Homotopy–perturbation method for pure nonlinear differential equation

In this paper, the homotopy–perturbation method proposed by J.-H. He is adopted for solving pure strong nonlinear second-order differential equation. For the oscillatory differential equation the initial approximate solution is assumed in the form of Jacobi elliptic function and the forementioned method is used for obtaining of the approximate analytic solution. Two types of differential equations are considered: with strong cubic and strong quadratic nonlinearity. The obtained solution is compared with exact numerical one. The difference between these solutions is negligible for a long time period. The method is found to work extremely well in the examples, but the theoretical reasons are not yet clear.     

Solution of Delay Differential Equation by Means of Homotopy Analysis Method

The algorithm of approximate analytical solution for delay differential equations (DDE) is obtained via homotopy analysis method (HAM) and modified homotopy analysis method (MHAM). Various examples of linear, nonlinear and system of initial value problems of DDE are solved and the results obtained show that these algorithms are accurate and efficient for the DDE. The convergence of this algorithm is also proved.     

Well-posedness of the transport equation by stochastic perturbation

We consider the linear transport equation with a globally Hölder continuous and bounded vector field, with an integrability condition on the divergence. While uniqueness may fail for the deterministic PDE, we prove that a multiplicative stochastic perturbation of Brownian type is enough to render the equation well-posed. This seems to be the first explicit example of a PDE of fluid dynamics that becomes well-posed under the influence of a (multiplicative) noise. The key tool is a differentiable stochastic flow constructed and analyzed by means of a special transformation of the drift of Itô-Tanaka type.     

Fast Linear Homotopy to Find Approximate Zeros of Polynomial Systems

We prove a new complexity bound, polynomial on the average, for the problem of finding an approximate zero of systems of polynomial equations. The average number of Newton steps required by this method is almost linear in the size of the input (dense encoding). We show that the method can also be used to approximate several or all the solutions of non-degenerate systems, and prove that this last task can be done in running time which is linear in the Bézout number of the system and polynomial in the size of the input, on the average.     

The heat kernel of the Laplacian defined on a uniform grid

We adapt a method originally developed by E.B. Davies for second order elliptic operators to obtain an upper heat kernel bound for the Laplacian defined on a uniform grid on the plane.     

Nonstationary nonlinear nonlocal problem of radiative–conductive heat transfer in a system of opaque bodies with properties depending on the radiation frequency

We consider a nonstationary nonlinear initial-boundary value problem governing radiative-conductive heat transfer in a periodic system of grey heat shields. The existence and uniqueness of a weak solution and a regular weak solution are established. Estimates for the solutions in terms of the data of the problem are obtained. Bibliography: 36 titles.     

Stationary nonlinear nonlocal problem of radiative–conductive heat transfer in a system of opaque bodies with properties depending on the radiation frequency

We consider a stationary nonlinear nonlocal boundary value problem governing radiative–conductive heat transfer in a system of opaque bodies with surfaces whose properties depend on the radiation frequency. We establish the existence and uniqueness of a solution and prove a comparison theorem. Results on the exponential summability and boundedness of the solution are also obtained. Bibliography: 32 titles.     

Hypercontractivity, Nash inequalities and subordination for classes of nonlinear semigroups

A suitable notion of hypercontractivity for a nonlinear semigroup {T _t } is shown to imply Nash-type inequalities for its generator H, provided a subhomogeneity property holds for the energy functional (u,Hu). We use this fact to prove that, for semigroups generated by operators of p-Laplacian-type, hypercontractivity implies ultracontractivity. Then we introduce the notion of subordinated nonlinear semigroups when the corresponding Bernstein function is f(x)=x ^α , and write an explicit formula for the associated generator. It is shown that hypercontractivity still holds for the subordinated semigroup and, hence, that Nash-type inequalities hold as well for the subordinated generator.     

On the complete integrability and linearization of a Burgers– Korteweg–de Vries-type nonlinear equation

In this work, on the basis of the Bogolyubov–Prykarpats’kyi gradient–holonomic algorithm for the investigation of the integrability of nonlinear dynamical systems on functional manifolds, the exact linearization of a Burgers–Korteweg–de Vries-type nonlinear dynamical system is established. As a result, we describe the linear structure of the space of solutions and show its relation to the convexity of certain functional subsets. The bi-Hamiltonian property of the Burgers–Korteweg–de Vries dynamical system is also established, and the infinite hierarchy of functionally independent invariants is constructed.     

