In order to control traffic congestion, many mathematical models have been used for several decades. In this paper, we study diffusion-type traffic flow model based on exponential velocity density relation, which prov...In order to control traffic congestion, many mathematical models have been used for several decades. In this paper, we study diffusion-type traffic flow model based on exponential velocity density relation, which provides a non-linear second-order parabolic partial differential equation. The analytical solution of the diffusion-type traffic flow model is very complicated to approximate the initial density of the Cauchy problem as a function of x from given data and it may cause a huge error. For the complexity of the analytical solution, the numerical solution is performed by implementing an explicit upwind, explicitly centered, and second-order Lax-Wendroff scheme for the numerical solution. From the comparison of relative error among these three schemes, it is observed that Lax-Wendroff scheme gives less error than the explicit upwind and explicit centered difference scheme. The numerical, analytical analysis and comparative result discussion bring out the fact that the Lax-Wendroff scheme with exponential velocity-density relation of diffusion type traffic flow model is suitable for the congested area and shows a better fit in traffic-congested regions.展开更多
In this paper, we implement energy equation coupled with viscous Burgers’ equation as a mathematical model for the estimation of thermal pollution of river water. The model is a nonlinear system of partial differenti...In this paper, we implement energy equation coupled with viscous Burgers’ equation as a mathematical model for the estimation of thermal pollution of river water. The model is a nonlinear system of partial differential equations (PDEs) that read as an initial and boundary value problem (IBVP). For the numerical solution of the IBVP, we investigate an explicit second-order Lax- Wendroff type scheme for nonlinear parabolic PDEs. We present the numerical solutions graphically as a temperature profile, which shows good qualitative agreement with natural phenomena of heat transfer. We estimate the thermal pollution of water caused by industrialization on the bank of a river.展开更多
Some new linear and nonlinear integral inequalities of Volterra-type in n independent variables are established. They improve and contain the main results of A. Corduneanu[7] which in turn extend a number of known res...Some new linear and nonlinear integral inequalities of Volterra-type in n independent variables are established. They improve and contain the main results of A. Corduneanu[7] which in turn extend a number of known results in the literature. An application to certain initial value problem of partial differential equation is also indicated.展开更多
In this paper, we use Hermite weighted essentially non-oscillatory （HWENO） schemes with a Lax-Wendroff time discretization procedure, termed HWENO-LW schemes, to solve Hamilton-Jacobi equations. The idea of the reco...In this paper, we use Hermite weighted essentially non-oscillatory （HWENO） schemes with a Lax-Wendroff time discretization procedure, termed HWENO-LW schemes, to solve Hamilton-Jacobi equations. The idea of the reconstruction in the HWENO schemes comes from the original WENO schemes, however both the function and its first derivative values are evolved in time and are used in the reconstruction. One major advantage of HWENO schemes is its compactness in the reconstruction. We explore the possibility in avoiding the nonlinear weights for part of the procedure, hence reducing the cost but still maintaining non-oscillatory properties for problems with strong discontinuous derivative. As a result, comparing with HWENO with Runge-Kutta time discretizations schemes （HWENO-RK） of Qiu and Shu [19] for Hamilton-Jacobi equations, the major advantages of HWENO-LW schemes are their saving of computational cost and their compactness in the reconstruction. Extensive numerical experiments are performed to illustrate the capability of the method.展开更多
文摘In order to control traffic congestion, many mathematical models have been used for several decades. In this paper, we study diffusion-type traffic flow model based on exponential velocity density relation, which provides a non-linear second-order parabolic partial differential equation. The analytical solution of the diffusion-type traffic flow model is very complicated to approximate the initial density of the Cauchy problem as a function of x from given data and it may cause a huge error. For the complexity of the analytical solution, the numerical solution is performed by implementing an explicit upwind, explicitly centered, and second-order Lax-Wendroff scheme for the numerical solution. From the comparison of relative error among these three schemes, it is observed that Lax-Wendroff scheme gives less error than the explicit upwind and explicit centered difference scheme. The numerical, analytical analysis and comparative result discussion bring out the fact that the Lax-Wendroff scheme with exponential velocity-density relation of diffusion type traffic flow model is suitable for the congested area and shows a better fit in traffic-congested regions.
文摘In this paper, we implement energy equation coupled with viscous Burgers’ equation as a mathematical model for the estimation of thermal pollution of river water. The model is a nonlinear system of partial differential equations (PDEs) that read as an initial and boundary value problem (IBVP). For the numerical solution of the IBVP, we investigate an explicit second-order Lax- Wendroff type scheme for nonlinear parabolic PDEs. We present the numerical solutions graphically as a temperature profile, which shows good qualitative agreement with natural phenomena of heat transfer. We estimate the thermal pollution of water caused by industrialization on the bank of a river.
文摘Some new linear and nonlinear integral inequalities of Volterra-type in n independent variables are established. They improve and contain the main results of A. Corduneanu[7] which in turn extend a number of known results in the literature. An application to certain initial value problem of partial differential equation is also indicated.
基金Research partially supported by NNSFC grant 10371118,SRF for ROCS,SEM and Nanjing University Talent Development Foundation.
文摘In this paper, we use Hermite weighted essentially non-oscillatory （HWENO） schemes with a Lax-Wendroff time discretization procedure, termed HWENO-LW schemes, to solve Hamilton-Jacobi equations. The idea of the reconstruction in the HWENO schemes comes from the original WENO schemes, however both the function and its first derivative values are evolved in time and are used in the reconstruction. One major advantage of HWENO schemes is its compactness in the reconstruction. We explore the possibility in avoiding the nonlinear weights for part of the procedure, hence reducing the cost but still maintaining non-oscillatory properties for problems with strong discontinuous derivative. As a result, comparing with HWENO with Runge-Kutta time discretizations schemes （HWENO-RK） of Qiu and Shu [19] for Hamilton-Jacobi equations, the major advantages of HWENO-LW schemes are their saving of computational cost and their compactness in the reconstruction. Extensive numerical experiments are performed to illustrate the capability of the method.
