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Error analysis and numerical solution of Burgers-Huxley equation using 3-scale Haar wavelets

ENGINEERING WITH COMPUTERS 2022年第1期38卷 3-11页

作者： Shukla, Shitesh Kumar, Manoj Motilal Nehru Natl Inst Technol Allahabad Dept Math Prayagraj 211004 UP India

In this article, we develop an efficient and accurate numerical scheme based on the Crank-Nicolson finite difference method and Haar wavelet analysis to evaluate the numerical solution of the Burgers-Huxley equation. The present method is extended form of Haar wavelet 2D scaling which shows that it is reliable for solving nonlinear partial differential equations. The numerical results are more accurate than other existing methods available in the literature and very close to the exact solution. The Haar basis function is generated from multi-resolution analysis and used to evaluate fast and accurate approximate solutions on the collocation points. The convergence of the proposed method is demonstrated by its error analysis. We compared numerical solutions with the exact solutions and solutions available in the literature. The proposed method is found to be straight forward, accurate with small computational cost and can be easily implemented in mathematical software MATLAB.

关键词： Haar operational matrix 3D scaling Quasilinearization technique piecewise continuous function Collocation points

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C*-ALGEBRAS OF POLY-BERGMAN TYPE OPERATORS WITH piecewise SLOWLY OSCILLATING COEFFICIENTS

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Journal of Mathematical Sciences (United States) 2022年第1期266卷 77-102页

作者： Espinoza-Loyola, Enrique Karlovich, Yuri I. Centro de Investigación en Ciencias Instituto de Investigación en Ciencias Básicas y Aplicadas Universidad Autónoma del Estado de Morelos Av. Universidad 1001 Col. Chamilpa Morelos Cuernavaca C.P. 62209 Mexico

Given n, m∈ N and a simply connected uniform domain U⊂ C with a sufficiently smooth boundary ∂U, we study the C*-algebra BU,n,m(L):=alg{aI,BU,1,…,BU,n,B~U,1,…,B~U,m:a∈X(L)},generated by the operators of multiplica... 详细信息

Given n, m∈ N and a simply connected uniform domain U⊂ C with a sufficiently smooth boundary ∂U, we study the C^*-algebra BU,n,m(L):=alg{aI,BU,1,…,BU,n,B~U,1,…,B~U,m:a∈X(L)},generated by the operators of multiplication by functions in X(L) , by the poly-Bergman projections B_U_,₁, … , B_U_,_n and by the anti-poly-Bergman projections B~ _U_,₁, … , B~ _U_,_m acting on the Lebesgue space L²(U). The C^*-algebra X(L) is generated by the set SO_∂(U) of all bounded continuous functions on U that slowly oscillate at points of ∂U and by the set PC(L) of all piecewise continuous functions on the closure U¯ of U with discontinuities on a finite union L of piecewise Dini-smooth curves that have one-sided tangents at every point z∈ L, possess a finite set Y= L∩ ∂U, do not form cusps, and are not tangent to ∂U at the points z∈ Y. Making use of the Allan-Douglas local principle, the limit operators techniques, quasicontinuous maps, and properties of SO_∂(U) functions, a Fredholm symbol calculus for the C^*-algebra B_U_,_n_,_m(L) is constructed and a Fredholm criterion for its operators is obtained. © 2022, The Author(s), under exclusive licence to Springer Nature Switzerland AG.

关键词： C*-algebra Fredholm symbol calculus Fredholmness piecewise continuous function Poly-Bergman and anti-poly-Bergman projections Slowly oscillating function

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INVERTIBILITY OF FOURIER CONVOLUTION OPERATORS WITH PC SYMBOLS ON VARIABLE LEBESGUE SPACES WITH KHVEDELIDZE WEIGHTS

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Journal of Mathematical Sciences (United States) 2022年第3期266卷 419-434页

作者： Fernandes, Cláudio Karlovych, Oleksiy Medalha, Samuel Centro de Matemática e Aplicações (CMA) Faculdade de Ciências e Tecnologia Universidade Nova de Lisboa Quinta da Torre Caparica 2829–516 Portugal Departamento de Matemática Faculdade de Ciências e Tecnologia Universidade Nova de Lisboa Quinta da Torre Caparica 2829–516 Portugal

Let p(·) : R→ (1 , ∞) be a sufficiently regular variable exponent and ϱ be a Khvedelidze weight on R. Suppose that a function a belongs to the algebra PCp(·),ϱ of piecewise continuous Fourier multipliers o... 详细信息

Let p(·) : R→ (1 , ∞) be a sufficiently regular variable exponent and ϱ be a Khvedelidze weight on R. Suppose that a function a belongs to the algebra PC_p₍_·₎_,_ϱ of piecewise continuous Fourier multipliers on the weighted variable Lebesgue space L^p⁽^·⁾(R, ϱ). We show that the Fourier convolution operator W(a) = F^{- 1}aF is invertible on the space L^p⁽^·⁾(R, ϱ) if and only if its symbol a is invertible in L^∞(R). © 2022, The Author(s), under exclusive licence to Springer Nature Switzerland AG.

