In this paper, existence and uniqueness of the generalized global solution and the classical global solution to the initial value problem for a class of fourth-order nonlinear wave equations are studied in the fractio...In this paper, existence and uniqueness of the generalized global solution and the classical global solution to the initial value problem for a class of fourth-order nonlinear wave equations are studied in the fractional order Sobolev space using the contraction mapping principle and the extension theorem. The sufficient conditions for the blow up of the solution to the initial value problem are given.展开更多
This paper proposes a new collocation method for initial value problems of second order ODEs based on the Laguerre-Gauss interpolation. It provides the global numerical solutions and possesses the spectral accuracy. N...This paper proposes a new collocation method for initial value problems of second order ODEs based on the Laguerre-Gauss interpolation. It provides the global numerical solutions and possesses the spectral accuracy. Numerical results demonstrate its high efficiency.展开更多
We employ the Duan-Rach-Wazwaz modified Adomian decomposition method for solving initial value problems for the systems of nonlinear ordinary differential equations numerically. In order to confirm practicality, robus...We employ the Duan-Rach-Wazwaz modified Adomian decomposition method for solving initial value problems for the systems of nonlinear ordinary differential equations numerically. In order to confirm practicality, robustness and reliability of the method, we compare the results from the modified Adomian decomposition method with those from the MATHEMATICA solutions and also from the fourth-order Runge Kutta method solutions in some cases. Furthermore, we apply Padé approximants technique to improve the solutions of the modified decomposition method whenever the exact solutions exist.展开更多
In this paper, using the differentiability of the solution with respect to the initial value and the parameter, we present a method which, different from Liapunov's direct method. will determine the stability oj t...In this paper, using the differentiability of the solution with respect to the initial value and the parameter, we present a method which, different from Liapunov's direct method. will determine the stability oj the non-stationary solution of the initial value problem when the non-stationary solution remains unknown.展开更多
Interest in the construction of efficient methods for solving initial value problems that have some peculiar properties with it or its solution is recently gaining wide popularity. Based on the assumption that the sol...Interest in the construction of efficient methods for solving initial value problems that have some peculiar properties with it or its solution is recently gaining wide popularity. Based on the assumption that the solution is representable by nonlinear trigonometric expressions, this work presents an explicit single-step nonlinear method for solving first order initial value problems whose solution possesses singularity. The stability and convergence properties of the constructed scheme are also presented. Implementation of the new method on some standard test problems compared with those discussed in the literature proved its accuracy and efficiency.展开更多
This paper focuses on the application of Mamadu-Njoseh polynomials(MNPs)as basis functions for the solution of singular initial value problems in the second-order ordinary differential equations in a perturbation by d...This paper focuses on the application of Mamadu-Njoseh polynomials(MNPs)as basis functions for the solution of singular initial value problems in the second-order ordinary differential equations in a perturbation by decomposition approach.Here,the proposed method is an hybrid of the perturbation theory and decomposition method.In this approach,the approximate solution is slihtly perturbed with the MNPs to ensure absolute convergence.Nonlinear cases are first treated by decomposition.The method is,easy to execute with well-posed mathematical formulae.The existence and convergence of the method is also presented explicitly.Resulting numerical evidences show that the proposed method,in comparison with the Adomian Decomposition Method(ADM),Homotpy Pertubation Method and the exact solution is reliable,efficient and accuarate.展开更多
The well-posedness of the initial value problem of the Euler equations was mainly discussed based on the stratification theory, and the necessary and sufficient conditions of well-posedness are presented for some repr...The well-posedness of the initial value problem of the Euler equations was mainly discussed based on the stratification theory, and the necessary and sufficient conditions of well-posedness are presented for some representative initial or boundary value problem, thus the structure of solution space for local (exact) solution of the Euler equations is determined. Moreover the computation formulas of the analytical solution of the well-posed problem are also given.展开更多