A new modification of He’s homotopy perturbation method for rapid convergence of nonlinear undamped oscillators

In this paper we present a new efficient modification of the homotopy perturbation method with x ³ force nonlinear undamped oscillators for the first time that will accurate and facilitate the calculations. The He’s homotopy perturbation method is modified by adding a term to linear operator depends on the equation and boundary conditions. We find that this modified homotopy perturbation method works very well for the wide range of time and boundary conditions for nonlinear oscillator. Only two or three iteration leads to high accuracy of the solutions. We then conduct a comparative study between the new modification and the homotopy perturbation method for strongly nonlinear oscillators. Numerical illustrations are investigated to show the accurate of the techniques. The new modified method accelerates the rapid convergence of the solution, reduces the error solution and increases the validity range. The new modification introduces a promising tool for many nonlinear problems.     

Approximate performance analysis of CONWIP disciplines in unreliable non homogeneous transfer lines

For a given choice of the maximum allowable total storage parameter, the performance of constant work-in-process (CONWIP) disciplines in unreliable transfer lines subjected to a constant rate of demand for parts, is characterized via a tractable approximate mathematical model. For a (n−1) machines CONWIP loop, the model consists of n multi-state machine single buffer building blocks, separately solvable once a total of (n−1)² unknown constants shared by the building blocks are initialized. The multi-state machine is common to all building blocks, and its n discrete states approximate the joint operating state of the machines within the CONWIP loop; each of the first (n−1) blocks maps into a single internal buffer dynamics, while the nth building block characterizes total work-in-process (wip) dynamics. The blocks correspond to linear n component state equations with boundary conditions. The unknown (shared) constants in the block dynamics are initialized and calculated by means of successive iterations. The performance estimates of interest—mean total wip, and probability of parts availability at the end buffer in the loop—are obtained from the model and validated against the results of Monte Carlo simulations.     

Rates of convergence for the homogenization of fully nonlinear uniformly elliptic pde in random media

We establish a logarithmic-type rate of convergence for the homogenization of fully nonlinear uniformly elliptic second-order pde in strongly mixing media with similar, i.e., logarithmic, decorrelation rate. The proof consists of two major steps. The first, which is actually the only place in the paper where probability plays a role, establishes the rate for special (quadratic) data using the methodology developed by the authors and Wang to study the homogenization of nonlinear uniformly elliptic pde in general stationary ergodic random media. The second is a general argument, based on the new notion of δ-viscosity solutions which is introduced in this paper, that shows that rates known for quadratic can be extended to general data. As an application of this we also obtain here rates of convergence for the homogenization in periodic and almost periodic environments. The former is algebraic while the latter depends on the particular equation.     

Theoretical analysis of an optimal control problem of conductive–convective–radiative heat transfer

The problem of optimal heat removal from a three-dimensional domain is considered. The specific of the study consist in accounting for the radiative heat transfer. The so-called P₁ approximation of the radiative heat transfer equation is used, which reduces the model to a nonlinear elliptic system. A problem of optimal boundary control of this system is considered. The solvability of the control problem is proved, and necessary optimality conditions of first order are derived. Examples of non-singularity of these conditions are given.     

Approximate analytical solutions of reaction–diffusion equations with exponential source term: Homotopy perturbation method (HPM)

In this letter, the solutions of some nonlinear differential equations have been obtained by means of the homotopy perturbation method (HPM). Applications of the homotopy method to some nonlinear reaction–diffusion equations with exponential source term show rapid convergence of the sequence constructed by this method to the exact solutions.     

