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Arash Fahim - One of the best experts on this subject based on the ideXlab platform.
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a stochastic approximation for fully nonlinear free boundary Parabolic problems
Numerical Methods for Partial Differential Equations, 2014Co-Authors: Erhan Bayraktar, Arash FahimAbstract:We present a stochastic numerical method for solving fully nonlinear free boundary problems of Parabolic Type and provide a rate of convergence under reasonable conditions on the nonlinearity. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 902–929, 2014
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a stochastic approximation for fully nonlinear free boundary Parabolic problems
arXiv: Numerical Analysis, 2011Co-Authors: Erhan Bayraktar, Arash FahimAbstract:We present a stochastic numerical method for solving fully non-linear free boundary problems of Parabolic Type and provide a rate of convergence under reasonable conditions on the non-linearity.
Tomomi Yokota - One of the best experts on this subject based on the ideXlab platform.
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boundedness in quasilinear keller segel systems of Parabolic Parabolic Type on non convex bounded domains
Journal of Differential Equations, 2014Co-Authors: Sachiko Ishida, Kiyotaka Seki, Tomomi YokotaAbstract:Abstract This paper deals with the quasilinear fully Parabolic Keller–Segel system { u t = ∇ ⋅ ( D ( u ) ∇ u ) − ∇ ⋅ ( S ( u ) ∇ v ) , x ∈ Ω , t > 0 , v t = Δ v − v + u , x ∈ Ω , t > 0 , under homogeneous Neumann boundary conditions in a bounded domain Ω ⊂ R N with smooth boundary, N ∈ N . The diffusivity D ( u ) is assumed to satisfy some further technical conditions such as algebraic growth and D ( 0 ) ⩾ 0 , which says that the diffusion is allowed to be not only non-degenerate but also degenerate. The global-in-time existence and uniform-in-time boundedness of solutions are established under the subcritical condition that S ( u ) / D ( u ) ⩽ K ( u + e ) α for u > 0 with α 2 / N , K > 0 and e ⩾ 0 . When D ( 0 ) > 0 , this paper represents an improvement of Tao and Winkler [17] , because the domain does not necessarily need to be convex in this paper. In the case Ω = R N and D ( 0 ) ⩾ 0 , uniform-in-time boundedness is an open problem left in a previous paper [7] . This paper also gives an answer to it in bounded domains.
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blow up in finite or infinite time for quasilinear degenerate keller segel systems of Parabolic Parabolic Type
Discrete and Continuous Dynamical Systems-series B, 2013Co-Authors: Sachiko Ishida, Tomomi YokotaAbstract:This paper gives a blow-up result for the quasilinear degenerate Keller-Segel systems of Parabolic-Parabolic Type. It is known that the system has a global solvability in the case where $q < m + \frac{2}{N}$ ($m$ denotes the intensity of diffusion and $q$ denotes the nonlinearity) without any restriction on the size of initial data, and where $q \geq m + \frac{2}{N}$ and the initial data are ``small''. However, there is no result when $q \geq m + \frac{2}{N}$ and the initial data are ``large''. This paper discusses such case and shows that there exist blow-up energy solutions from initial data having large negative energy.
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global existence of weak solutions to quasilinear degenerate keller segel systems of Parabolic Parabolic Type with small data
Journal of Differential Equations, 2012Co-Authors: Sachiko Ishida, Tomomi YokotaAbstract:Abstract This paper deals with the quasilinear degenerate Keller–Segel system (KS) of “Parabolic–Parabolic” Type. The global existence of weak solutions to (KS) with small initial data is established when q ⩾ m + 2 N (m denotes the intensity of diffusion and q denotes the nonlinearity). In the system of “Parabolic–elliptic” Type, Sugiyama and Kunii (2006) [13, Theorem 3] and Sugiyama (2007) [12, Theorem 2] state the similar result; note that q = m + 2 N corresponds to generalized Fujitaʼs critical exponent. However, the super-critical case where q ⩾ m + 2 N has been unsolved for “Parabolic–Parabolic” Type. Therefore this paper gives an answer to the unsolved problem.
