The paper consists of two parts. The first part deals with the solvability of new boundary-value problems for the model quasihyperbolic equations (-1)(p)D(l)(2p)u = Au + f(x,t), where p > 1, for a self-adjoint seco...
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The paper consists of two parts. The first part deals with the solvability of new boundary-value problems for the model quasihyperbolic equations (-1)(p)D(l)(2p)u = Au + f(x,t), where p > 1, for a self-adjoint second-order elliptic operator A. For the problems under study, the existence and uniqueness theorems are proved for regular solutions. In the second part, the results obtained in the first part are somewhat sharpened and generalized.
Closed-form solutions are obtained for some non-stationary boundary-value problems of filtration dynamics in fractured-porous formations, posed within the framework of fractional-differential mathematical models, taki...
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Closed-form solutions are obtained for some non-stationary boundary-value problems of filtration dynamics in fractured-porous formations, posed within the framework of fractional-differential mathematical models, taking into account the space-time nonlocality of the process. The mathematical models of anomalous filtration dynamics are formulated using the Hilfer or Caputo derivatives with respect to the time variable and the Riemann-Liouville derivative with respect to the geometric variable. Along with direct filtration problems, the authors also consider the inverse boundary-value problem of determining the unknown source function that depends only on the geometric variable. Conditions of the existence of regular solutions to the considered problems are given.
The first fundamental problem for a nonlinear anisotropic body containing an arbitrary crack with a fracture-process zone near its front is stated in terms of the covariant components of the displacement vector. It is...
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The first fundamental problem for a nonlinear anisotropic body containing an arbitrary crack with a fracture-process zone near its front is stated in terms of the covariant components of the displacement vector. It is assumed that the body is described by a tensor linear constitutive equation. This equation is derived from the proposed relations between the covariant components of the strain tensor and the contravariant components of the stress tensor, which generalize Reiner's relations. These relations are analyzed from the point of view of the first and second laws of thermodynamics. As a result, the algebraic invariants of the stress and tensors that appear in the constitutive equations are related. In the case of plane stress state, a system of equations for discretized variables is derived. A method for solving this system is proposed.
Closed-form solutions are obtained to some boundary-value problems of fractional-differential filtration-consolidation dynamics with respect to the non-classical mathematical model taking into account the space-time n...
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Closed-form solutions are obtained to some boundary-value problems of fractional-differential filtration-consolidation dynamics with respect to the non-classical mathematical model taking into account the space-time nonlocality of the process. This mathematical model is formulated using the Caputo-Fabrizio derivative for the time variable and the Riemann-Liouville derivative for the geometric variable. Along with the direct consolidation problem for a soil mass of finite thickness, the inverse boundary-value problems are considered to determine the unknown source functions that only depend on the geometric or time variable. Conditions for the existence of regular solutions to the considered problems are given.
The homogeneous differential equations of Donnell’s theory of thin cylindrical shells are integrated. Expressions are obtained in closed form for the displacements, membrane stresses, moments, and shear forces.
The homogeneous differential equations of Donnell’s theory of thin cylindrical shells are integrated. Expressions are obtained in closed form for the displacements, membrane stresses, moments, and shear forces.
The paper is devoted to local and nonlocal boundary-value problems for a loaded differential equation with variable coefficients and the Gerasimov-Kaputo fractional derivative. To solve the considered problems we obta...
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The paper is devoted to local and nonlocal boundary-value problems for a loaded differential equation with variable coefficients and the Gerasimov-Kaputo fractional derivative. To solve the considered problems we obtain a priori estimates in differential and difference interpretations, from which follow the uniqueness and stability of solution with respect to initial data and the right-hand side, as well as convergence of a solution to the difference problem to a solution of the differential problem.
In this paper, we consider the following boundary-value problems for second-order three-point nonlinear impulsive integrodifferential equation of mixed type in a real Banach space x ''(t) + f(t, x(t), x'(t...
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In this paper, we consider the following boundary-value problems for second-order three-point nonlinear impulsive integrodifferential equation of mixed type in a real Banach space x ''(t) + f(t, x(t), x'(t), (Ax)(t), (Bx)(t)) = theta, t is an element of J, t not equal t(k), Delta x vertical bar(t=tk) = I-k(x(t(k))), Delta x'vertical bar(t=tk) = (I) over bar (k) (x(t(k)), x'(t(k))), k = 1, 2, ..., m, x(0) = theta, x(1) = rho x(eta), where theta is the zero element of E, (Ax)(t) = integral(t)(0) g(t,s)x(s)ds, (Bx)(t) = integral(1)(0)h(t, s)x(s)ds, g is an element of C[D, R+], D = {(t, s) is an element of J x J : t >= s}, h is an element of C[J x J, R], and Delta x vertical bar(t=tk) denotes the jump of x(t) at t = t(k), Delta x'vertical bar(t=tk) denotes the jump of x'(t) at t = t(k). Some new results are obtained for the existence and multiplicity of positive solutions of the above problems by using the fixed-point index theory and fixed-point theorem in the cone of strict set contraction operators. Meanwhile, an example is worked out to demonstrate the main results. (C) 2007 Elsevier Ltd. All rights reserved.
boundary-value problems with generalized nonlinear boundary conditions could be reduced to an equivalent finite-dimensional problem. This reduction can be achieved with a class of projection operators that includes su...
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boundary-value problems with generalized nonlinear boundary conditions could be reduced to an equivalent finite-dimensional problem. This reduction can be achieved with a class of projection operators that includes subspace consisting of splines. This class of approximation of subspaces possesses properties useful for the analysis of existence of solutions to boundary-value problems with non-linear boundary conditions. Thus, a result based on topological degree arguments which establishes sufficient condition for the existence of a solution to boundary-value problems based on the existence of a Galerkin approximation is presented.
A B-spline collocation method is presented for nonlinear singularly-perturbed boundary-value problems with mixed boundary conditions. The quasilinearization technique is used to linearize the original nonlinear singul...
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A B-spline collocation method is presented for nonlinear singularly-perturbed boundary-value problems with mixed boundary conditions. The quasilinearization technique is used to linearize the original nonlinear singular perturbation problem into a sequence of linear singular perturbation problems. The B-spline collocation method on piecewise uniform mesh is derived for the linear case and is used to solve each linear singular perturbation problem obtained through quasilinearization. The fitted mesh technique is employed to generate a piecewise uniform mesh, condensed in the neighborhood of the boundary layers. The convergence analysis is given and the method is shown to have second-order uniform convergence. The stability of the B-spline collocation system is discussed. Numerical experiments are conducted to demonstrate the efficiency of the method.
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