关键词： Fourier convolution operator Fourier multiplier Invertibility Khvedelidze weight piecewise continuous function Variable Lebesgue space

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Algebras of continuous Fourier Multipliers on Variable Lebesgue Spaces

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MEDITERRANEAN JOURNAL OF MATHEMATICS 2020年第4期17卷 1-19页

作者： Karlovich, Alexei Yu. Univ Nova Lisboa Dept Matemat Ctr Matemat & Aplicacoes Fac Ciencias & Tecnol P-2829516 Caparica Portugal

We show that several definitions of algebras of continuous Fourier multipliers on variable Lebesgue spaces over the real line are equivalent under some natural assumptions on variable exponents. Some of our results are new even in the case of standard Lebesgue spaces and give answers on two questions about algebras of continuous Fourier multipliers on Lebesgue spaces over the real line posed by Mascarenhas, Santos and Seidel.

关键词： continuous Fourier multiplier variable Lebesgue space Stechkin's inequality piecewise continuous function slowly oscillating function 42B15 46E30

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Numerical solution for singular differential equations using Haar wavelet

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International Journal of Modeling, Simulation, and Scientific Computing 2020年第5期11卷 31-45页

作者： Shitesh Shukla Manoj Kumar Department of Mathematics Motilal Nehru National Institute of Technology Allahabad Prayagraj-211004(U.P) India

The aim of this paper is to obtain the numerical solution of singular ordinary differential equations using the Haar-wavelet *** proposed method is mathematically simple and provides highly accurate *** this method,we derive the Haar operational matrix using Haar *** operational matrix is a basic tool and applied in system analysis to evaluate the numerical solution of differential *** convergence of the proposed method is discussed through its error *** illustrate the efficiency of this method,solutions of four singular differential equations are obtained.

关键词： Haar operational matrix singularity quasilinearization technique piecewise continuous function

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C*-algebras of Bergman Type Operators with piecewise continuous Coefficients Over Domains with Dini-Smooth Corners

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COMPLEX ANALYSIS AND OPERATOR THEORY 2019年第1期13卷 151-192页

作者： Espinoza-Loyola, Enrique Karlovich, Yuri I. Univ Nacl Autonoma Mexico Unidad Cuernavaca Inst Matemat Av Univ S-N Cuernavaca 62210 Morelos Mexico Univ Autonoma Estado Morelos Inst Invest Ciencias Basicas & Aplicadas Ctr Invest Ciencias Av Univ 1001 Cuernavaca 62209 Morelos Mexico

For any simply connected domain UC with a piecewise Dini-smooth boundary U which admits Dini-smooth corners of openings lying in (0,2], the C-algebra BU=alg{aI,BU,BU:aPC(L)} generated by the operators of multiplication by piecewise continuous functions with discontinuities on a finite union LU of piecewise Dini-smooth curves that have one-sided tangents at every point, do not form cusps and are not tangent to U at the points of LU, and by the Bergman projection BU and anti-Bergman projection BU acting on the Lebesgue space L2(U) is studied. A Fredholm symbol calculus for the C-algebra BU is constructed and a Fredholm criterion for the operators ABU in terms of their symbols is established. Results essentially depend on openings of corners and angles between curves of discontinuity of coefficients at the points z is an element of partial derivative U boolean AND L.

关键词： Bergman and anti-Bergman projections piecewise continuous function Dini-smooth corner C*-algebra Fredholm symbol calculus Fredholmness

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C*-algebras of Bergman Type Operators with piecewise Slowly Oscillating Coefficients over Domains with Dini-Smooth Corners

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INTEGRAL EQUATIONS AND OPERATOR THEORY 2019年第5期91卷 1-30页

作者： Espinoza-Loyola, Enrique Karlovich, Yuri, I Univ Nacl Autonoma Mexico Inst Matemat Unidad Cuernavaca Av Univ S-N Cuernavaca 62210 Morelos Mexico Univ Autonoma Estado Morelos Inst Invest Ciencias Basicas & Aplicadas Ctr Invest Ciencias Av Univ 1001 Cuernavaca 62209 Morelos Mexico

Given a simply connected domain U subset of C with a piecewise Dini-smooth boundary partial derivative U which admits a finite set of Dini-smooth corners of openings lying in (0, 2 pi], we study the C*-algebra B-U = {aI, B-U, (B) over tilde (U) : a is an element of X(L)} generated by the operators of multiplication by functions in X(L), and by the Bergman projection B-U and anti-Bergman projection (B) over tilde (U) acting on the Lebesgue space L-2(U). The C*-algebra X(L) is generated by all piecewise continuous functions on the closure (U) over bar of U with discontinuities on a finite union L of piecewise Dini-smooth curves that have one-sided tangents at every point, do not form cusps and are not tangent to partial derivative U at the points of L boolean AND partial derivative U, and by all bounded continuous functions on U that slowly oscillate at points of partial derivative U. Making use of the Allan-Douglas local principle, the limit operators techniques and the Kehe Zhu results on the class Q = VMO partial derivative(D) boolean AND L-infinity (D), a Fredholm symbol calculus for the C*-algebra B-U is constructed and a Fredholm criterion for the operators A is an element of B-U is obtained.

