will, in general, be residually stressed. However, when a certain curvature tensor vanishes, there actually exists a compatible and stress-free configuration, and we show that the traditional treatment of residual stresses in arteries, using the opening-angle method, relates to such a situation. Boundary value problems of nonlinear elasticity are preferably formulated on a fixed integration domain. For residually stressed bodies, three such formulations naturally appear: (i) a formulation relating to ... with a non-Euclidean metric structure; (ii) a formulation relating to ... with a Euclidean metric structure; and (iii) a formulation relating to the incompatible configuration ... . We state these formulations, show that (i) and (ii) coincide in the incompressible case, and that an extra term appears in a formulation on ... , due to the incompatibility.
3
Content available remote Modelling and Investigation of Temperature Influence on Normal Mode of the Layer Oscillations
Nahirnyj T., Tchervinka K.
Vibrations in Physical Systems
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2004
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Vol. 21
279--282
EN
In this paper the boundary value problems for local gradient body are formulated and investigated. Using averaging over oscillation period operation there is obtained the set of differential equations for determining the averaged and wave components of thermo-elastic fields. The methods for approximate integration of this set are proposed (using expansion over the small parameter of the problem). The normal mode layer oscillations are studied for various mechanical conditions at the layer surfaces (fixed surfaces, one surface is fixed, both surfaces are free). The analysis of equations for normal mode frequency is carried out. The frequencies dependences on temperature and parameters being characteristics of interface nonhomogeneity are studied.
4
Content available remote Generalized Riemann-Hilbert transmission and boundary value problems, Fredholm pairs and bordisms
Bojarski B., Weber A.
Bulletin of the Polish Academy of Sciences. Mathematics
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2002
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Vol. 50, no 4
479-496
EN
We present classical and generalized Riemann-Hilbert problem in several contexts arising from K-theory and bordism theory. The language of Fredholm pairs turns out to be useful and unavoidable. We propose an abstract formulation of a notion of bordism in the context of Hilbert spaces equipped with splittings.
5
Content available remote Green functions and periodic solution to the m-parabolic equation with boundary-value conditions of Riquier type
Barański F., Zdankiewicz Z.
Fasciculi Mathematici
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2001
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Nr 32
5-13
EN
The subject of the paper is the construction of the periodic solutions to the w-caloric equation P"'u(x,t) = 0, P=D(2)*(2)-D, P(2) =P(P), Pm = P(P(m~l) in the strip (mathemical formula), satisfying the periodic boundary-value conditions (mathematical formula), where h(11)(t), h1,2(t) are the periodic functions with the period p > 0.
6
Content available remote Mixed algorithm for solving boundary value problem
Paláncz B., Popper G.
Computer Assisted Mechanics and Engineering Sciences
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1999
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Vol. 6, No. 3-4
479-486
EN
Symbolic computation has been applied to Runge-Kutta technique in order to solve two-point boundary value problem. The unknown initial values are considered as symbolic variables, therefore they will appear in a system of algebraic equations, after the integration of the ordinary differential equations. Then this algebraic equation system can be solved for the unknown initial values and substituted into the solution. Consequently, only one integration pass is enough to solve the problem instead of using iteration technique like shooting-method. This procedure is illustrated by solving the boundary value problem of the mechanical analysis of a liquid storage tank. Computation was carried out by MAPLE V. Power Edition package.
7
Content available remote Filtration in cohesive soils : numerical approach
Schaefer R., Sędziwy S.
Computer Assisted Mechanics and Engineering Sciences
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1999
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Vol. 6, No. 1
15-26
EN
Paper presents a numerical method for solving the initial boundary-value problem for a certain quasilinear parabolic equation describing the low velocity filtration problem. The convergence of the method is proved.
8
Content available remote Filtration in cohesive soils : mathematical model
Schaefer R., Sędziwy S.
Computer Assisted Mechanics and Engineering Sciences
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1999
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Vol. 6, No. 1
1-13
EN
The paper discusses the physical basis of the process of filtration of water in a case of very low velocities and presents the mathematical model of the process, based on a new constitutive formula. The existence and uniqueness of a weak solution to the resulting nonhomogeneous initial boundary-value problem is then proven.
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