Applications of Mathematics最新文献

Conjugate gradient methods are widely used for solving large-scale unconstrained optimization problems, because they do not need the storage of matrices. Based on the self-scaling memoryless Broyden-Fletcher-Goldfarb-Shanno (SSML-BFGS) method, new conjugate gradient algorithms CG-DESCENT and CGOPT have been proposed by W. Hager, H. Zhang (2005) and Y. Dai, C. Kou (2013), respectively. It is noted that the two conjugate gradient methods perform more efficiently than the SSML-BFGS method. Therefore, C. Kou, Y. Dai (2015) proposed some suitable modifications of the SSML-BFGS method such that the sufficient descent condition holds. For the sake of improvement of modified SSML-BFGS method, in this paper, we present an efficient SSML-BFGS-type three-term conjugate gradient method for solving unconstrained minimization using Ford-Moghrabi secant equation instead of the usual secant equations. The method is shown to be globally convergent under certain assumptions. Numerical results compared with methods using the usual secant equations are reported.

We introduce a new scaling parameter for the Dai-Kou family of conjugate gradient algorithms (2013), which is one of the most numerically efficient methods for unconstrained optimization. The suggested parameter is based on eigenvalue analysis of the search direction matrix and minimizing the measure function defined by Dennis and Wolkowicz (1993). The corresponding search direction of conjugate gradient method has the sufficient descent property and the extended conjugacy condition. The global convergence of the proposed algorithm is given for both uniformly convex and general nonlinear objective functions. Also, numerical experiments on a set of test functions of the CUTER collections and the practical problem of the manipulator of robot movement control show that the proposed method is effective.

We consider pressure-driven flow between adjacent surfaces, where the fluid is assumed to have constant density. The main novelty lies in using implicit algebraic constitutive relations to describe the fluid’s response to external stimuli, enabling the modeling of fluids whose responses cannot be accurately captured by conventional methods. When the implicit algebraic constitutive relations cannot be solved for the Cauchy stress in terms of the symmetric part of the velocity gradient, the traditional approach of inserting the expression for the Cauchy stress into the equation for the balance of linear momentum to derive the governing equation for the velocity becomes inapplicable. Instead, a non-standard system of first-order equations governs the flow. This system is highly complex, making it important to develop simplified models. Our primary contribution is the development of a framework for achieving this. Additionally, we apply our findings to a fluid that exhibits an S-shaped curve in the shear stress versus shear rate plot, as observed in some colloidal solutions.

We present a precise anisotropic interpolation error estimate for the Morley finite element method (FEM) and apply it to fourth-order elliptic equations. We do not impose the shape-regularity mesh condition in the analysis. Anisotropic meshes can be used for this purpose. The main contributions of this study include providing a new proof of the term consistency. This enables us to obtain an anisotropic consistency error estimate. The core idea of the proof involves using the relationship between the Raviart-Thomas and Morley finite-element spaces. Our results indicate optimal convergence rates and imply that the modified Morley FEM may be effective for errors.

We construct a new initial data to prove the ill-posedness of both Navier-Stokes and Euler equations in weaker Besov spaces in the sense that the solution maps to these equations starting from u₀ are discontinuous at t = 0.

Contact and Lie point symmetries of a certain class of second order differential equations using the Lie symmetry theory are obtained. Generators of these symmetries are used to obtain first integrals and exact solutions of the equations. This class of equations is transformed into the so-called generalized Lane-Emden equations of the second kind

Then we consider two types of functions g(x) and present first integrals and exact solutions of the Lane-Emden equation for them. One of the considered cases is new.

The problem of an approximate solution of thermo-viscous fluid flow in a porous slab bounded between two impermeable parallel plates in relative motion is examined in this paper. The two plates are kept at two different temperatures and the flow is generated by a constant pressure gradient together with the motion of one of the plates relative to the other. The velocity and temperature distributions have been obtained by a four-stage algorithm approach. It is worth mentioning that reverse effects are noticed on velocity and temperature distributions. These effects can be attributed to Darcy’s friction offered by the medium. The approximation results obtained in the present paper are in good agreement with the earlier numerical results of thermo-viscous fluid flows in plane geometry.

We investigate a generalized class of fractional hemivariational inequalities involving the time-fractional aspect. The existence result is established by employing the Rothe method in conjunction with the surjectivity of multivalued pseudomonotone operators and the properties of the Clarke generalized gradient. We are also exploring a numerical approach to address the problem, utilizing both spatially semi-discrete and fully discrete finite elements, along with a discrete approximation of the fractional derivative. All these results are applied to the analysis and numerical approximations of a frictional contact model that describes the quasi-static contact between a viscoelastic body and a solid foundation. The constitutive relation is modeled using the fractional Kelvin-Voigt law. The contact and friction are described by the subdifferential boundary conditions. The variational formulation of this problem leads to a fractional hemivariational inequality. The error estimates for this problem are derived. Finally, here’s a second concrete example to illustrate the application. We propose a frictional contact model that incorporates normal compliance and Coulomb friction to describe the quasi-static contact between a viscoelastic body and a solid foundation.