This paper deals with the forward and backward problems for the nonlinear fractional pseudo-parabolic equation ut+(-Δ)^(s1)ut+β(-Δ)^(s2)u=F(u,x,t)subject o random Gaussian white noise for initial and final data.Und...This paper deals with the forward and backward problems for the nonlinear fractional pseudo-parabolic equation ut+(-Δ)^(s1)ut+β(-Δ)^(s2)u=F(u,x,t)subject o random Gaussian white noise for initial and final data.Under the suitable assumptions s1,s2andβ,we first show the ill-posedness of mild solutions for forward and backward problems in the sense of Hadamard,which are mainly driven by random noise.Moreover,we propose the Fourier truncation method for stabilizing the above ill-posed problems.We derive an error estimate between the exact solution and its regularized solution in an E‖·‖Hs22norm,and give some numerical examples illustrating the effect of above method.展开更多
By defining fuzzy valued simple functions and giving L1(μ) approximations of fuzzy valued integrably bounded functions by such simple functions, the paper analyses by L1(μ)-norm the approximation capability of four-...By defining fuzzy valued simple functions and giving L1(μ) approximations of fuzzy valued integrably bounded functions by such simple functions, the paper analyses by L1(μ)-norm the approximation capability of four-layer feedforward regular fuzzy neural networks to the fuzzy valued integrably bounded function F : Rn → FcO(R). That is, if the transfer functionσ: R→R is non-polynomial and integrable function on each finite interval, F may be innorm approximated by fuzzy valued functions defined as to anydegree of accuracy. Finally some real examples demonstrate the conclusions.展开更多
Four layer feedforward regular fuzzy neural networks are constructed. Universal approximations to some continuous fuzzy functions defined on F 0 (R) n by the four layer fuzzy neural networks are shown. At f...Four layer feedforward regular fuzzy neural networks are constructed. Universal approximations to some continuous fuzzy functions defined on F 0 (R) n by the four layer fuzzy neural networks are shown. At first,multivariate Bernstein polynomials associated with fuzzy valued functions are empolyed to approximate continuous fuzzy valued functions defined on each compact set of R n . Secondly,by introducing cut preserving fuzzy mapping,the equivalent conditions for continuous fuzzy functions that can be arbitrarily closely approximated by regular fuzzy neural networks are shown. Finally a few of sufficient and necessary conditions for characterizing approximation capabilities of regular fuzzy neural networks are obtained. And some concrete fuzzy functions demonstrate our conclusions.展开更多
In this article,we prove a regularity result for weak solutions away from singular set of stationary Navier-Stokes systems with subquadratic growth under controllable growth condition.The proof is based on the A-harmo...In this article,we prove a regularity result for weak solutions away from singular set of stationary Navier-Stokes systems with subquadratic growth under controllable growth condition.The proof is based on the A-harmonic approximation technique.In this article,we extend the result of Shuhong Chen and Zhong Tan[7]and Giaquinta and Modica[18]to the stationary Navier-Stokes system with subquadratic growth.展开更多
How to get the rapid and stable inversion results and reconstruct the clear subsurface resistivity structures is a focus problem in current magnetotelluric inversion. A stable solution of an ill-posed inverse problem ...How to get the rapid and stable inversion results and reconstruct the clear subsurface resistivity structures is a focus problem in current magnetotelluric inversion. A stable solution of an ill-posed inverse problem was obtained by the regularization methods in which some desired structures were imposed to stabilize the inverse problem. By the smoothness-constrained model and approximate sensitivity method, the stable subsurface resistivity structures were reconstructed. The synthetic examples show that the smoothness-constrained regularized inversion method is effective and can be reasonable to reconstruct three-dimensional subsurface resistivity structures.展开更多
3D inversion of borehole-surface electrical data for complex geo-electrical models is still a challenging problem in geophysical exploration. We have developed a program for 3D inversion to borehole-surface electrical...3D inversion of borehole-surface electrical data for complex geo-electrical models is still a challenging problem in geophysical exploration. We have developed a program for 3D inversion to borehole-surface electrical data based on the quasi-analytical approximation （QA） and re-weighted regularized conjugate gradient method （RRCG） algorithms using Visual Fortran 6.5. Application of the QA approximation to forward modeling and Frechet derivative computations speeds up the calculation dramatically. The trial calculation for synthetic data of theoretical model showed that the program is fast and highly precise.展开更多
