The Experts below are selected from a list of 172335 Experts worldwide ranked by ideXlab platform
Stanislav Yu Lukashchuk - One of the best experts on this subject based on the ideXlab platform.
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conservation laws for time fractional subdiffusion and diffusion Wave Equations
Nonlinear Dynamics, 2015Co-Authors: Stanislav Yu LukashchukAbstract:A new technique for constructing conservation laws for fractional differential Equations not having a Lagrangian is proposed. The technique is based on the methods of Lie group analysis and employs the concept of nonlinear self-adjointness which is enhanced to the certain class of fractional evolution Equations. The proposed approach is demonstrated on subdiffusion and diffusion-Wave Equations with the Riemann–Liouville and Caputo time-fractional derivatives. It is shown that these Equations are nonlinearly self-adjoint, and therefore, desired conservation laws can be calculated using the appropriate formal Lagrangians. The explicit forms of fractional generalizations of the Noether operators are also proposed for the Equations with the Riemann–Liouville and Caputo time-fractional derivatives of order \(\alpha \in (0,2)\). Using these operators and formal Lagrangians, new conservation laws are constructed for the linear and nonlinear time-fractional subdiffusion and diffusion-Wave Equations by their Lie point symmetries.
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conservation laws for time fractional subdiffusion and diffusion Wave Equations
arXiv: Mathematical Physics, 2014Co-Authors: Stanislav Yu LukashchukAbstract:The concept of nonlinear self-adjointness is employed to construct the conservation laws for fractional evolution Equations using its Lie point symmetries. The approach is demonstrated on subdiffusion and diffusion-Wave Equations with the Riemann-Liouville and Caputo time-fractional derivatives. It is shown that these Equations are nonlinearly self-adjoint and therefore desired conservation laws can be obtained using appropriate formal Lagrangians. Fractional generalizations of the Noether operators are also proposed for the Equations with the Riemann-Liouville and Caputo time-fractional derivatives of order $\alpha \in (0,2)$. Using these operators and formal Lagrangians, new conserved vectors have been constructed for the linear and nonlinear fractional subdiffusion and diffusion-Wave Equations corresponding to its Lie point symmetries.
Sven Peter Nasholm - One of the best experts on this subject based on the ideXlab platform.
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comparison of fractional Wave Equations for power law attenuation in ultrasound and elastography
Ultrasound in Medicine and Biology, 2014Co-Authors: Sverre Holm, Sven Peter NasholmAbstract:Abstract A set of Wave Equations with fractional loss operators in time and space are analyzed. The fractional Szabo equation, the power law Wave equation and the causal fractional Laplacian Wave equation are all found to be low-frequency approximations of the fractional Kelvin-Voigt Wave equation and the more general fractional Zener Wave equation. The latter two Equations are based on fractional constitutive Equations, whereas the former Wave Equations have been derived from the desire to model power law attenuation in applications like medical ultrasound. This has consequences for use in modeling and simulation, especially for applications that do not satisfy the low-frequency approximation, such as shear Wave elastography. In such applications, the Wave Equations based on constitutive Equations are the viable ones.
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deriving fractional acoustic Wave Equations from mechanical and thermal constitutive Equations
Computers & Mathematics With Applications, 2013Co-Authors: Sverre Holm, Sven Peter Nasholm, Fabrice Prieur, Ralph SinkusAbstract:It is argued that fractional acoustic Wave Equations come in two kinds. The first kind is constructed ad hoc to have loss operators that fit power law measurements. The second kind is more fundamental as they in addition are based on underlying physical Equations. Here that means constitutive Equations. These Equations are the fractional Kelvin-Voigt and the more general fractional Zener stress-strain relationships as well as a fractional version of the Fourier heat law. The properties of the Wave Equations are given in terms of attenuation, and phase/group velocities for low-, intermediate- and high-frequency regions. In the most general case, the attenuation exhibits power law behavior in all frequency ranges while the phase and group velocities increase sharply in the intermediate frequency range and converge to a constant, finite value for high frequencies. It is also shown that the fractional Zener Wave equation is equivalent to the multiple relaxation model for attenuation.
