Auxiliary principle and three-step iterative algorithms for generalized set-valued strongly nonlinear mixed variational-like inequalities

An auxiliary principle technique to study a class of generalized set-valued strongly nonlinear mixed variational-like inequalities is extended. The existence and uniqueness of the solution of the auxiliary problem for the generalized set-valued strongly nonlinear mixed variational-like inequalities are proved, a novel and innovative three-step iterative algorithm to compute approximate solution is constructed, and the existence of the solution of the generalized set-valued strongly nonlinear mixed variational-like inequality is shown using the auxiliary principle iterative sequences generated by the algorithm technique. The convergence of three-step is also proved.

Status of research on non-conventional technology assisted single-point diamond turning

With the increasing use of difficult-to-machine materials in aerospace applications,machining requirements are becoming ever more rigorous.However,traditional single-point diamond turning(SPDT)can cause surface damage and tool wear.Thus,it is difficult for SPDT to meet the processing requirements,and it has significant limitations.Research indicates that supplementing SPDT with unconventional techniques can,importantly,solve problems due to the high cutting forces and poor surface quality for difficult-to-machine materials.This paper first introduces SPDT and reviews research into unconventional techniques for use with SPDT.The machining mechanism is discussed,and the main advantages and disadvantages of various methods are investigated.Second,hybrid SPDT is briefly described,which encompasses ultrasonic-vibration magnetic-field SPDT,ultrasonic-vibration laser SPDT,and ultrasonic-vibration cold-plasma SPDT.Compared with the traditional SPDT method,hybrid SPDT produces a better optical surface quality.The current status of research into unconventional techniques to supplement SPDT is then summarized.Finally,future development trends and the application prospects of unconventional assisted SPDT are discussed.

A Posteriori Error Estimate of Weak Galerkin FEM for Stokes Problem Using Auxiliary Subspace Techniques

Based on the auxiliary subspace techniques,a posteriori error estimator of nonconforming weak Galerkin finite element method(WGFEM)for Stokes problem in two and three dimensions is presented.Without saturation assumption,we prove that the WGFEM approximation error is bounded by the error estimator up to an oscillation term.The computational cost of the approximation and the error problems is considered in terms of size and sparsity of the system matrix.To reduce the computational cost of the error problem,an equivalent error problem is constructed by using diagonalization techniques,which needs to solve only two diagonal linear algebraic systems corresponding to the degree of freedom(d.o.f)to get the error estimator.Numerical experiments are provided to demonstrate the effectiveness and robustness of the a posteriori error estimator.

EXISTENCE AND ALGORITHM OF SOLUTIONS FOR GENERAL MULTIVALUED MIXED IMPLICIT QUASI-VARIATIONAL INEQUALITIES

A new class of general multivalued mixed implicit quasi-variational inequalities in a real Hilbert space was introduced, which includes the known class of generalized mixed implicit quasi-variational inequalities as a special case , introduced and studied by Ding Xie-ping . The auxiliary variational principle technique was applied to solve this class of general multivalued mixed implicit quasi-variational inequalities. Firstly, a new auxiliary variational inequality with a proper convex , lower semicontinuous , binary functional was defined and a suitable functional was chosen so that its unique minimum point is equivalent to the solution of such an auxiliary variational inequality . Secondly , this auxiliary variational inequality was utilized to construct a new iterative algorithm for computing approximate solutions to general multivalued mixed implicit quasi-variational inequalities . Here , the equivalence guarantees that the algorithm can generate a sequence of approximate solutions. Finally, the existence of solutions and convergence of approximate solutions for general multivalued mixed implicit quasi-variational inequalities are proved. Moreover, the new convergerce criteria for the algorithm were provided. Therefore, the results give an affirmative answer to the open question raised by M. A . Noor, and extend and improve the earlier and recent results for various variational inequalities and complementarity problems including the corresponding results for mixed variational inequalities, mixed quasi-variational inequalities and quasi-complementarity problems involving the single-valued and set- valued mappings in the recent literature .

Existence and Algorithm of Solutions for Generalized Mixed Implicit Quasi-Variational-Like Inequalities

By applying the auxiliary variational principle technique, the existence of solutions for a new class of generalized mixed implicit quasi-variational-like inequalities and the convergence criteria of a new iterative algorithm to compute approximate solutions are proved in Hilbert spaces. The obtained result is a improvement over and generalization of the main theorem proposed by Ding.

A variety of solitons on the oceans exposed by the Kadomtsev Petviashvili-modified equal width equation adopting different techniques

The diverse patterns of waves on the oceans yielded by the Kadomtsev Petviashvili-modified equal width(KP-mEW)equation are highlighted in this paper.Two recent established approaches such as the im-proved auxiliary equation technique and the enhanced rational(G'/G)-expansion scheme are utilized to construct wave solutions of the proposed governing model.Numerous rational,trigonometric,exponen-tial,and hyperbolic wave solutions bearing many free parameters are successfully acquired in appropriate form.The obtained solutions are plotted in various profiles as three-dimension,two-dimension,and con-tour to illustrate their physical appearances.The plotting outlines appear in the shapes of singular kink,anti-kink,kink,compacton,anti-compacton,bell,anti-bell,periodic,singular periodic etc.The computa-tional software Maple is used for plotting and checking the validity of the found solutions.This paper claims to be novel for generating new results regarding the earlier results.

Iterative Algorithm for Finding Approximate Solutions of a Class of Mixed Variational-like Inequalities

The purpose of this paper is to investigate the iterative algorithm for finding approximate solutions of a class of mixed variational-like inequalities in a real Hilbert space, where the iterative algorithm is presented by virtue of the auxiliary principle technique. On one hand, the existence of approximate solutions of this class of mixed variational-like inequalities is proven. On the other hand, it is shown that the approximate solutions converge strongly to the exact solution of this class of mixed variational-like inequalities.