Impulsive boundary value problem for nonlinear differential equations of fractional order

Axisymmetric deformation of a three-layer circular plate under repeated alternating loading from the plastic region by a local load is considered. To describe kinematics of asymmetrical on the thickness of the plate pack is adopted the hypothesis of a broken line. In a thin elastic-plastic load-bearing layers are used the hypothesis of Kirchhoff. A non-linearly elastic relatively thick filler is incompressible in thickness. It is taken to be a hypothesis of Tymoshenko regarding the straightness and the incompressibility of the deformed normals with linear approximation of the displacements through the thickness layer. The work of the filler in the tangential direction is taken into account. The physical relations of stress-strain relations correspond to the theory of small elastic-plastic deformations. The effect of heat flow is taken into account. The temperature field in the plate was calculated by the formula obtained by averaging the thermophysical parameters over the thickness of the package. The system of differential equations of equilibrium under loading of the plate from the natural state is obtained by the Lagrange variational method. Boundary conditions on the plate contour are formulated. The solution of the corresponding boundary value problem is reduced to finding the three desired functions: deflection, shear and radial displacement of the shear surface of the filler. A non-uniform system of ordinary nonlinear differential equations is written for these functions. Its analytical iterative solution is obtained in Bessel functions by the method of elastic solutions of Ilyushin. In case of repeated alternating loading of the plate, the solution of the boundary value problem is constructed using the theory of variable loading of Moskvitin. In this case, the hypothesis of similarity of plasticity functions at each loading step is used. Their analytical form is taken independent of the point of unloading. However, the material constants included in the approximation formulas will be different. The cyclic hardening of the material of the bearing layers is taken into account. The parametric analysis of the obtained solutions under different boundary conditions in the case of a local load distributed in a circle is carried out. The influence of temperature and nonlinearity of layer materials on the displacements in the plate is numerically investigated.

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Anti-periodic fractional boundary value problems for nonlinear differential equations of fractional order

Advances in Difference Equations ◽

10.1186/1687-1847-2012-116 ◽

2012 ◽

Vol 2012 (1) ◽

Cited By ~ 6

Author(s):

Fang Wang ◽

Zhenhai Liu

Keyword(s):

Differential Equations ◽

Boundary Value Problems ◽

Fractional Order ◽

Nonlinear Differential Equations ◽

Boundary Value ◽

Fractional Boundary Value Problems

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A boundary value problem on the half-line for higher-order nonlinear differential equations

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210508000577 ◽

2009 ◽

Vol 139 (5) ◽

pp. 1017-1035 ◽

Cited By ~ 2

Author(s):

Ch. G. Philos

Keyword(s):

Differential Equations ◽

Boundary Value Problem ◽

Ordinary Differential Equations ◽

Existence And Uniqueness ◽

Nonlinear Differential Equations ◽

Boundary Value ◽

Higher Order ◽

Half Line ◽

Nonlinear Ordinary Differential Equations ◽

Specific Class

This article is devoted to the study of the existence of solutions as well as the existence and uniqueness of solutions to a boundary-value problem on the half-line for higher-order nonlinear ordinary differential equations. An existence result is obtained by the use of the Schauder–Tikhonov theorem. Furthermore, an existence and uniqueness criterion is established using the Banach contraction principle. These two results are applied, in particular, to the specific class of higher-order nonlinear ordinary differential equations of Emden–Fowler type and to the special case of higher-order linear ordinary differential equations, respectively. Moreover, some (general or specific) examples demonstrating the applicability of our results are given.

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