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Sandra Carillo - One of the best experts on this subject based on the ideXlab platform.
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A 3-dimensional Singular Kernel problem in viscoelasticity: an existence result
arXiv: Mathematical Physics, 2018Co-Authors: Sandra CarilloAbstract:Materials with memory, namely those materials whose mechanical and/or thermodynamical behaviour depends on time not only via the present time, but also through its past history, are considered. Specifically, a three dimensional viscoelastic body is studied. Its mechanical behaviour is described via an integro-differential equation, whose Kernel represents the relaxation modulus, characteristic of the viscoelastic material under investigation. According to the classical model, to guarantee the thermodynamical compatibility of the model itself, such a Kernel satisfies regularity conditions which include the integrability of its time derivative. To adapt the model to a wider class of materials, this condition is relaxed; that is, conversely to what is generally assumed, no integrability condition is imposed on the time derivative of the relaxation modulus. Hence, the case of a relaxation modulus which is unbounded at the initial time t = 0, is considered, so that a Singular Kernel integro-differential equation, is studied. In this framework, the existence of a weak solution is proved in the case of a three dimensional Singular Kernel initial boundary value problem.
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Regular and Singular Kernel problems in magneto-viscoelasticity
Meccanica, 2017Co-Authors: Sandra CarilloAbstract:Magneto-viscoelastic materials find their interest in a variety of applications in which mechanical properties are coupled with magnetic ones. In particular, new materials such as magneto-rheological elastomers or, in general, magneto-sensitive polymeric composites are more and more widely employed in new materials. The deformation evolution is assumed to be viscoelastic, that is, the stress–strain relation depends on the deformation history of the material further to on the deformation at the present time. This is a characteristic feature of all materials with memory, namely those materials whose mechanical and/or thermodynamical response depends on time not only via the present time, but also through the whole past history. To describe this behaviour integro-differential model equations are adopted subject to the fading memory assumption which corresponds to require that, asymptotically, effects of past deformation events become negligible. Magneto-viscoelastic materials are modelled aiming to describe viscoelastic materials whose mechanical response is also influenced by the presence of a magnetic field. Thus, the model system is obtained on coupling the viscoelasticity linear integro-differential equation with a nonlinear partial differential equation which describes magnetic effects. The attention is focussed on the Kernel of the integro-differential equation: both regular as well as a Singular Kernel, at \(t=0\), problems are analysed. Indeed, Singular Kernel models allow to describe a wider class of materials and are also connected to materials modelled via Kernels described via fractional derivatives.
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Regular and Singular Kernel Problems in Materials with Memory
2015Co-Authors: Sandra CarilloAbstract:The term material with memory was introduced to characterize those materials which exhibit the crucial physical property that their behaviour depends on time not only through the present time but also through their past history. This peculiarity leads, under the analytical point of view to study integro-differential model problems. This is the case of both rigid thermodynamics with memory as well as of isothermal viscoelasticity. The model of a rigid linear heat conductor with memory is considered and, specifically, evolution problems which describe the time evolution of the temperature distribution are studied. The attention is focussed on the thermodynamical state of such a rigid heat conductor which, according to the model proposed by Fabrizio, Gentili and Reynolds [6] and the constitutive equations therein, depends on the history of the material. When initial and boundary conditions are assigned, the model evolution problem is represented by an integro-differential one whose Kernel depends on time through the present time as well as the past history of the material. As a consequence, the choice of suitable expressions of the minimum free energy and of the thermal work [1, 5] turns out to be crucial in this study. Indeed, the functional spaces where solutions are looked for, are obtained by the requirement to be meaningful both under the physical as well as the analytic viewpoint. Recent results, concerning both rigid thermodynamics and isothermal viscoelasticity, obtained together with V. Valente and G. Vergara Caffarelli, in the case, in turn, of magneto-viscoelasticity problems, [2, 3], and of Singular Kernel problems [4, 7, 8] are presented. [1] Amendola G. and Carillo S., Thermal work and minimum free energy in a heat conductor with memory, Quarterly Journal Mech. Appl. Math., (57), n. 3, (2004), 429-446; doi: 10 . 