A discrete monotone iterative method is reported here to solve a space-fractional nonlinear diffu... more A discrete monotone iterative method is reported here to solve a space-fractional nonlinear diffusion-reaction equation. More precisely, we propose a Crank-Nicolson discretization of a reaction-diffusion system with fractional spatial derivative of the Riesz type. The finite-difference scheme is based on the use of fractional-order centered differences, and it is solved using a monotone iterative technique. The existence and uniqueness of solutions of the numerical model are analyzed using this approach, along with the technique of upper and lower solutions. This methodology is employed also to prove the main numerical properties of the technique, namely, the consistency, stability, and convergence. As an application, the particular case of the space-fractional Fisher's equation is theoretically analyzed in full detail. In that case, the monotone iterative method guarantees the preservation of the positivity and the boundedness of the numerical approximations. Various numerical examples are provided to illustrate the validity of the numerical approximations. More precisely, we provide an extensive series of comparisons against other numerical methods available in the literature, we show detailed numerical analyses of convergence in time and in space against fractional and integer-order models, and we provide studies on the robustness and the numerical performance of the discrete monotone method.
In this work, we investigate numerically a nonlinear hyperbolic partial differential equation wit... more In this work, we investigate numerically a nonlinear hyperbolic partial differential equation with space fractional derivatives of the Riesz type. The model under consideration generalizes various nonlinear wave equations, including the sine-Gordon and the nonlinear Klein-Gordon models. The system considered in this work is conservative when homogeneous Dirichlet boundary conditions are imposed. Motivated by this fact, we propose a finite-difference method based on fractional centered differences that is capable of preserving the discrete energy of the system. The method under consideration is a nonlinear implicit scheme which has various numerical properties. Among the most interesting numerical features, we show that the methodology is consistent of second order in time and fourth order in space. Moreover, we show that the technique is stable and convergent. Some numerical simulations show that the method is capable of preserving the energy of the discrete system. This characteristic of the technique is in obvious agreement with the properties of its continuous counterpart.
Departing from an initial-boundary-value problem governed by a Klein-Gordon-Zakarov system with f... more Departing from an initial-boundary-value problem governed by a Klein-Gordon-Zakarov system with fractional derivatives in the spatial variable, we provide an explicit finite-difference scheme to approximate its solutions. In agreement with the continuous system, the method proposed in this work is also capable of preserving the energy of the system, and the energy quantities are nonnegative under flexible parameter conditions. The boundedness, the consistency, the stability and the convergence of the technique are also established rigorously. The method is easy to implement, and the computer simulations confirm the main analytical and numerical properties of the new model.
Journal of Computational and Applied Mathematics, 2018
In this work, we consider a general class of damped wave equations in two spatial dimensions. The... more In this work, we consider a general class of damped wave equations in two spatial dimensions. The model considers the presence of Weyl space-fractional derivatives as well as a generic nonlinear potential. The system has an associated positive energy functional when damping is not present, in which case, the model is capable of preserving the energy throughout time. Meanwhile, the energy of the system is dissipated in the damped scenario. In this work, the Weyl space-fractional derivatives are approximated through second-order accurate fractional centered differences. A highorder compact difference scheme with fourth order accuracy in space and second order in time is proposed. Some associated discrete quantities are introduced to estimate the energy functional. We prove that the numerical method is capable of conserving the discrete variational structure under the same conditions for which the continuous model is conservative. The positivity of the discrete energy of the system is also discussed. The properties of consistency, solvability, stability and convergence of the proposed method are rigorously proved. We provide some numerical simulations that illustrate the agreement between the physical properties of the continuous and the discrete models.
Ubiquitin is a small modifier protein which is usually conjugated to substrate proteins for degra... more Ubiquitin is a small modifier protein which is usually conjugated to substrate proteins for degradation. In recent years, a number of ubiquitin-like proteins have been identified; however, their roles in eukaryotes are largely unknown. Here, we describe a ubiquitinlike protein URM1, and found it plays important roles in the development and infection process of the rice blast fungus, Magnaporthe oryzae. Targeted deletion of URM1 in M. oryzae resulted in slight reduction in vegetative growth and significant decrease in conidiation. More importantly, the urm1 mutant also showed evident reduction in virulence to host plants. Infection process observation demonstrated that the mutant was arrested in invasive growth and resulted in accumulation of massive host reactive oxygen species (ROS). Further, we found the urm1 mutant was sensitive to the cell wall disturbing reagents, thiol oxidizing agent diamide and rapamycin. We also showed that URM1-mediated modification was responsive to oxidative stresses, and the thioredoxin peroxidase Ahp1 was one of the important urmylation targets. These results suggested that URM1-mediated urmylation plays important roles in detoxification of host oxidative stress to facilitate invasive growth in M. oryzae.