关键词： Bergman and anti-Bergman projections piecewise continuous function Slowly oscillating function Dini-smooth corner C*-algebra Fredholm symbol calculus Fredholmness

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Linear extensions of some Baire-one functions

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ARCHIV DER MATHEMATIK 2018年第2期110卷 175-182页

作者： Sieg, Waldemar UKW Bydgoszcz Bydgoszcz Poland

Let X be a Hausdorff topological space, and let denote the space of all real Baire-one functions defined on X. Let A be a nonempty subset of X endowed with the topology induced from X, and let be the set of functions with a property making a linear subspace of . We give a sufficient condition for the existence of a linear extension operator , where means to be piecewise continuous on a sequence of closed and subsets of X and is denoted by . We show that restricted to bounded elements of endowed with the supremum norm is an isometry. As a consequence of our main theorem, we formulate the conclusion about existence of a linear extension operator for the classes of Baire-one-star and piecewise continuous functions.

关键词： Baire one function Baire one star function piecewise continuous function Extension Zero set Normal space Perfectly normal space Tietze extension theorem Kuratowski extension theorem Borsuk-Dugundji extension

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Closed-Loop Optimal System Synthesis in the Mode Of “Special” Control of a continuous Induction Heater

Closed-Loop Optimal System Synthesis in the Mode Of “Specia...

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International Conference on Complex Systems: Control and Modeling Problems (CSCMP)

作者： Aleksander Danilushkin Samara State Technical University Samara Russian Federation

ISBN: (纸本)9781728167015

The article deals with the problem of synthesizing the system of the optimal control of transient modes of a through-type induction heater. An algorithm to control the inductor power is obtained for the case of varying the performance of the technological complex. It is shown that the optimal algorithm for the transient mode has the form of a periodic decaying piecewise continuous function of time with discontinuities of the first kind. The number of intervals of continuity is infinite. It is noted that under conditions of random noise and a wide variety of process parameters, the synthesis of a closed-loop optimal system with temperature feedback seems to be the most appropriate. A specific feature of optimal control algorithms for non-stationary modes is the most using “special” control intervals. The conditions of absolute equality between the current value of the metal temperature at the exit from the heater and the given value are satisfied on “special” intervals of the trajectory. In this case, the temperature of the metal at the exit from the heater remains constant during the all transient modes. Therefore, the synthesis of closed-loop systems with feedback on the metal temperature at the exit from the heater is, in principle, impossible. At the same time, the metal temperature inside the heater varies according to certain regularity in accordance with the optimal algorithm. This established behavior allows us to synthesize a closed-loop system on the “special” interval of optimal control as a system with feedback on temperature at some “internal” point of the heater. Analyzing the nature of the metal temperature distribution along the length of the heater in a transient mode allows to determine the coordinate of the internal point at which the temperature changes with a maximum deviation from the steady-state value. This coordinate is used to set the feedback sensor. A relationship has been established between the temperature change at the control point and

关键词： Optimization Operation Synthesis Phase trajectory Distributed system Heaters Distributed Systems control point transient model Optimal control trajectory Feedback synthesis EXITS induction heater temperature feedback Closed loop systems piecewise continuous function temperature change Optimization algorithms

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Lyapunov exponents of a class of piecewise continuous systems of fractional order

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NONLINEAR DYNAMICS 2015年第1-2期81卷 227-237页

作者： Danca, Marius-F. Avram Iancu Univ Dept Math & Comp Sci Cluj Napoca 400380 Romania Romanian Inst Sci & Technol Cluj Napoca 400487 Romania

In this paper, we prove that a class of piecewise continuous autonomous systems of fractional order has well-defined Lyapunov exponents. To do so, based on some known results from differential inclusions of integer order and fractional order, as well as differential equations with discontinuous right-hand sides, the corresponding discontinuous initial value problem is approximated by a continuous one with fractional order. Then, the Lyapunov exponents are numerically determined using, for example, Wolf's algorithm. Three examples of piecewise continuous chaotic systems of fractional order are simulated and analyzed: Sprott's system, Chen's system, and Simizu-Morioka's system.

关键词： piecewise continuous function Fractional-order system piecewise continuous system of fractional order Lyapunov exponent Chaotic system

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