1093 /qjmam/ 57 . 3 . 429 ; [2] Carillo S., Valente V. and Vergara Caffarelli G., A result of existence and uniqueness for an integro- differential system in magneto-viscoelasticity , Applicable Analisys: An International Journal, (90) n. 12, (2011) 1791–1802, ISSN: 0003-6811, doi: 10.1080/00036811003735832. [3] Carillo S., Valente V. and Vergara Caffarelli G., An existence theorem for the magneto-viscoelastic problem , Discrete and Continuous Dynamical Systems Series S. , (5) n. 3, (2012), 435 – 447, doi: 10 . 3934 /dcdss. 2012 . 5 . 435 ; [4]Carillo S., Valente V. and Vergara CaffarelliG., A linear viscoelasticity problem with a Singular memory Kernel: an existence and uniqueness result, Differential and Integral Equations, 26 , n.ro 9/10 (2013), 1115-1125; [5] S. Carillo, Existence, Uniqueness and Exponential Decay: an Evolution Problem in Heat Conduction with Memory, Quarterly of Appl. Math., . 635–649, LXIX, n. 4, S 0033-569X(2011)01223-1, Article electronically published on July 7, 2011; [6] M. Fabrizio, G. Gentili, D.W.Reynolds, On rigid heat conductors with memory, Int. J. Eng. Sci. , 36 (1998), 765–782. [7] S. Carillo, V. Valente and G. Vergara Caffarelli, Heat Conduction with Memory: a Singular Kernel Problem, Evolution Equations and Control Theory, 3 , (3), (2014), 399-410. doi:10.3934/eect.2014.399 [8] S. Carillo, Singular Kernel Problems in Materials with Memory , Meccanica, 50 , (3), (2015), 603-615 doi : 10 . 1007 /s 11012 − 014 − 0083 − y :
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Singular Kernel problems in materials with memory
Meccanica, 2015Co-Authors: Sandra CarilloAbstract:In recent years the interest on devising and study new materials is growing since they are widely used in different applications which go from rheology to bio-materials or aerospace applications. In this framework, there is also a growing interest in understanding the behaviour of materials with memory , here considered. The name of the model aims to emphasize that the behaviour of such materials crucially depends on time not only through the present time but also through the past history. Under the analytical point of view, this corresponds to model problems represented by integro-differential equations which exhibit a Kernel non local in time. This is the case of rigid thermodynamics with memory as well as of isothermal viscoelasticity; in the two different models the Kernel represents, in turn, the heat flux relaxation function and the relaxation modulus. In dealing with classical materials with memory these Kernels are regular function of both the present time as well as the past history. Aiming to study new materials integro-differential problems admitting Singular Kernels are compared. Specifically, on one side the temperature evolution in a rigid heat conductor with memory characterized by a heat flux relaxation function Singular at the origin, and, on the other, the displacement evolution within a viscoelastic model characterized by a relaxation modulus which is unbounded at the origin, are considered. One dimensional problems are examined; indeed, even if the results are valid also in three dimensional general cases, here the attention is focussed on pointing out analogies between the two different materials with memory under investigation. Notably, the method adopted has a wider interest since it can be applied in the cases of other evolution problems which are modeled by analogue integro-differential equations. An initial boundary value problem with homogenous Neumann boundary conditions is studied.
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Heat Conduction with Memory: a Singular Kernel problem
Evolution Equations & Control Theory, 2014Co-Authors: Sandra Carillo, Vanda Valente, Giorgio Vergara CaffarelliAbstract:The existence and uniqueness of solution to an integro-differential problem arising in heat conduction with memory is here considered. Specifically, a Singular Kernel problem is analyzed in the case of a multi-dimensional rigid heat conductor. The choice to investigate a Singular Kernel material is suggested by applications to model a wider variety of materials and, in particular, new materials whose heat flux relaxation function may be superiorly unbounded at the initial time $t=0$. The present study represents a generalization to higher dimensions of a previous one concerning a $1$-dimensional problem in the framework of linear viscoelasticity with memory. Specifically, an existence theorem is here proved when initial homogeneous data are assumed. Indeed, the choice of homogeneous data is needed to obtain the a priori estimate in Section 2 on which the subsequent results, are based.
Haixiang Zhang - One of the best experts on this subject based on the ideXlab platform.