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linking multiple relaxation power law attenuation and fractional Wave Equations
Journal of the Acoustical Society of America, 2011Co-Authors: Sven Peter Nasholm, Sverre HolmAbstract:The acoustic Wave attenuation is described by an experimentally established frequency power law in a variety of complex media, e.g., biological tissue, polymers, rocks, and rubber. Recent papers present a variety of acoustical fractional derivative Wave Equations that have the ability to model power-law attenuation. On the other hand, a multiple relaxation model is widely recognized as a physically based description of the acoustic loss mechanisms as developed by Nachman et al. [J. Acoust. Soc. Am. 88, 1584–1595 (1990)]. Through assumption of a continuum of relaxation mechanisms, each with an effective compressibility described by a distribution related to the Mittag-Leffler function, this paper shows that the Wave equation corresponding to the multiple relaxation approach is identical to a given fractional derivative Wave equation. This work therefore provides a physically based motivation for use of fractional Wave Equations in acoustic modeling.
Sverre Holm - One of the best experts on this subject based on the ideXlab platform.
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comparison of fractional Wave Equations for power law attenuation in ultrasound and elastography
Ultrasound in Medicine and Biology, 2014Co-Authors: Sverre Holm, Sven Peter NasholmAbstract:Abstract A set of Wave Equations with fractional loss operators in time and space are analyzed. The fractional Szabo equation, the power law Wave equation and the causal fractional Laplacian Wave equation are all found to be low-frequency approximations of the fractional Kelvin-Voigt Wave equation and the more general fractional Zener Wave equation. The latter two Equations are based on fractional constitutive Equations, whereas the former Wave Equations have been derived from the desire to model power law attenuation in applications like medical ultrasound. This has consequences for use in modeling and simulation, especially for applications that do not satisfy the low-frequency approximation, such as shear Wave elastography. In such applications, the Wave Equations based on constitutive Equations are the viable ones.
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deriving fractional acoustic Wave Equations from mechanical and thermal constitutive Equations
Computers & Mathematics With Applications, 2013Co-Authors: Sverre Holm, Sven Peter Nasholm, Fabrice Prieur, Ralph SinkusAbstract:It is argued that fractional acoustic Wave Equations come in two kinds. The first kind is constructed ad hoc to have loss operators that fit power law measurements. The second kind is more fundamental as they in addition are based on underlying physical Equations. Here that means constitutive Equations. These Equations are the fractional Kelvin-Voigt and the more general fractional Zener stress-strain relationships as well as a fractional version of the Fourier heat law. The properties of the Wave Equations are given in terms of attenuation, and phase/group velocities for low-, intermediate- and high-frequency regions. In the most general case, the attenuation exhibits power law behavior in all frequency ranges while the phase and group velocities increase sharply in the intermediate frequency range and converge to a constant, finite value for high frequencies. It is also shown that the fractional Zener Wave equation is equivalent to the multiple relaxation model for attenuation.
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linking multiple relaxation power law attenuation and fractional Wave Equations
Journal of the Acoustical Society of America, 2011Co-Authors: Sven Peter Nasholm, Sverre HolmAbstract:The acoustic Wave attenuation is described by an experimentally established frequency power law in a variety of complex media, e.g., biological tissue, polymers, rocks, and rubber. Recent papers present a variety of acoustical fractional derivative Wave Equations that have the ability to model power-law attenuation. On the other hand, a multiple relaxation model is widely recognized as a physically based description of the acoustic loss mechanisms as developed by Nachman et al. [J. Acoust. Soc. Am. 88, 1584–1595 (1990)]. Through assumption of a continuum of relaxation mechanisms, each with an effective compressibility described by a distribution related to the Mittag-Leffler function, this paper shows that the Wave equation corresponding to the multiple relaxation approach is identical to a given fractional derivative Wave equation. This work therefore provides a physically based motivation for use of fractional Wave Equations in acoustic modeling.
Timo Welti - One of the best experts on this subject based on the ideXlab platform.
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weak convergence rates for spatial spectral galerkin approximations of semilinear stochastic Wave Equations with multiplicative noise
Applied Mathematics and Optimization, 2021Co-Authors: Ladislas Jacobe De Naurois, Arnulf Jentzen, Timo WeltiAbstract:Stochastic Wave Equations appear in several models for evolutionary processes subject to random forces, such as the motion of a strand of DNA in a liquid or heat flow around a ring. Semilinear stochastic Wave Equations can typically not be solved explicitly, but the literature contains a number of results which show that numerical approximation processes converge with suitable rates of convergence to solutions of such Equations. In the case of approximation results for strong convergence rates, semilinear stochastic Wave Equations with both additive or multiplicative noise have been considered in the literature. In contrast, the existing approximation results for weak convergence rates assume that the diffusion coefficient of the considered semilinear stochastic Wave equation is constant, that is, it is assumed that the considered Wave equation is driven by additive noise, and no approximation results for multiplicative noise are known. The purpose of this work is to close this gap and to establish essentially sharp weak convergence rates for spatial spectral Galerkin approximations of semilinear stochastic Wave Equations with multiplicative noise. In particular, our weak convergence result establishes as a special case essentially sharp weak convergence rates for the continuous version of the hyperbolic Anderson model. Our method of proof makes use of the Kolmogorov equation and the Holder-inequality for Schatten norms.