We propose and analyze a single-interval Legendre-Gauss-Radau(LGR)spectral collocation method for nonlinear second-order initial value problems of ordinary differential equations.We design an efficient iterative algor...We propose and analyze a single-interval Legendre-Gauss-Radau(LGR)spectral collocation method for nonlinear second-order initial value problems of ordinary differential equations.We design an efficient iterative algorithm and prove spectral convergence for the single-interval LGR collocation method.For more effective implementation,we propose a multi-interval LGR spectral collocation scheme,which provides us great flexibility with respect to the local time steps and local approximation degrees.Moreover,we combine the multi-interval LGR collocation method in time with the Legendre-Gauss-Lobatto collocation method in space to obtain a space-time spectral collocation approximation for nonlinear second-order evolution equations.Numerical results show that the proposed methods have high accuracy and excellent long-time stability.Numerical comparison between our methods and several commonly used methods are also provided.展开更多
This paper is devoted to studying the representation of measures of non-generalized compactness,in particular,measures of noncompactness,of non-weak compactness and of non-super weak compactness,defined on Banach spac...This paper is devoted to studying the representation of measures of non-generalized compactness,in particular,measures of noncompactness,of non-weak compactness and of non-super weak compactness,defined on Banach spaces and its applications.With the aid of a three-time order-preserving embedding theorem,we show that for every Banach space X,there exist a Banach function space C(K)for some compact Hausdorff space K and an order-preserving affine mapping T from the super space B of all the nonempty bounded subsets of X endowed with the Hausdorff metric to the positive cone C(K)^(+) of C(K),such that for every convex measure,in particular,the regular measure,the homogeneous measure and the sublinear measure of non-generalized compactnessμon X,there is a convex function F on the cone V=T(B)which is Lipschitzian on each bounded set of V such that F(T(B))=μ(B),■B∈B.As its applications,we show a class of basic integral inequalities related to an initial value problem in Banach spaces,and prove a solvability result of the initial value problem,which is an extension of some classical results due to Bana′s and Goebel(1980),Goebel and Rzymowski(1970)and Rzymowski(1971).展开更多
This paper is devoted to studying the initial value problem of the modified nonlinear Kawahara equation the first partial dervative of u to t ,the second the third +α the second partial dervative of u to x ,the seco...This paper is devoted to studying the initial value problem of the modified nonlinear Kawahara equation the first partial dervative of u to t ,the second the third +α the second partial dervative of u to x ,the second the third +β the third partial dervative of u to x ,the second the thire +γ the fifth partial dervative of u to x = 0,(x,t)∈R^2.We first establish several Strichartz type estimates for the fundamental solution of the corresponding linear problem. Then we apply such estimates to prove local and global existence of solutions for the initial value problem of the modified nonlinear Karahara equation. The results show that a local solution exists if the initial function uo(x) ∈ H^s(R) with s ≥ 1/4, and a global solution exists if s ≥ 2.展开更多
This paper is devoted to studying the initial value problems of the nonlinear Kaup Kupershmidt equations δu/δt + α1 uδ^2u/δx^2 + βδ^3u/δx^3 + γδ^5u/δx^5 = 0, (x,t)∈ E R^2, and δu/δt + α2 δu/δx ...This paper is devoted to studying the initial value problems of the nonlinear Kaup Kupershmidt equations δu/δt + α1 uδ^2u/δx^2 + βδ^3u/δx^3 + γδ^5u/δx^5 = 0, (x,t)∈ E R^2, and δu/δt + α2 δu/δx δ^2u/δx^2 + βδ^3u/δx^3 + γδ^5u/δx^5 = 0, (x, t) ∈R^2. Several important Strichartz type estimates for the fundamental solution of the corresponding linear problem are established. Then we apply such estimates to prove the local and global existence of solutions for the initial value problems of the nonlinear Kaup- Kupershmidt equations. The results show that a local solution exists if the initial function u0(x) ∈ H^s(R), and s ≥ 5/4 for the first equation and s≥301/108 for the second equation.展开更多
Based on the theory of stratification, the well-posedness of the init ial and boundary value problems for the system of two-dimensional non-hydrosta ti c Boussinesq equations was discussed. The sufficient and necessa...Based on the theory of stratification, the well-posedness of the init ial and boundary value problems for the system of two-dimensional non-hydrosta ti c Boussinesq equations was discussed. The sufficient and necessary conditions of the existence and uniqueness for the solution of the equations were given for s ome representative initial and boundary value problems. Several special cases we re discussed.展开更多
The initial value problems and the first boundary problems for the quasilinear wave equation u_(tt)-[a_0+na_1(u_x)^(n-1)]u_(xx)-a_2u_(xxtt)=0 are considered,where a_0,a_2>0 are constants,a_1 is an arbitrary real nu...The initial value problems and the first boundary problems for the quasilinear wave equation u_(tt)-[a_0+na_1(u_x)^(n-1)]u_(xx)-a_2u_(xxtt)=0 are considered,where a_0,a_2>0 are constants,a_1 is an arbitrary real number,n is a natural number.The existence and uniqueness of the classical solutions for the initial value problems and the first boundary problems of the equation (1) are proved by the Galerkin method.展开更多