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global existence of weak solutions to quasilinear degenerate keller segel systems of Parabolic Parabolic Type
Journal of Differential Equations, 2012Co-Authors: Sachiko Ishida, Tomomi YokotaAbstract:Abstract This paper deals with the quasilinear degenerate Keller–Segel system (KS) of Parabolic–Parabolic Type. The global existence of weak solutions to (KS) is established when q m + 2 N (m denotes the intensity of diffusion and q denotes the nonlinearity) without restriction on the size of initial data; note that q = m + 2 N corresponds to generalized Fujitaʼs exponent. The result improves both Sugiyama (2007) [14, Theorem 1] and Sugiyama and Kunii (2006) [15, Theorem 1] in which it is assumed that q ⩽ m .
Mechthild Thalhammer - One of the best experts on this subject based on the ideXlab platform.
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Convergence analysis of high-order commutator-free quasi-Magnus exponential integrators for nonautonomous linear evolution equations of Parabolic Type
'Oxford University Press (OUP)', 2018Co-Authors: Blanes Zamora Sergio, Casas Fernando, Mechthild ThalhammerAbstract:[EN] The main objective of this work is to provide a stability and error analysis of high-order commutator-free quasi-Magnus (CFQM) exponential integrators. These time integration methods for nonautonomous linear evolution equations are formed by products of exponentials involving linear combinations of the defining operator evaluated at certain times. In comparison with other classes of time integration methods, such as Magnus integrators, an inherent advantage of CFQM exponential integrators is that structural properties of the operator are well preserved by the arising linear combinations. Employing the analytical framework of sectorial operators in Banach spaces, evolution equations of Parabolic Type and dissipative quantum systems are included in the scope of applications. In this context, however, numerical experiments show that CFQM exponential integrators of nonstiff order five or higher proposed in the literature suffer from poor stability properties. The given analysis delivers insight that CFQM exponential integrators are well defined and stable only if the coefficients occurring in the linear combinations satisfy a positivity condition and that an alternative approach for the design of stable high-order schemes relies on the consideration of complex coefficients. Together with suitable local error expansions, this implies that a high-order CFQM exponential integrator retains its nonstiff order of convergence under appropriate regularity and compatibility requirements on the exact solution. Numerical examples confirm the theoretical result and illustrate the favourable behaviour of novel schemes involving complex coefficients in stability and accuracy.Ministerio de Economia y Competitividad (Spain) through projects MTM2013-46553-C3 and MTM2016-77660-P (AEI/FEDER, UE) to S.B. and F.C.Blanes Zamora, S.; Casas, F.; Mechthild Thalhammer (2018). Convergence analysis of high-order commutator-free quasi-Magnus exponential integrators for nonautonomous linear evolution equations of Parabolic Type. IMA Journal of Numerical Analysis. 38(2):743-778. https://doi.org/10.1093/imanum/drx012S74377838
Casas Fernando - One of the best experts on this subject based on the ideXlab platform.