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crank nicolson quasi wavelets method for solving fourth order partial integro differential equation with a weakly Singular Kernel
Journal of Computational Physics, 2013Co-Authors: Xuehua Yang, Da Xu, Haixiang ZhangAbstract:In this paper, we study a novel numerical scheme for the fourth order partial integro-differential equation with a weakly Singular Kernel. In the time direction, a Crank-Nicolson time-stepping is used to approximate the differential term and the product trapezoidal method is employed to treat the integral term, and the quasi-wavelets numerical method for space discretization. Our interest in the present paper is a continuation of the investigation in Yang et al. [33], where we study discretization in time by using the forward Euler scheme. The comparisons of present results with the previous ones show that the present scheme is more stable and efficient for numerically solving the fourth order partial integro-differential equation with a weakly Singular Kernel. We also tested the method proposed on several one and two dimensional problems with very promising results. Besides, in order to demonstrate the power of the quasi-wavelets method in comparison with standard discretization methods we also consider the high-frequency oscillation problems with the integro-differential term.
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quasi wavelet based numerical method for fourth order partial integro differential equations with a weakly Singular Kernel
International Journal of Computer Mathematics, 2011Co-Authors: Xuehua Yang, Haixiang ZhangAbstract:In this paper, we study the numerical solution of initial boundary-value problem for the fourth-order partial integro-differential equations with a weakly Singular Kernel. We use the forward Euler scheme for time discretization and the quasi-wavelet based numerical method for space discretization. Detailed discrete formulations are given to the treatment of three different boundary conditions, including clamped-type condition, simply supported-type condition and a transversely supported-type condition. Some numerical experiments are included to demonstrate the validity and applicability of the discrete technique. The comparisons of present results with analytical solutions show that the quasi-wavelet based numerical method has a distinctive local property. Especially, the method is easy to implement and produce very accurate results.
Jun Huang - One of the best experts on this subject based on the ideXlab platform.
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legendre wavelets method for the numerical solution of fractional integro differential equations with weakly Singular Kernel
Applied Mathematical Modelling, 2016Co-Authors: Lifeng Wang, Jun HuangAbstract:Abstract In this paper, numerical solutions of the linear and nonlinear fractional integro- differential equations with weakly Singular Kernel where fractional derivatives are considered in the Caputo sense, have been obtained by Legendre wavelets method. The block pulse functions and their properties are employed to derive a general procedure for forming the operational matrix of fractional integration for Legendre wavelets. The application of this matrix for solving initial problem is explained. The mentioned equations are transformed into a system of algebraic equations. The error analysis of the proposed method is investigated. Finally, some numerical examples are shown to illustrate the efficiency of the approach.
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CAS wavelet method for solving the fractional integro-differential equation with a weakly Singular Kernel
International Journal of Computer Mathematics, 2014Co-Authors: Jun HuangAbstract:In this paper, a computational method for numerical solution of a class of integro-differential equations with a weakly Singular Kernel of fractional order which is based on Cos and Sin CAS wavelets and block pulse functions is introduced. Approximation of the arbitrary order weakly Singular integral is also obtained. The fractional integro-differential equations with weakly Singular Kernel are transformed into a system of algebraic equations by using the operational matrix of fractional integration of CAS wavelets. The error analysis of CAS wavelets is given. Finally, the results of some numerical examples support the validity and applicability of the approach.
Abeer Majed Al-bugami - One of the best experts on this subject based on the ideXlab platform.
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The product NystrÖm method and Volterra-Hammerstien Integral Equation with A Generalized Singular Kernel
Journal of Advances in Mathematics, 2014Co-Authors: Abeer Majed Al-bugamiAbstract:In this work, the existence of a unique solution of Volterra-Hammerstein integral equation of the second kind ( V-HIESK ) is discussed. The Volterra integral term ( VIT ) is considered in time with a continuous Kernel, while the Fredholm integral term ( FIT ) is considered in position with a generalized Singular Kernel. Using a numerical technique, V-HIESK is reduced to a nonlinear system of Fredholm integral equations ( SFIEs ). Using product Nystrom method we have a nonlinear algebraic system of equations. Finally, some numerical examples when the Kernel takes the logarithmic, and Carleman forms, are considered.
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Error Analysis for Numerical Solutions of Hammerstein Integral Equation With A Generalized Singular Kernel
Progress in Applied Mathematics, 2013Co-Authors: Abeer Majed Al-bugamiAbstract:In this work, the existence and uniqueness solution of the Hammerstein integral equation (HIE), with a generalized Singular Kernel, is discussed and solved numerically using Toeplitz matrix method and Product Nystrom method. Moreover, the error analysis for these methods is discussed. Finally, numerical results when the Kernel takes a generalized logarithmic form, Carleman function and Cauchy Kernel function are investigated. Also the error, in each case, is estimated.
