The 'trial function method' ( TFM for short) and a routine way in finding traveling,wave solutions to some nonlinear partial differential equations( PDE for short), wer explained. Two types of evolution equati...The 'trial function method' ( TFM for short) and a routine way in finding traveling,wave solutions to some nonlinear partial differential equations( PDE for short), wer explained. Two types of evolution equations are studied, one is a generalized Burgers or KdV equation, the other is the Fisher equation with special nonlinear forms of its reaction rate term. One can see that this method is simple, fast and allowing further extension.展开更多
In[20],a semi-implicit spectral deferred correction(SDC)method was proposed,which is efficient for highly nonlinear partial differential equations(PDEs).The semi-implicit SDC method in[20]is based on first-order time ...In[20],a semi-implicit spectral deferred correction(SDC)method was proposed,which is efficient for highly nonlinear partial differential equations(PDEs).The semi-implicit SDC method in[20]is based on first-order time integration methods,which are corrected iteratively,with the order of accuracy increased by one for each additional iteration.In this paper,we will develop a class of semi-implicit SDC methods,which are based on second-order time integration methods and the order of accuracy are increased by two for each additional iteration.For spatial discretization,we employ the local discontinuous Galerkin(LDG)method to arrive at fully-discrete schemes,which are high-order accurate in both space and time.Numerical experiments are presented to demonstrate the accuracy,efficiency and robustness of the proposed semi-implicit SDC methods for solving complex nonlinear PDEs.展开更多
In this paper, the nonlinear Hunter–Saxton equation, which is a famous partial differential equation,is solved by using a hybrid numerical method based on the quasilinearization method and the bivariate generalized f...In this paper, the nonlinear Hunter–Saxton equation, which is a famous partial differential equation,is solved by using a hybrid numerical method based on the quasilinearization method and the bivariate generalized fractional order of the Chebyshev functions(B-GFCF) collocation method. First, using the quasilinearization method,the equation is converted into a sequence of linear partial differential equations(LPD), and then these LPDs are solved using the B-GFCF collocation method. A very good approximation of solutions is obtained, and comparisons show that the obtained results are more accurate than the results of other researchers.展开更多
With the aid of a nonlinear transformation, a class of nonlinear convection-diffusion PDE in one space dimension is converted into a linear one, the unique solution of a nonlinear boundary-initial value problem for th...With the aid of a nonlinear transformation, a class of nonlinear convection-diffusion PDE in one space dimension is converted into a linear one, the unique solution of a nonlinear boundary-initial value problem for the nonlinear PDE can be exactly expressed by the nonlinear transformation, and several illustrative examples are given.展开更多
In this paper, a numerical solution of nonlinear partial differential equation, Benjamin-Bona-Mahony (BBM) and Cahn-Hilliard equation is presented by using Adomain Decomposition Method (ADM) and Variational Iteration ...In this paper, a numerical solution of nonlinear partial differential equation, Benjamin-Bona-Mahony (BBM) and Cahn-Hilliard equation is presented by using Adomain Decomposition Method (ADM) and Variational Iteration Method (VIM). The results reveal that the two methods are very effective, simple and very close to the exact solution.展开更多
The purpose of this study is to present an application of a novel enhancement technique for enhancing medical images generated from X-rays. The method presented in this study is based on a nonlinear partial differenti...The purpose of this study is to present an application of a novel enhancement technique for enhancing medical images generated from X-rays. The method presented in this study is based on a nonlinear partial differential equation （PDE） model, Kramer＇s PDE model. The usefulness of this method is investigated by experimental results. We apply this method to a medical X-ray image. For comparison, the X-ray image is also processed using classic Perona-Malik PDE model and Catte PDE model. Although the Perona-Malik model and Catte PDE model could also enhance the image, the quality of the enhanced images is considerably inferior compared with the enhanced image using Kramer＇s PDE model. The study suggests that the Kramer＇s PDE model is capable of enhancing medical X-ray images, which will make the X-ray images more reliable.展开更多
文摘The 'trial function method' ( TFM for short) and a routine way in finding traveling,wave solutions to some nonlinear partial differential equations( PDE for short), wer explained. Two types of evolution equations are studied, one is a generalized Burgers or KdV equation, the other is the Fisher equation with special nonlinear forms of its reaction rate term. One can see that this method is simple, fast and allowing further extension.
基金supported by NSFC(Grant No.11601490).Research of Y.Xu is supported by NSFC(Grant No.12071455).
文摘In[20],a semi-implicit spectral deferred correction(SDC)method was proposed,which is efficient for highly nonlinear partial differential equations(PDEs).The semi-implicit SDC method in[20]is based on first-order time integration methods,which are corrected iteratively,with the order of accuracy increased by one for each additional iteration.In this paper,we will develop a class of semi-implicit SDC methods,which are based on second-order time integration methods and the order of accuracy are increased by two for each additional iteration.For spatial discretization,we employ the local discontinuous Galerkin(LDG)method to arrive at fully-discrete schemes,which are high-order accurate in both space and time.Numerical experiments are presented to demonstrate the accuracy,efficiency and robustness of the proposed semi-implicit SDC methods for solving complex nonlinear PDEs.
文摘In this paper, the nonlinear Hunter–Saxton equation, which is a famous partial differential equation,is solved by using a hybrid numerical method based on the quasilinearization method and the bivariate generalized fractional order of the Chebyshev functions(B-GFCF) collocation method. First, using the quasilinearization method,the equation is converted into a sequence of linear partial differential equations(LPD), and then these LPDs are solved using the B-GFCF collocation method. A very good approximation of solutions is obtained, and comparisons show that the obtained results are more accurate than the results of other researchers.
基金Natural Science Foundation of Gansu Province of China
文摘With the aid of a nonlinear transformation, a class of nonlinear convection-diffusion PDE in one space dimension is converted into a linear one, the unique solution of a nonlinear boundary-initial value problem for the nonlinear PDE can be exactly expressed by the nonlinear transformation, and several illustrative examples are given.
文摘In this paper, a numerical solution of nonlinear partial differential equation, Benjamin-Bona-Mahony (BBM) and Cahn-Hilliard equation is presented by using Adomain Decomposition Method (ADM) and Variational Iteration Method (VIM). The results reveal that the two methods are very effective, simple and very close to the exact solution.
文摘The purpose of this study is to present an application of a novel enhancement technique for enhancing medical images generated from X-rays. The method presented in this study is based on a nonlinear partial differential equation （PDE） model, Kramer＇s PDE model. The usefulness of this method is investigated by experimental results. We apply this method to a medical X-ray image. For comparison, the X-ray image is also processed using classic Perona-Malik PDE model and Catte PDE model. Although the Perona-Malik model and Catte PDE model could also enhance the image, the quality of the enhanced images is considerably inferior compared with the enhanced image using Kramer＇s PDE model. The study suggests that the Kramer＇s PDE model is capable of enhancing medical X-ray images, which will make the X-ray images more reliable.