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Convergence analysis of high-order commutator-free quasi-Magnus exponential integrators for nonautonomous linear evolution equations of Parabolic Type
Oxford University Press, 2018Co-Authors: Blanes Sergio, Casas Fernando, Thalhammer MechthildAbstract:The main objective of this work is to provide a stability and error analysis of high-order commutator-free quasi-Magnus (CFQM) exponential integrators. These time integration methods for nonautonomous linear evolution equations are formed by products of exponentials involving linear combinations of the defining operator evaluated at certain times. In comparison with other classes of time integration methods, such as Magnus integrators, an inherent advantage of CFQM exponential integrators is that structural properties of the operator are well preserved by the arising linear combinations. Employing the analytical framework of sectorial operators in Banach spaces, evolution equations of Parabolic Type and dissipative quantum systems are included in the scope of applications. In this context, however, numerical experiments show that CFQM exponential integrators of nonstiff order five or higher proposed in the literature suffer from poor stability properties. The given analysis delivers insight that CFQM exponential integrators are well defined and stable only if the coefficients occurring in the linear combinations satisfy a positivity condition and that an alternative approach for the design of stable high-order schemes relies on the consideration of complex coefficients. Together with suitable local error expansions, this implies that a high-order CFQM exponential integrator retains its nonstiff order of convergence under appropriate regularity and compatibility requirements on the exact solution. Numerical examples confirm the theoretical result and illustrate the favourable behaviour of novel schemes involving complex coefficients in stability and accuracy
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Convergence analysis of high-order commutator-free quasi-Magnus exponential integrators for nonautonomous linear evolution equations of Parabolic Type
'Oxford University Press (OUP)', 2018Co-Authors: Blanes Zamora Sergio, Casas Fernando, Mechthild ThalhammerAbstract:[EN] The main objective of this work is to provide a stability and error analysis of high-order commutator-free quasi-Magnus (CFQM) exponential integrators. These time integration methods for nonautonomous linear evolution equations are formed by products of exponentials involving linear combinations of the defining operator evaluated at certain times. In comparison with other classes of time integration methods, such as Magnus integrators, an inherent advantage of CFQM exponential integrators is that structural properties of the operator are well preserved by the arising linear combinations. Employing the analytical framework of sectorial operators in Banach spaces, evolution equations of Parabolic Type and dissipative quantum systems are included in the scope of applications. In this context, however, numerical experiments show that CFQM exponential integrators of nonstiff order five or higher proposed in the literature suffer from poor stability properties. The given analysis delivers insight that CFQM exponential integrators are well defined and stable only if the coefficients occurring in the linear combinations satisfy a positivity condition and that an alternative approach for the design of stable high-order schemes relies on the consideration of complex coefficients. Together with suitable local error expansions, this implies that a high-order CFQM exponential integrator retains its nonstiff order of convergence under appropriate regularity and compatibility requirements on the exact solution. Numerical examples confirm the theoretical result and illustrate the favourable behaviour of novel schemes involving complex coefficients in stability and accuracy.Ministerio de Economia y Competitividad (Spain) through projects MTM2013-46553-C3 and MTM2016-77660-P (AEI/FEDER, UE) to S.B. and F.C.Blanes Zamora, S.; Casas, F.; Mechthild Thalhammer (2018). Convergence analysis of high-order commutator-free quasi-Magnus exponential integrators for nonautonomous linear evolution equations of Parabolic Type. IMA Journal of Numerical Analysis. 38(2):743-778. https://doi.org/10.1093/imanum/drx012S74377838
Thalhammer Mechthild - One of the best experts on this subject based on the ideXlab platform.
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Convergence analysis of high-order commutator-free quasi-Magnus exponential integrators for nonautonomous linear evolution equations of Parabolic Type
Oxford University Press, 2018Co-Authors: Blanes Sergio, Casas Fernando, Thalhammer MechthildAbstract:The main objective of this work is to provide a stability and error analysis of high-order commutator-free quasi-Magnus (CFQM) exponential integrators. These time integration methods for nonautonomous linear evolution equations are formed by products of exponentials involving linear combinations of the defining operator evaluated at certain times. In comparison with other classes of time integration methods, such as Magnus integrators, an inherent advantage of CFQM exponential integrators is that structural properties of the operator are well preserved by the arising linear combinations. Employing the analytical framework of sectorial operators in Banach spaces, evolution equations of Parabolic Type and dissipative quantum systems are included in the scope of applications. In this context, however, numerical experiments show that CFQM exponential integrators of nonstiff order five or higher proposed in the literature suffer from poor stability properties. The given analysis delivers insight that CFQM exponential integrators are well defined and stable only if the coefficients occurring in the linear combinations satisfy a positivity condition and that an alternative approach for the design of stable high-order schemes relies on the consideration of complex coefficients. Together with suitable local error expansions, this implies that a high-order CFQM exponential integrator retains its nonstiff order of convergence under appropriate regularity and compatibility requirements on the exact solution. Numerical examples confirm the theoretical result and illustrate the favourable behaviour of novel schemes involving complex coefficients in stability and accuracy