In this paper, we study the well-posedness of initial value problem for n-dimensional gener-alized Tricomi equation in the mixed-type domain {(t,x):t∈[1,+∞),x∈Rn} with the initial data given on the line t=1 in...In this paper, we study the well-posedness of initial value problem for n-dimensional gener-alized Tricomi equation in the mixed-type domain {(t,x):t∈[1,+∞),x∈Rn} with the initial data given on the line t=1 in Hadamard's sense. By taking partial Fourier transformation, we obtain the explicit expression of the solution in terms of two integral operators and further establish the global estimate of such a solution for a class of initial data and source term. Finally, we establish the global solution in time direction for a semilinear problem used the estimate.展开更多
We propose a collocation method for solving initial value problems of secondorder ODEs by using modified Laguerre functions.This new process provides global numerical solutions.Numerical results demonstrate the effici...We propose a collocation method for solving initial value problems of secondorder ODEs by using modified Laguerre functions.This new process provides global numerical solutions.Numerical results demonstrate the efficiency of the proposed algorithm.展开更多
We present a windowing technique of waveform relaxation for dynamic systems. An effective estimation on window length is derived by an iterative error expression provided here. Relaxation processes can be speeded up i...We present a windowing technique of waveform relaxation for dynamic systems. An effective estimation on window length is derived by an iterative error expression provided here. Relaxation processes can be speeded up if one takes the windowing technique in advance. Numerical experiments are given to further illustrate the theoretical analysis.展开更多
In this paper,a second-order singularly perturbed initial value problem is considered.A hybrid scheme which is a combination of a cubic spline and a modified midpoint upwind scheme is proposed on various types of laye...In this paper,a second-order singularly perturbed initial value problem is considered.A hybrid scheme which is a combination of a cubic spline and a modified midpoint upwind scheme is proposed on various types of layer-adapted meshes.The error bounds are established for the numerical solution and for the scaled numerical derivative in the discrete maximum norm.It is observed that the numerical solution and the scaled numerical derivative are of second-order convergence on the layer-adapted meshes irrespective of the perturbation parameter.To show the performance of the proposed method,it is applied on few test examples which are in agreement with the theoretical results.Furthermore,existing results are also compared to show the robustness of the proposed scheme.展开更多
Under the sign assumptions we investigate the global existence of solutions of the initial value problem x' =f(t, x, x'), x(0) = A, where the scalar function f(t, x,p) may be singular at x = A.
This paper studies the global existence of the classical solutions to the following problem:This problem describes the nonlinear vibrations of finite rods with nonlinear vis-coelasticity. Under certain conditions on ...This paper studies the global existence of the classical solutions to the following problem:This problem describes the nonlinear vibrations of finite rods with nonlinear vis-coelasticity. Under certain conditions on σand f , we obtained the unique existence of the global classical solution of this problem.展开更多
Existence and uniqueness conditions for nonnegative solutions to initial value prob- lems of general sublinear-linear differential equations are obtained.They extend the uniqueness theorem due to H.Murakami~[6] and th...Existence and uniqueness conditions for nonnegative solutions to initial value prob- lems of general sublinear-linear differential equations are obtained.They extend the uniqueness theorem due to H.Murakami~[6] and the main results of H.G.Kaper and M.K.Kwong~[4].展开更多
基金supported by the National Natural Science Foundation of China (No. 10671182)
文摘In this paper, existence and uniqueness of the generalized global solution and the classical global solution to the initial value problem for a class of fourth-order nonlinear wave equations are studied in the fractional order Sobolev space using the contraction mapping principle and the extension theorem. The sufficient conditions for the blow up of the solution to the initial value problem are given.
基金supported by the National Natural Science Foundation of China(No.11171227)the Ph.D.Programs Foundation of Ministry of Education of China(No.20080270001)+2 种基金the Shanghai Leading Academic Discipline Project(No.S30405)the Fund for E-Institute of Shanghai Universities(No.E03004)the Foundation for Distinguished Young Talents in Higher Education of Guangdong of China(No.LYM09138)
文摘This paper proposes a new collocation method for initial value problems of second order ODEs based on the Laguerre-Gauss interpolation. It provides the global numerical solutions and possesses the spectral accuracy. Numerical results demonstrate its high efficiency.
文摘We employ the Duan-Rach-Wazwaz modified Adomian decomposition method for solving initial value problems for the systems of nonlinear ordinary differential equations numerically. In order to confirm practicality, robustness and reliability of the method, we compare the results from the modified Adomian decomposition method with those from the MATHEMATICA solutions and also from the fourth-order Runge Kutta method solutions in some cases. Furthermore, we apply Padé approximants technique to improve the solutions of the modified decomposition method whenever the exact solutions exist.
