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1.
The objectives of this paper are twofold. Firstly, we formulate a system of partial differential equations that models the contamination of groundwater due to migration of dissolved contaminants through unsaturated to saturated zone. A closed form solution using the singular perturbation techniques for the flow and solute transport equations in the unsaturated zone is obtained. Indeed, the solution can be used as a tool to verify the accuracy of numerical models of water flow and solute transport. The second part of this paper, deals with how the water level in a water reserve drops due to pumping water out of a well that is some distance away. 相似文献
2.
Soyoon Bak 《Numerical Methods for Partial Differential Equations》2019,35(5):1756-1776
In this study, new high‐order backward semi‐Lagrangian methods are developed to solve nonlinear advection–diffusion type problems, which are realized using high‐order characteristic‐tracking strategies. The proposed characteristic‐tracking strategies are second‐order L‐stable and third‐order L(α)‐stable methods, which are based on a classical implicit multistep method combined with a error‐correction method. We also use backward differentiation formulas and the fourth‐order finite‐difference scheme for diffusion problem discretization in the temporal and spatial domains, respectively. To demonstrate the adaptability and efficiency of these time‐discretization strategies, we apply these methods to nonlinear advection–diffusion type problems such as the viscous Burgers' equation. Through simulations, not only the temporal and spatial accuracies are numerically evaluated but also the proposed methods are shown to be superior to the compared existing characteristic‐tracking methods under the same rates of convergence in terms of accuracy and efficiency. Finally, we have shown that the proposed method well preserves the energy and mass when the viscosity coefficient becomes zero. 相似文献
3.
Min Zhang Yang Liu Hong Li 《Numerical Methods for Partial Differential Equations》2019,35(4):1588-1612
In this article, a local discontinuous Galerkin (LDG) method is studied for numerically solving the fractal mobile/immobile transport equation with a new time Caputo–Fabrizio fractional derivative. The stability of the LDG scheme is proven, and a priori error estimates with the second‐order temporal convergence rate and the (k + 1) th order spatial convergence rate are derived in detail. Finally, numerical experiments based on Pk, k = 0, 1, 2, 3, elements are provided to verify our theoretical results. 相似文献
4.
Davood Domiri Ganji Mehdi Safari Reza Ghayor 《Numerical Methods for Partial Differential Equations》2011,27(4):887-897
In this article, the Sawada–Kotera–Ito seventh‐order equation is studied. He's variational iteration method and Adomian's decomposition method (ADM) are applied to obtain solution of this equation. We compare these methods together. The study highlights the significant features of the employed methods and its capability of handling completely integrable equations. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 887–897, 2011 相似文献
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S. van Veldhuizen C. Vuik C.R. Kleijn 《Numerical Methods for Partial Differential Equations》2008,24(3):1037-1054
Two‐dimensional transient simulations are presented of the transport phenomena and multispecies, multireaction chemistry in chemical vapor deposition (CVD). The transient simulations are run until steady state, such that the steady state can be validated against the steady state solutions from literature. We compare various time integration methods in terms of efficiency and robustness. Besides stability, which is important due to the stiffness of the problem, preservation of non‐negativity is crucial. It appears that this latter condition on a time integration method is much more restrictive toward the time step size than stability. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008 相似文献
6.
Chengchuang Zhang Chuanzhong Li Jingsong He 《Mathematical Methods in the Applied Sciences》2015,38(11):2411-2425
In this paper, the Darboux transformation of the Kundu–nonlinear Schrödinger equation is derived and generalized to the matrix of n‐fold Darboux transformation. From known solution Q, the determinant representation of n‐th new solutions of Q[n] are obtained by the n‐fold Darboux transformation. Then soliton solutions and positon solutions are generated from trivial seed solutions, breather solutions and rogue wave solutions that are obtained from periodic seed solutions. After that, the higher order rogue wave solutions of the Kundu–nonlinear Schrödinger equation are given. We show that free parameters in eigenfunctions can adjust the patterns of the higher order rogue waves. Meanwhile, the third‐order rogue waves are given explicitly. Copyright © 2014 John Wiley & Sons, Ltd. 相似文献
7.
WenYi Tian Weihua Deng Yujiang Wu 《Numerical Methods for Partial Differential Equations》2014,30(2):514-535
This article discusses the spectral collocation method for numerically solving nonlocal problems: one‐dimensional space fractional advection–diffusion equation; and two‐dimensional linear/nonlinear space fractional advection–diffusion equation. The differentiation matrixes of the left and right Riemann–Liouville and Caputo fractional derivatives are derived for any collocation points within any given bounded interval. Several numerical examples with different boundary conditions are computed to verify the efficiency of the numerical schemes and confirm the exponential convergence; the physical simulations for Lévy–Feller advection–diffusion equation and space fractional Fokker–Planck equation with initial δ‐peak and reflecting boundary conditions are performed; and the eigenvalue distributions of the iterative matrix for a variety of systems are displayed to illustrate the stabilities of the numerical schemes in more general cases. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 514–535, 2014 相似文献
8.
In this work we use the sine–cosine and the tanh methods for solving the Rosenau–KdV and Rosenau–Kawahara equations. The two methods reveal solitons and periodic solutions. The study confirms the power of the two schemes. 相似文献
9.
In this work, we study a completely integrable dissipative equation. The Burgers equation is extended by using the sense of the Kadomtsev–Petviashvili (KP) equation. The new established Burgers–KP equation is studied by using the tanh–coth method to obtain kink solutions and periodic solutions. We also apply the powerful Hirota’s bilinear method to establish exact N-soliton solutions for the derived integrable equation. 相似文献
10.