Using aggregate fill rate for dynamic scheduling of multi-class systems

For dynamic scheduling of multi-class systems where backorder cost is incurred per unit backordered regardless of the time needed to satisfy backordered demand, the following models are considered: the cost model to minimize the sum of expected average inventory holding and backorder costs and the service model to minimize expected average inventory holding cost under an aggregate fill rate constraint. Use of aggregate fill rate constraint in the service model instead of an individual fill rate constraint for each class is justified by deriving equivalence relations between the considered cost and service models. Based on the numerical investigation that the optimal policy for the cost model is a base-stock policy with switching curves and fixed base-stock levels, an alternative service model is considered over the class of base-stock controlled dynamic scheduling policies to minimize the total inventory (base-stock) investment under an aggregate fill rate constraint. The policy that solves this alternative model is proposed as an approximation of the optimal policy of the original cost and the equivalent service models. Very accurate heuristics are devised to approximate the proposed policy for given base-stock levels. Comparison with base-stock controlled First Come First Served (FCFS) and Longest Queue (LQ) policies and an extension of LQ policy (Δ policy) shows that the proposed policy performs much better to solve the service models under consideration, especially when the traffic intensity is high.     

The computation and perturbation analysis for weighted group inverse of rectangular matrices

Cen (Math. Numer. Sin. 29(1):39–48, 2007) has defined a weighted group inverse of rectangular matrices. Let A∈C ^m×n ,W∈C ^n×m , if X∈C ^m×n satisfies the system of matrix equations $$(W_{1})\quad AWXWA=A,\quad\quad (W_{2})\quad XWAWX=X,\quad\quad (W_{3})\quad AWX=XWA$$ X is called the weighted group inverse of A with W, and denoted by A _W ^# . In this paper, we will study the algebra perturbation and analytical perturbation of this kind weighted group inverse A _W ^# . Under some conditions, we give a decomposition of B _W ^# ?A _W ^# . As a results, norm estimate of ‖B _W ^# ?A _W ^# ‖ is presented (where B=A+E). An expression of algebra of perturbation is also obtained. In order to compute this weighted group inverse with ease, we give a new representation of this inverse base on Gauss-elimination, then we can calculate this weighted inverse by Gauss-elimination. In the end, we use a numerical example to show our results.     

Nonsmooth exclusion test for finding all solutions of nonlinear equations

A new approach is proposed for finding all real solutions of systems of nonlinear equations with bound constraints. The zero finding problem is converted to a global optimization problem whose global minima with zero objective value, if any, correspond to all solutions of the original problem. A branch-and-bound algorithm is used with McCormick’s nonsmooth convex relaxations to generate lower bounds. An inclusion relation between the solution set of the relaxed problem and that of the original nonconvex problem is established which motivates a method to generate automatically, starting points for a local Newton-type method. A damped-Newton method with natural level functions employing the restrictive monotonicity test is employed to find solutions robustly and rapidly. Due to the special structure of the objective function, the solution of the convex lower bounding problem yields a nonsmooth root exclusion test which is found to perform better than earlier interval-analysis based exclusion tests. Both the componentwise Krawczyk operator and interval-Newton operator with Gauss-Seidel based root inclusion and exclusion tests are also embedded in the proposed algorithm to refine the variable bounds for efficient fathoming of the search space. The performance of the algorithm on a variety of test problems from the literature is presented, and for most of them, the first solution is found at the first iteration of the algorithm due to the good starting point generation.     

Existence and iterative algorithm of solutions for a system of generalized set-valued strongly nonlinear mixed variational-like inequalities problems in Banach spaces

In this paper, we introduce and study a new system of generalized set-valued strongly nonlinear mixed variational-like inequalities problems and its related auxiliary problems in reflexive Banach spaces. The auxiliary principle technique is applied to study the existence and iterative algorithm of solutions for the system of generalized set-valued strongly nonlinear mixed variational-like inequalities problems. Firstly, we prove the existence and uniqueness of solutions of the auxiliary problems for the system of generalized set-valued strongly nonlinear mixed variational-like inequalities problems. Secondly, an iterative algorithm for solving the system of generalized set-valued strongly nonlinear mixed variational-like inequalities problems is constructed by using this existence and uniqueness result. Finally, we show the existence of solutions of the system of generalized set-valued strongly nonlinear mixed variational-like inequalities problems and discuss the convergence analysis of this algorithm. These results improve, unify and generalize many corresponding known results given in literatures.