This paper presents a meshless method for the nonlinear generalized regularized long wave （GRLW） equation based on the moving least-squares approximation. The nonlinear discrete scheme of the GRLW equation is obtain...This paper presents a meshless method for the nonlinear generalized regularized long wave （GRLW） equation based on the moving least-squares approximation. The nonlinear discrete scheme of the GRLW equation is obtained and is solved using the iteration method. A theorem on the convergence of the iterative process is presented and proved using theorems of the infinity norm. Compared with numerical methods based on mesh, the meshless method for the GRLW equation only requires the.scattered nodes instead of meshing the domain of the problem. Some examples, such as the propagation of single soliton and the interaction of two solitary waves, are given to show the effectiveness of the meshless method.展开更多
An intensity-based non-rigid registration algorithm is discussed, which uses Gaussian smoothing to constrain the transformation to be smooth, and thus preserves the topology of images. In view of the insufficiency of ...An intensity-based non-rigid registration algorithm is discussed, which uses Gaussian smoothing to constrain the transformation to be smooth, and thus preserves the topology of images. In view of the insufficiency of the uniform Gaussian filtering of the deformation field, an automatic and accurate non-rigid image registration method based on B-splines approximation is proposed. The regularization strategy is adopted by using multi-level B-splines approximation to regularize the displacement fields in a coarse-to-fine manner. Moreover, it assigns the different weights to the estimated displacements according to their reliabilities. In this way, the level of regularity can be adapted locally. Experiments were performed on both synthetic and real medical images of brain, and the results show that the proposed method improves the registration accuracy and robustness.展开更多
Approximate entropy (ApEn), a measure quantifying regularity and complexity, is believed to be an effective analyzing method of diverse settings that include both deterministic chaotic and stochastic processes, partic...Approximate entropy (ApEn), a measure quantifying regularity and complexity, is believed to be an effective analyzing method of diverse settings that include both deterministic chaotic and stochastic processes, particularly operative in the analysis of physiological signals that involve relatively small amount of data. However, the similarity definition of vectors based on Heaviside function, of which the boundary is discontinuous and hard, may cause some problems in the validity and accuracy of ApEn. To overcome these problems, a modified ApEn based on fuzzy similarity (mApEn) was proposed. The performance on the MIX stochastic model, as well as those on the Logistic map and the Hennon map with noise, shows that the fuzzy similarity-based ApEn gets more satisfying results than the standard ApEn when characterizing systems with different regularities.展开更多
For a subset K of a metric space(X,d)and x∈X,Px(x)={y∈K:d(x,y)=d(x,K)≡inf{d(x,k):k∈K}}is called the set of best K-approximant to x.An element go E K is said to be a best simulta-neous approximation of the pair y1,...For a subset K of a metric space(X,d)and x∈X,Px(x)={y∈K:d(x,y)=d(x,K)≡inf{d(x,k):k∈K}}is called the set of best K-approximant to x.An element go E K is said to be a best simulta-neous approximation of the pair y1,y2 E∈if max{d(y1,go),d(y2,go)}=inf g∈K max{d(y1,g),d(y2,g)}.In this paper,some results on the existence of common fixed points for Banach operator pairs in the framework of convex metric spaces have been proved.For self mappings T and S on K,results are proved on both T-and S-invariant points for a set of best simultaneous approximation.Some results on best K-approximant are also deduced.The results proved generalize and extend some results of I.Beg and M.Abbas^[1],S.Chandok and T.D.Narang^[2],T.D.Narang and S.Chandok^[11],S.A.Sahab,M.S.Khan and S.Sessa^[14],P.Vijayaraju^[20]and P.Vijayaraju and M.Marudai^[21].展开更多
A new method for approximating the inerse Laplace transform is presented. We first change our Laplace transform equation into a convolution type integral equation, where Tikhonov regularization techniques and the Four...A new method for approximating the inerse Laplace transform is presented. We first change our Laplace transform equation into a convolution type integral equation, where Tikhonov regularization techniques and the Fourier transformation are easily applied. We finally obtain a regularized approximation to the inverse Laplace transform as finite sum展开更多
This article is concerned with the partial regularity for the weak solutions of stationary Navier-Stokes system under the controllable growth condition.By A-harmonic approximation technique,the optimal regularity is o...This article is concerned with the partial regularity for the weak solutions of stationary Navier-Stokes system under the controllable growth condition.By A-harmonic approximation technique,the optimal regularity is obtained.展开更多