文摘In this paper, using the differentiability of the solution with respect to the initial value and the parameter, we present a method which, different from Liapunov's direct method. will determine the stability oj the non-stationary solution of the initial value problem when the non-stationary solution remains unknown.
文摘Interest in the construction of efficient methods for solving initial value problems that have some peculiar properties with it or its solution is recently gaining wide popularity. Based on the assumption that the solution is representable by nonlinear trigonometric expressions, this work presents an explicit single-step nonlinear method for solving first order initial value problems whose solution possesses singularity. The stability and convergence properties of the constructed scheme are also presented. Implementation of the new method on some standard test problems compared with those discussed in the literature proved its accuracy and efficiency.
文摘This paper focuses on the application of Mamadu-Njoseh polynomials(MNPs)as basis functions for the solution of singular initial value problems in the second-order ordinary differential equations in a perturbation by decomposition approach.Here,the proposed method is an hybrid of the perturbation theory and decomposition method.In this approach,the approximate solution is slihtly perturbed with the MNPs to ensure absolute convergence.Nonlinear cases are first treated by decomposition.The method is,easy to execute with well-posed mathematical formulae.The existence and convergence of the method is also presented explicitly.Resulting numerical evidences show that the proposed method,in comparison with the Adomian Decomposition Method(ADM),Homotpy Pertubation Method and the exact solution is reliable,efficient and accuarate.
文摘The well-posedness of the initial value problem of the Euler equations was mainly discussed based on the stratification theory, and the necessary and sufficient conditions of well-posedness are presented for some representative initial or boundary value problem, thus the structure of solution space for local (exact) solution of the Euler equations is determined. Moreover the computation formulas of the analytical solution of the well-posed problem are also given.
基金supported in part by the National Natural Science Foundation of China(Grant Nos.12171322,11771298 and 11871043)the Natural Science Foundation of Shanghai(Grant Nos.21ZR1447200,20ZR1441200 and 22ZR1445500)the Science and Technology Innovation Plan of Shanghai(Grant No.20JC1414200).
文摘We propose and analyze a single-interval Legendre-Gauss-Radau(LGR)spectral collocation method for nonlinear second-order initial value problems of ordinary differential equations.We design an efficient iterative algorithm and prove spectral convergence for the single-interval LGR collocation method.For more effective implementation,we propose a multi-interval LGR spectral collocation scheme,which provides us great flexibility with respect to the local time steps and local approximation degrees.Moreover,we combine the multi-interval LGR collocation method in time with the Legendre-Gauss-Lobatto collocation method in space to obtain a space-time spectral collocation approximation for nonlinear second-order evolution equations.Numerical results show that the proposed methods have high accuracy and excellent long-time stability.Numerical comparison between our methods and several commonly used methods are also provided.
基金supported by National Natural Science Foundation of China(Grant No.11731010)。
文摘This paper is devoted to studying the representation of measures of non-generalized compactness,in particular,measures of noncompactness,of non-weak compactness and of non-super weak compactness,defined on Banach spaces and its applications.With the aid of a three-time order-preserving embedding theorem,we show that for every Banach space X,there exist a Banach function space C(K)for some compact Hausdorff space K and an order-preserving affine mapping T from the super space B of all the nonempty bounded subsets of X endowed with the Hausdorff metric to the positive cone C(K)^(+) of C(K),such that for every convex measure,in particular,the regular measure,the homogeneous measure and the sublinear measure of non-generalized compactnessμon X,there is a convex function F on the cone V=T(B)which is Lipschitzian on each bounded set of V such that F(T(B))=μ(B),■B∈B.As its applications,we show a class of basic integral inequalities related to an initial value problem in Banach spaces,and prove a solvability result of the initial value problem,which is an extension of some classical results due to Bana′s and Goebel(1980),Goebel and Rzymowski(1970)and Rzymowski(1971).
文摘This paper is devoted to studying the initial value problem of the modified nonlinear Kawahara equation the first partial dervative of u to t ,the second the third +α the second partial dervative of u to x ,the second the third +β the third partial dervative of u to x ,the second the thire +γ the fifth partial dervative of u to x = 0,(x,t)∈R^2.We first establish several Strichartz type estimates for the fundamental solution of the corresponding linear problem. Then we apply such estimates to prove local and global existence of solutions for the initial value problem of the modified nonlinear Karahara equation. The results show that a local solution exists if the initial function uo(x) ∈ H^s(R) with s ≥ 1/4, and a global solution exists if s ≥ 2.