Seakweng Vong Chenyang Shi Pin Lyu 《Numerical Methods for Partial Differential Equations》2019,35(2):493-508
Second order finite difference schemes for fractional advection–diffusion equations are considered in this paper. We note that, when studying these schemes, advection terms with coefficients having the same sign as those of diffusion terms need additional estimates. In this paper, by comparing generating functions of the corresponding discretization matrices, we find that sufficiently strong diffusion can dominate the effects of advection. As a result, convergence and stability of schemes are obtained in this situation. 相似文献
11.
In this paper, we investigate the classical Drinfel’d–Sokolov–Wilson equation (DSWE)where p, q, r, s are some nonzero parameters. Some explicit expressions of solutions for the equation are obtained by using the bifurcation method and qualitative theory of dynamical systems. These solutions contain solitary wave solutions, blow-up solutions, periodic solutions, periodic blow-up solutions and kink-shaped solutions. Some previous results are extended. 相似文献
12.
Jian Chen Zhongying Chen Sirui Cheng Jiemin Zhan 《Numerical Methods for Partial Differential Equations》2015,31(5):1665-1691
In this article, multilevel augmentation method (MAM) for solving the Burgers' equation is developed. The Crank–Nicolson–Galerkin scheme of the Burgers' equation results in nonlinear algebraic systems at each time step, the computational cost for solving these nonlinear systems is huge. The MAM allows us to solve the nonlinear system at a fixed initial lower level and then compensate the error by solving a linear system at the higher level. We prove that the method has the same optimal convergence order as the projection method, while reducing the computational complexity greatly. Finally, numerical experiments are presented to confirm the theoretical analysis and illustrate the efficiency of the proposed method. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1665–1691, 2015 相似文献
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14.
In this work we derive a new completely integrable dispersive equation. The equation is obtained by combining the Sawada–Kotera (SK) equation with the sense of the Kadomtsev–Petviashvili (KP) equation. The newly derived Sawada–Kotera–Kadomtsev–Petviashvili (SK–KP) equation is studied by using the tanh–coth method, to obtain single-soliton solution, and by the Hirota bilinear method, to determine the N-soliton solutions. The study highlights the significant features of the employed methods and its capability of handling completely integrable equations. 相似文献
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16.
On the basis of the F‐expansion method with a new sub‐equation and Exp‐function method, an improved F‐expansion method is introduced. As illustrative examples, the exact solutions expressed by exponential function, hyperbolic function of Kudryashov–Sinelshchikov equation for arbitrary α,β are derived. Some previous results are extended. The method is straightforward, concise and is a promising and powerful method for other nonlinear evolution equations in mathematical physics. Copyright © 2013 John Wiley & Sons, Ltd. 相似文献
17.
By application of Green's function and some fixed‐point theorems, that is, Leray–Schauder alternative principle and Schauder's fixed‐point theorem, we establish two new existence results of positive periodic solutions for nonlinear fourth‐order singular differential equation, which extend and improve significantly existing results in the literature. Copyright © 2015 John Wiley & Sons, Ltd. 相似文献
18.
We use the bifurcation method of dynamical systems to study the (2+1)‐dimensional Broer–Kau–Kupershmidt equation. We obtain some new nonlinear wave solutions, which contain solitary wave solutions, blow‐up wave solutions, periodic smooth wave solutions, periodic blow‐up wave solutions, and kink wave solutions. When the initial value vary, we also show the convergence of certain solutions, such as the solitary wave solutions converge to the kink wave solutions and the periodic blow‐up wave solutions converge to the solitary wave solutions. Copyright © 2014 John Wiley & Sons, Ltd. 相似文献
19.
The nitrogen uptake model for plant roots is an advection-diffusion equation subject to double Robin boundary conditions in Cartesian coordinates and its analytical method is expected to accurately estimate the quantity of nutrient uptake and fertilization. Firstly, the Michaelis-Menten (MM) kinetics function in the left boundary condition is changed into a function of time by numerical fitting and the nonlinear left Robin boundary condition then becomes a linear one in order to use traditional analytical methods. Based on the eigenfunction expansion method originally built by Golz and Dorroh, the nitrogen uptake model is homogenized and its eigenvalues are obtained from the Sturm-Liouville problem. Because the convergence of this eigenfunction expansion method is slow around the left boundary, i.e., root surface, we additionally consider the Laplace transform to solve the nitrogen uptake model. However, the solution after Laplace transform involves composite functions and numerical inverse Laplace transforms are introduced to obtain the final solutions. The analytical and numerical solutions show that the nitrogen concentration profiles along the distance from the root surface are convex upward and almost horizontal in the middle part with large gradients at both ends. The numerical simulation demonstrates that the eigenfunction expansion method can reach a satisfactory accuracy and the Laplace transform method with Stehfest inversion has higher calculation efficiency. 相似文献
20.
Ya‐nan Zhang Zhi‐zhong Sun Ting‐chun Wang 《Numerical Methods for Partial Differential Equations》2013,29(5):1487-1503
A linearized Crank–Nicolson‐type scheme is proposed for the two‐dimensional complex Ginzburg–Landau equation. The scheme is proved to be unconditionally convergent in the L2 ‐norm by the discrete energy method. The convergence order is begin{align*}mathcal{O}(tau^2+h_1^2+h^2_2)end{align*}, where τ is the temporal grid size and h1,h2 are spatial grid sizes in the x ‐ and y ‐directions, respectively. A numerical example is presented to support the theoretical result. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013 相似文献