In this article, we consider nonlinear elliptic systems of divergence type with Dini continuous coefficients. The authors use a new method introduced by Duzaar and Grotowski, to prove partial regularity for weak solut...In this article, we consider nonlinear elliptic systems of divergence type with Dini continuous coefficients. The authors use a new method introduced by Duzaar and Grotowski, to prove partial regularity for weak solutions, based on a generalization of the technique of harmonic approximation and directly establish the optimal HSlder exponent for the derivative of a weak solution on its regular set.展开更多
In this article, we consider the partial regularity of stationary Navier-Stokes system under the natural growth condition. Applying the method of A-harmonic approximation,we obtain some results about the partial regul...In this article, we consider the partial regularity of stationary Navier-Stokes system under the natural growth condition. Applying the method of A-harmonic approximation,we obtain some results about the partial regularity and establish the optimal Holder exponent for the derivative of a weak solution on its regular set.展开更多
Two new regularization algorithms for solving the first-kind Volterra integral equation, which describes the pressure-rate deconvolution problem in well test data interpretation, are developed in this paper. The main ...Two new regularization algorithms for solving the first-kind Volterra integral equation, which describes the pressure-rate deconvolution problem in well test data interpretation, are developed in this paper. The main features of the problem are the strong nonuniform scale of the solution and large errors (up to 15%) in the input data. In both algorithms, the solution is represented as decomposition on special basic functions, which satisfy given a priori information on solution, and this idea allow us significantly to improve the quality of approximate solution and simplify solving the minimization problem. The theoretical details of the algorithms, as well as the results of numerical experiments for proving robustness of the algorithms, are presented.展开更多
This paper is devoted to the long time behavior of the solution to the initial boundary value problems for a class of the Kirchhoff wave equations with nonlinear strongly damped terms: . Firstly, in order to prove the...This paper is devoted to the long time behavior of the solution to the initial boundary value problems for a class of the Kirchhoff wave equations with nonlinear strongly damped terms: . Firstly, in order to prove the smoothing effect of the solution, we make efficient use of the analytic property of the semigroup generated by the principal operator of the equation in the phase space. Then we obtain the regularity of the global attractor and construct the approximate inertial manifold of the equation. Finally, we prove that arbitrary trajectory of the Kirchhoff wave equations goes into a small neighbourhood of the approximate inertial manifold after large time.展开更多
In this article,we consider interior regularity for weak solutions to nonlinear elliptic systems of divergence type with Dini continuous coefficients under natural growth condition for the case 1〈m〈 2.All estimates ...In this article,we consider interior regularity for weak solutions to nonlinear elliptic systems of divergence type with Dini continuous coefficients under natural growth condition for the case 1〈m〈 2.All estimates in the case of m≥2 is no longer suitable,and we can’t obtain the Caccioppoli’s second inequality by using these techniques developed in the case of m≥2.But the Caccioppoli’s second inequality is the key to use A-harmonic approximation method.Thus,we adopt another technique introduced by Acerbi and Fcsco to overcome the difficulty and we also overcome those difficulties due to Dini condition.And then we apply the A-harmonic approximation method to prove partial regularity of weak solutions.展开更多
Hyperbolic conservation laws arise in the context of continuum physics,and are mathematically presented in differential form and understood in the distributional(weak)sense.The formal application of the Gauss-Green th...Hyperbolic conservation laws arise in the context of continuum physics,and are mathematically presented in differential form and understood in the distributional(weak)sense.The formal application of the Gauss-Green theorem results in integral balance laws,in which the concept of flux plays a central role.This paper addresses the spacetime viewpoint of the flux regularity,providing a rigorous treatment of integral balance laws.The established Lipschitz regularity of fluxes(over time intervals)leads to a consistent flux approximation.Thus,fully discrete finite volume schemes of high order may be consistently justified with reference to the spacetime integral balance laws.展开更多