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Toeplitz Matrix Method and Volterra-Hammerstien Integral Equation with a Generalized Singular Kernel
Progress in Applied Mathematics, 2013Co-Authors: Abeer Majed Al-bugamiAbstract:In this work, the existence of a unique solution of Volterra-Hammerstein integral equation of the second kind (V-HIESK) is discussed. The Volterra integral term (VIT) is considered in time with a continuous Kernel, while the Fredholm integral term (FIT) is considered in position with a generalized Singular Kernel. Using a numerical technique, V-HIESK is reduced to a nonlinear system of Fredholm integral equations (SFIEs). Using Toeplitz matrix method we have a nonlinear algebraic system of equations. Also, many important theorems related to the existence and uniqueness of the produced algebraic system are derived. Finally, some numerical examples when the Kernel takes the logarithmic, Carleman, and Cauchy forms, are considered.
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Hammerstein integral equation with a generalized Singular Kernel
International Journal of Basic and Applied Sciences, 2012Co-Authors: Abeer Majed Al-bugamiAbstract:In this work, the Hammerstein integral equation (HIE), with a generalized Singular Kernel, is considered and solved numerically, using Product Trapezoidal rule, Toeplitz matrix method and Product Nystrom method. The existence and uniqueness solution, under certain conditions, are considered. Moreover, numerical results when the Kernel takes a generalized logarithmic form, Carleman function and Cauchy Kernel are investigated. Also the error, in each case, is estimated.
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Numerical Solution for Volterra-Ferdholm Integral Equation with A Generalized Singular Kernel
Journal of Modern Methods in Numerical Mathematics, 2011Co-Authors: Mohamed A. Abdou, I. L. El Kalla, Abeer Majed Al-bugamiAbstract:In this paper, the existence of a unique solution of Volterra-Fredholm integral equation of the second kind (V-FIESK) is discussed. The Volterra integral term (VIT) is considered in time with a continuous Kernel, while the Fredholm integral term (FIT) is considered in position with a generalized Singular Kernel. Using a numerical technique,� V-FIESK� is reduced to a system of Fredholm integral equations ( SFIEs ). Using Toeplitz matrix method and Product Nystr�m method we have a linear algebraic system of equations ( LAS ). Finally, some numerical examples when the Kernel takes the logarithmic, Carleman, Cauchy and Hilbert forms, are considered.
Xuehua Yang - One of the best experts on this subject based on the ideXlab platform.
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crank nicolson quasi wavelets method for solving fourth order partial integro differential equation with a weakly Singular Kernel
Journal of Computational Physics, 2013Co-Authors: Xuehua Yang, Da Xu, Haixiang ZhangAbstract:In this paper, we study a novel numerical scheme for the fourth order partial integro-differential equation with a weakly Singular Kernel. In the time direction, a Crank-Nicolson time-stepping is used to approximate the differential term and the product trapezoidal method is employed to treat the integral term, and the quasi-wavelets numerical method for space discretization. Our interest in the present paper is a continuation of the investigation in Yang et al. [33], where we study discretization in time by using the forward Euler scheme. The comparisons of present results with the previous ones show that the present scheme is more stable and efficient for numerically solving the fourth order partial integro-differential equation with a weakly Singular Kernel. We also tested the method proposed on several one and two dimensional problems with very promising results. Besides, in order to demonstrate the power of the quasi-wavelets method in comparison with standard discretization methods we also consider the high-frequency oscillation problems with the integro-differential term.
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quasi wavelet based numerical method for fourth order partial integro differential equations with a weakly Singular Kernel
International Journal of Computer Mathematics, 2011Co-Authors: Xuehua Yang, Haixiang ZhangAbstract:In this paper, we study the numerical solution of initial boundary-value problem for the fourth-order partial integro-differential equations with a weakly Singular Kernel. We use the forward Euler scheme for time discretization and the quasi-wavelet based numerical method for space discretization. Detailed discrete formulations are given to the treatment of three different boundary conditions, including clamped-type condition, simply supported-type condition and a transversely supported-type condition. Some numerical experiments are included to demonstrate the validity and applicability of the discrete technique. The comparisons of present results with analytical solutions show that the quasi-wavelet based numerical method has a distinctive local property. Especially, the method is easy to implement and produce very accurate results.