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weak convergence rates for spatial spectral galerkin approximations of semilinear stochastic Wave Equations with multiplicative noise
arXiv: Probability, 2015Co-Authors: Ladislas Jacobe De Naurois, Arnulf Jentzen, Timo WeltiAbstract:Stochastic Wave Equations appear in several models for evolutionary processes subject to random forces, such as the motion of a strand of DNA in a liquid or heat flow around a ring. Semilinear stochastic Wave Equations can typically not be solved explicitly, but the literature contains a number of results which show that numerical approximation processes converge with suitable rates of convergence to solutions of such Equations. In the case of approximation results for strong convergence rates, semilinear stochastic Wave Equations with both additive or multiplicative noise have been considered in the literature. In contrast, the existing approximation results for weak convergence rates assume that the diffusion coefficient of the considered semilinear stochastic Wave equation is constant, that is, it is assumed that the considered Wave equation is driven by additive noise, and no approximation results for multiplicative noise are known. The purpose of this work is to close this gap and to establish sharp weak convergence rates for semilinear stochastic Wave Equations with multiplicative noise. In particular, our weak convergence result establishes as a special case essentially sharp weak convergence rates for the hyperbolic Anderson model. Our method of proof makes use of the Kolmogorov equation, the H\"older-inequality for Schatten norms, and the mild It\^o formula.
Arnulf Jentzen - One of the best experts on this subject based on the ideXlab platform.
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weak convergence rates for spatial spectral galerkin approximations of semilinear stochastic Wave Equations with multiplicative noise
Applied Mathematics and Optimization, 2021Co-Authors: Ladislas Jacobe De Naurois, Arnulf Jentzen, Timo WeltiAbstract:Stochastic Wave Equations appear in several models for evolutionary processes subject to random forces, such as the motion of a strand of DNA in a liquid or heat flow around a ring. Semilinear stochastic Wave Equations can typically not be solved explicitly, but the literature contains a number of results which show that numerical approximation processes converge with suitable rates of convergence to solutions of such Equations. In the case of approximation results for strong convergence rates, semilinear stochastic Wave Equations with both additive or multiplicative noise have been considered in the literature. In contrast, the existing approximation results for weak convergence rates assume that the diffusion coefficient of the considered semilinear stochastic Wave equation is constant, that is, it is assumed that the considered Wave equation is driven by additive noise, and no approximation results for multiplicative noise are known. The purpose of this work is to close this gap and to establish essentially sharp weak convergence rates for spatial spectral Galerkin approximations of semilinear stochastic Wave Equations with multiplicative noise. In particular, our weak convergence result establishes as a special case essentially sharp weak convergence rates for the continuous version of the hyperbolic Anderson model. Our method of proof makes use of the Kolmogorov equation and the Holder-inequality for Schatten norms.
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weak convergence rates for spatial spectral galerkin approximations of semilinear stochastic Wave Equations with multiplicative noise
arXiv: Probability, 2015Co-Authors: Ladislas Jacobe De Naurois, Arnulf Jentzen, Timo WeltiAbstract:Stochastic Wave Equations appear in several models for evolutionary processes subject to random forces, such as the motion of a strand of DNA in a liquid or heat flow around a ring. Semilinear stochastic Wave Equations can typically not be solved explicitly, but the literature contains a number of results which show that numerical approximation processes converge with suitable rates of convergence to solutions of such Equations. In the case of approximation results for strong convergence rates, semilinear stochastic Wave Equations with both additive or multiplicative noise have been considered in the literature. In contrast, the existing approximation results for weak convergence rates assume that the diffusion coefficient of the considered semilinear stochastic Wave equation is constant, that is, it is assumed that the considered Wave equation is driven by additive noise, and no approximation results for multiplicative noise are known. The purpose of this work is to close this gap and to establish sharp weak convergence rates for semilinear stochastic Wave Equations with multiplicative noise. In particular, our weak convergence result establishes as a special case essentially sharp weak convergence rates for the hyperbolic Anderson model. Our method of proof makes use of the Kolmogorov equation, the H\"older-inequality for Schatten norms, and the mild It\^o formula.