文摘This paper is devoted to studying the initial value problems of the nonlinear Kaup Kupershmidt equations δu/δt + α1 uδ^2u/δx^2 + βδ^3u/δx^3 + γδ^5u/δx^5 = 0, (x,t)∈ E R^2, and δu/δt + α2 δu/δx δ^2u/δx^2 + βδ^3u/δx^3 + γδ^5u/δx^5 = 0, (x, t) ∈R^2. Several important Strichartz type estimates for the fundamental solution of the corresponding linear problem are established. Then we apply such estimates to prove the local and global existence of solutions for the initial value problems of the nonlinear Kaup- Kupershmidt equations. The results show that a local solution exists if the initial function u0(x) ∈ H^s(R), and s ≥ 5/4 for the first equation and s≥301/108 for the second equation.
基金Project supported by the National Natural Science Foundation of China (Grant No.40175014)
文摘Based on the theory of stratification, the well-posedness of the init ial and boundary value problems for the system of two-dimensional non-hydrosta ti c Boussinesq equations was discussed. The sufficient and necessary conditions of the existence and uniqueness for the solution of the equations were given for s ome representative initial and boundary value problems. Several special cases we re discussed.
文摘The initial value problems and the first boundary problems for the quasilinear wave equation u_(tt)-[a_0+na_1(u_x)^(n-1)]u_(xx)-a_2u_(xxtt)=0 are considered,where a_0,a_2>0 are constants,a_1 is an arbitrary real number,n is a natural number.The existence and uniqueness of the classical solutions for the initial value problems and the first boundary problems of the equation (1) are proved by the Galerkin method.
基金Supported by National Natural Science Foundation of China (Grant No.11001122)Scientific Research Fund of Nanjing Institute of Technology (Grant No.YKJ201113)
文摘In this paper, we study the well-posedness of initial value problem for n-dimensional gener-alized Tricomi equation in the mixed-type domain {(t,x):t∈[1,+∞),x∈Rn} with the initial data given on the line t=1 in Hadamard's sense. By taking partial Fourier transformation, we obtain the explicit expression of the solution in terms of two integral operators and further establish the global estimate of such a solution for a class of initial data and source term. Finally, we establish the global solution in time direction for a semilinear problem used the estimate.
基金supported in part by Foundation for Distinguished Young Talents in Higher Education of Guangdong,China,N.LYM09138supported in part by NSF of China N.10871131+2 种基金Fund for Doctor Authority of Chinese Educational Ministry N.20080270001Shanghai Leading Academic Discipline Project N.S30405Fund for E-institutes of Shanghai Universities N.E03004.
文摘We propose a collocation method for solving initial value problems of secondorder ODEs by using modified Laguerre functions.This new process provides global numerical solutions.Numerical results demonstrate the efficiency of the proposed algorithm.
基金Supported by the National Natural Science Foundation of China(No.60472003)the 863 Program of China(No.2001AA111042)
文摘We present a windowing technique of waveform relaxation for dynamic systems. An effective estimation on window length is derived by an iterative error expression provided here. Relaxation processes can be speeded up if one takes the windowing technique in advance. Numerical experiments are given to further illustrate the theoretical analysis.
基金This research work is supported by the Department of Science and Technology,Government of India Under Research Grant No.EMR/2016/005805.
文摘In this paper,a second-order singularly perturbed initial value problem is considered.A hybrid scheme which is a combination of a cubic spline and a modified midpoint upwind scheme is proposed on various types of layer-adapted meshes.The error bounds are established for the numerical solution and for the scaled numerical derivative in the discrete maximum norm.It is observed that the numerical solution and the scaled numerical derivative are of second-order convergence on the layer-adapted meshes irrespective of the perturbation parameter.To show the performance of the proposed method,it is applied on few test examples which are in agreement with the theoretical results.Furthermore,existing results are also compared to show the robustness of the proposed scheme.
文摘Under the sign assumptions we investigate the global existence of solutions of the initial value problem x' =f(t, x, x'), x(0) = A, where the scalar function f(t, x,p) may be singular at x = A.
文摘This paper studies the global existence of the classical solutions to the following problem:This problem describes the nonlinear vibrations of finite rods with nonlinear vis-coelasticity. Under certain conditions on σand f , we obtained the unique existence of the global classical solution of this problem.
文摘Existence and uniqueness conditions for nonnegative solutions to initial value prob- lems of general sublinear-linear differential equations are obtained.They extend the uniqueness theorem due to H.Murakami~[6] and the main results of H.G.Kaper and M.K.Kwong~[4].