A discrete monotone iterative method is reported here to solve a space-fractional nonlinear diffu... more A discrete monotone iterative method is reported here to solve a space-fractional nonlinear diffusion-reaction equation. More precisely, we propose a Crank-Nicolson discretization of a reaction-diffusion system with fractional spatial derivative of the Riesz type. The finite-difference scheme is based on the use of fractional-order centered differences, and it is solved using a monotone iterative technique. The existence and uniqueness of solutions of the numerical model are analyzed using this approach, along with the technique of upper and lower solutions. This methodology is employed also to prove the main numerical properties of the technique, namely, the consistency, stability, and convergence. As an application, the particular case of the space-fractional Fisher's equation is theoretically analyzed in full detail. In that case, the monotone iterative method guarantees the preservation of the positivity and the boundedness of the numerical approximations. Various numerical examples are provided to illustrate the validity of the numerical approximations. More precisely, we provide an extensive series of comparisons against other numerical methods available in the literature, we show detailed numerical analyses of convergence in time and in space against fractional and integer-order models, and we provide studies on the robustness and the numerical performance of the discrete monotone method.
In this work, we investigate numerically a nonlinear hyperbolic partial differential equation wit... more In this work, we investigate numerically a nonlinear hyperbolic partial differential equation with space fractional derivatives of the Riesz type. The model under consideration generalizes various nonlinear wave equations, including the sine-Gordon and the nonlinear Klein-Gordon models. The system considered in this work is conservative when homogeneous Dirichlet boundary conditions are imposed. Motivated by this fact, we propose a finite-difference method based on fractional centered differences that is capable of preserving the discrete energy of the system. The method under consideration is a nonlinear implicit scheme which has various numerical properties. Among the most interesting numerical features, we show that the methodology is consistent of second order in time and fourth order in space. Moreover, we show that the technique is stable and convergent. Some numerical simulations show that the method is capable of preserving the energy of the discrete system. This characteristic of the technique is in obvious agreement with the properties of its continuous counterpart.
Departing from an initial-boundary-value problem governed by a Klein-Gordon-Zakarov system with f... more Departing from an initial-boundary-value problem governed by a Klein-Gordon-Zakarov system with fractional derivatives in the spatial variable, we provide an explicit finite-difference scheme to approximate its solutions. In agreement with the continuous system, the method proposed in this work is also capable of preserving the energy of the system, and the energy quantities are nonnegative under flexible parameter conditions. The boundedness, the consistency, the stability and the convergence of the technique are also established rigorously. The method is easy to implement, and the computer simulations confirm the main analytical and numerical properties of the new model.
Journal of Computational and Applied Mathematics, 2018
In this work, we consider a general class of damped wave equations in two spatial dimensions. The... more In this work, we consider a general class of damped wave equations in two spatial dimensions. The model considers the presence of Weyl space-fractional derivatives as well as a generic nonlinear potential. The system has an associated positive energy functional when damping is not present, in which case, the model is capable of preserving the energy throughout time. Meanwhile, the energy of the system is dissipated in the damped scenario. In this work, the Weyl space-fractional derivatives are approximated through second-order accurate fractional centered differences. A highorder compact difference scheme with fourth order accuracy in space and second order in time is proposed. Some associated discrete quantities are introduced to estimate the energy functional. We prove that the numerical method is capable of conserving the discrete variational structure under the same conditions for which the continuous model is conservative. The positivity of the discrete energy of the system is also discussed. The properties of consistency, solvability, stability and convergence of the proposed method are rigorously proved. We provide some numerical simulations that illustrate the agreement between the physical properties of the continuous and the discrete models.
Ubiquitin is a small modifier protein which is usually conjugated to substrate proteins for degra... more Ubiquitin is a small modifier protein which is usually conjugated to substrate proteins for degradation. In recent years, a number of ubiquitin-like proteins have been identified; however, their roles in eukaryotes are largely unknown. Here, we describe a ubiquitinlike protein URM1, and found it plays important roles in the development and infection process of the rice blast fungus, Magnaporthe oryzae. Targeted deletion of URM1 in M. oryzae resulted in slight reduction in vegetative growth and significant decrease in conidiation. More importantly, the urm1 mutant also showed evident reduction in virulence to host plants. Infection process observation demonstrated that the mutant was arrested in invasive growth and resulted in accumulation of massive host reactive oxygen species (ROS). Further, we found the urm1 mutant was sensitive to the cell wall disturbing reagents, thiol oxidizing agent diamide and rapamycin. We also showed that URM1-mediated modification was responsive to oxidative stresses, and the thioredoxin peroxidase Ahp1 was one of the important urmylation targets. These results suggested that URM1-mediated urmylation plays important roles in detoxification of host oxidative stress to facilitate invasive growth in M. oryzae.
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Papers by Ahmed Hendy