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EN
This paper describes the influence of some parameters significant to biomass pyrolysis on the numerical solutions of the non-isothermal nth order distributed activation energy model (DAEM) using the Gamma distribution and discusses the special case for the positive integer value of the scale parameter (𝜆), i.e. the Erlang distribution. Investigated parameters are the integral upper limit, the frequency factor, the heating rate, the reaction order, and the shape and rate parameters of the Gamma distribution. Influence of these parameters has been considered for the determination of the kinetic parameters of the non-isothermal nth order Gamma distribution from the experimentally derived thermoanalytical data of biomass pyrolysis. Mathematically, the effect of parameters on numerical solution is also used for predicting the behaviour of the unpyrolysized fraction of biomass with respect to temperature. Analysis of the mathematical model is based upon asymptotic expansions, which leads to the systematic methods for efficient way to determine the accurate approximations. The proposed method, therefore, provides a rapid and highly effective way for estimating the kinetic parameters and the distribution of activation energies.
EN
This article is concerned with the study of the Borel summability of divergent power series solutions for certain singular first-order linear partial differential equations of nilpotent type. Our main purpose is to obtain conditions which coefficients of equations should satisfy in order to ensure the Borel summability of divergent solutions. We will see that there is a close affinity between the Borel summability of divergent solutions and global analytic continuation properties for coefficients of equations.
EN
This paper is devoted to the study of the following perturbed system of nonlinear functional equations (…) , where ε is a small parameter, aijk; bijk are the given real constants, Rijk; Sijk; Xijk : (…) are the given continuous functions and (…) are unknown functions. First, by using the Banach fixed point theorem, we find sufficient conditions for the unique existence and stability of a solution of (E). Next, in the case of (…) ; we investigate the quadratic convergence of (E). Finally, in the case of (…) and ε sufficiently small, we establish an asymptotic expansion of the solution of (E) up to order N + 1 in ε. In order to illustrate the results obtained, some examples are also given.
EN
We propose a novel approach to adaptive refinement in FEM based on local sensitivities for node insertion. To this end, we consider refinement as a continuous graph operation, for instance by splitting nodes along edges. Thereby, we introduce the concept of the topological mesh derivative for a given objective function. For its calculation, we rely on the first-order asymptotic expansion of the Galerkin solution of a symmetric linear second-order elliptic PDE. In this work, we apply this concept to the total potential energy, which is related to the approximation error in the energy norm. In fact, our approach yields local sensitivities for minimization of the energy error by refinement. Moreover, we prove that our indicator is equivalent to the classical explicit a posteriori error estimator in a certain sense. Numerical results suggest that our method leads to efficient and competitive adaptive refinement.
5
Content available remote A note on speed of convergence to the quasi-stationary distribution
EN
In this note we show that for Z being a birth and death process on Z or Brownian motion with drift and (…), the speed of convergence to the quasi-stationary distribution is of order 1/t. The corresponding version that X is the number of calls in M/M/1 queue or the reflected Brownian motion is also considered. The result is obtained by asymptotic expansions of some transition functions. For this we use some new asymptotic expansion of the Bessel function.
EN
We study the initial-boundary value problem for a nonlinear wave equation given by [...] where p > 2, q > l, K, lambda are given constants and uo, u1, F are given functions, the unknown function u(x,t) and the unknown boundary value P (t) satisfy the following nonlinear integral equation [...] where K1, alpha, beta are given constants and g, k arę given functions. In Part 1 we prove a theorem of existence and uniqueness of a weak solution (u, P) of problem (1), (2). The proof is based on the Faedo-Galerkin method associated with a priori estimates, weak convergence and compactness techniques. In Part 3 we obtain an asymptotic expansion of the solution (u, P) of the problem (1), (2) up to order N+1 in three small parameters K, lambda, K1.
EN
We give a generic framework to analyze the average-case running time for computing the so called recurrent properties for pairs of binary search trees. Recurrent properties are algorithms that operate on pairs of trees testing some characteristic on nodes by performing a preorder traversal on both trees. Analysis of recurrent properties using the probability model associated with randomly grown binary search trees leads to wave equations. We use a "normalized" integral equation as a pattern to model a specific wave equation and investigate the asymptotic behavior of its solution. This methodology is applied to some particular cases of recurrent properties like testing equality, detecting direct occurrences and clashes or pattern matching.
9
Content available remote Asymptotic formulas for unified elliptic-type integrals
EN
The object of this article is to present a unification and generalisation of certain families of elliptic-type integrals which were studied in a number of earlier works on the subject due to their importance for possible applications in certain problems arising in radiation physics and nuclear technology. The results obtained are of general character and include the investigations carried out by several authors including Srivastava and Siddiqi, Kalla and Tuan, Al-Zamel et al and Saxena et al.
EN
Singularly perturbed linear Volterra integral equations are solved in this paper. To improve the results which has been published earlier, formal solutions of systems of equations are determined and rigorously proved to be asymptotic to the exact solutions.
12
Content available Topological derivative for optimal control problems
EN
The topological derivative is introduced for the extremal values of cost functionals for control problems. The optimal control problem considered in the paper is defined for the elliptic equation which models the deflection of an elastic membrane. The derivative measures the sensitivity of the optimal value of the cost with respect to changes in topology. A change in topology means removing a small ball from the interior of the domain of integration. The topological derivative can be used for obtaining the numerical solutions of the shape optimization problems.
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