Computing bifurcations behavior of mixed type singular time-fractional partial integrodifferential equations of Dirichlet functions types in hilbert space with error analysis

In this article, we propose and analyze a computational method for the numerical solutions of mixed type singular time-fractional partial integrodifferential equations of Dirichlet functions types. The method provide appropriate representation of the solutions in infinite series formula with accurately computable structures. By interrupting the n-term of exact solutions, numerical solutions of linear and nonlinear singular time-fractional equations of nonhomogeneous function type are studied from mathematical viewpoint. The utilized results show that the present method and simulated annealing provide a good scheduling methodology to such singular integrodifferential equations.

Interaction Between an Edge Dislocation and a Crack With Surface Elasticity

We undertake an analytical study of the interaction of an edge dislocation with a finite crack whose faces are assumed to have separate surface elasticity. The surface elasticity on the faces of the crack is described by a version of the continuum-based surface/interface theory of Gurtin and Murdoch. By using the Green's function method, we obtain a complete exact solution by reducing the problem to three Cauchy singular integrodifferential equations of the first-order, which are solved by means of Chebyshev polynomials and a collocation method. The correctness of the solution is rigorously verified by comparison with existing analytical solutions. Our analysis shows that the stresses and the image force acting on the edge dislocation are size-dependent and that the stresses exhibit both the logarithmic and square root singularities at the crack tips when the surface tension is neglected.

Quasilinearization for the Boundary Value Problem of Second-Order Singular Differential System

We study the boundary value problems of second-order singular differential equations. At first, we reduce the BVPs to initial value problems of first-order singular integrodifferential equations and then we employ the quasilinearization method in studying the IVPs and obtain two monotone iterative sequences, which converge uniformly and quadratically to the unique solution of the IVPs. Finally, we get the similar result for the given BVPs.

Numerical Solution of Weakly Singular Integrodifferential Equations on Closed Smooth Contour in Lebesgue Spaces

The present paper deals with the justification of solvability conditions and properties of solutions for weakly singular integro-differential equations by collocation and mechanical quadrature methods. The equations are defined on an arbitrary smooth closed contour of the complex plane. Error estimates and convergence for the investigated methods are established in Lebesgue spaces.

Excitation of shear waves in a piezoceramic medium with partially electrodized tunnel openings strengthened by a rigid stringer

An antiplane mixed boundary electroelasticity of a stationary wave process in an unbounded piezoceramic medium containing tunnel heterogeneities of opening or thin rigid inclusion (stringer) type is considered. The excitation of an electric field occurs at the expense of differences of electric potentials applied to the system of electrodes located on a free from stresses opening surface. Using the correct integral representations of the solutions, the boundary problem is reduced to the system of singular integrodifferential equations of the second type with resolvent kernels. The results of the parametric investigations characterizing the behavior of the components of an electroelastic field in the medium area and on the opening surface are given. A system of singular integrodifferential equations is obtained for investigation of a conjugated electroelastic field in a piezomedium with a tunnel along the material axis opening a rigid curvilinear inclusion, excited by a system of active electrodes, located on the opening surface. The solvable system of equations of the boundary problem is reduced to two differential equations of Helmhöltz and Laplace with respect to the amplitude of shear displacement and some auxiliary functions. The obtained system is solved numerically by a special scheme of the method of quadrature.