The concept of two-direction refinable functions and two-direction wavelets is introduced.We investigate the existence of distributional(or L2-stable) solutions of the two-direction refinement equation: φ(x)=∑p+kφ(...The concept of two-direction refinable functions and two-direction wavelets is introduced.We investigate the existence of distributional(or L2-stable) solutions of the two-direction refinement equation: φ(x)=∑p+kφ(mx-k)+∑p-kφ(k-mx) where m ≥ 2 is an integer. Based on the positive mask {pk+} and negative mask {p-k}, the conditions that guarantee the above equation has compactly distributional solutions or L2-stable solutions are established. Furthermore, the condition that the L2-stable solution of the above equation can generate a two-direction MRA is given. The support interval of φ(x) is discussed amply. The definition of orthogonal two-direction refinable function and orthogonal two-direction wavelets is presented, and the orthogonality criteria for two-direction refinable functions are established. An algorithm for constructing orthogonal two-direction refinable functions and their two-direction wavelets is presented. Another construction algorithm for two-direction L2-refinable functions, which have nonnegative symbol masks and possess high approximation order and regularity, is presented. Finally, two construction examples are given.展开更多
Abstract In the present paper, we construct two approximate inertial manifolds for the generalized symmetric regularized long wave equations with damping term. The orders of approximations of these manifolds to the gl...Abstract In the present paper, we construct two approximate inertial manifolds for the generalized symmetric regularized long wave equations with damping term. The orders of approximations of these manifolds to the global attractor are derived.展开更多
基金supported by the Natural Science Foundation of China(11801108)the Natural Science Foundation of Guangdong Province(2021A1515010314)the Science and Technology Planning Project of Guangzhou City(202201010111)。
文摘This paper deals with the forward and backward problems for the nonlinear fractional pseudo-parabolic equation ut+(-Δ)^(s1)ut+β(-Δ)^(s2)u=F(u,x,t)subject o random Gaussian white noise for initial and final data.Under the suitable assumptions s1,s2andβ,we first show the ill-posedness of mild solutions for forward and backward problems in the sense of Hadamard,which are mainly driven by random noise.Moreover,we propose the Fourier truncation method for stabilizing the above ill-posed problems.We derive an error estimate between the exact solution and its regularized solution in an E‖·‖Hs22norm,and give some numerical examples illustrating the effect of above method.
基金Supported by the National Natural Science Foundation of China(No:69872039)
文摘By defining fuzzy valued simple functions and giving L1(μ) approximations of fuzzy valued integrably bounded functions by such simple functions, the paper analyses by L1(μ)-norm the approximation capability of four-layer feedforward regular fuzzy neural networks to the fuzzy valued integrably bounded function F : Rn → FcO(R). That is, if the transfer functionσ: R→R is non-polynomial and integrable function on each finite interval, F may be innorm approximated by fuzzy valued functions defined as to anydegree of accuracy. Finally some real examples demonstrate the conclusions.
基金This work was supported by National Natural Science Foundation(699740 4 1 699740 0 6)
文摘Four layer feedforward regular fuzzy neural networks are constructed. Universal approximations to some continuous fuzzy functions defined on F 0 (R) n by the four layer fuzzy neural networks are shown. At first,multivariate Bernstein polynomials associated with fuzzy valued functions are empolyed to approximate continuous fuzzy valued functions defined on each compact set of R n . Secondly,by introducing cut preserving fuzzy mapping,the equivalent conditions for continuous fuzzy functions that can be arbitrarily closely approximated by regular fuzzy neural networks are shown. Finally a few of sufficient and necessary conditions for characterizing approximation capabilities of regular fuzzy neural networks are obtained. And some concrete fuzzy functions demonstrate our conclusions.
文摘In this article,we prove a regularity result for weak solutions away from singular set of stationary Navier-Stokes systems with subquadratic growth under controllable growth condition.The proof is based on the A-harmonic approximation technique.In this article,we extend the result of Shuhong Chen and Zhong Tan[7]and Giaquinta and Modica[18]to the stationary Navier-Stokes system with subquadratic growth.
基金Project(20110162120064)supported by Higher School Doctor Subject Special Scientific Research Foundation of ChinaProject(10JJ6059)supported by the Natural Science Foundation of Hunan Province,China
文摘How to get the rapid and stable inversion results and reconstruct the clear subsurface resistivity structures is a focus problem in current magnetotelluric inversion. A stable solution of an ill-posed inverse problem was obtained by the regularization methods in which some desired structures were imposed to stabilize the inverse problem. By the smoothness-constrained model and approximate sensitivity method, the stable subsurface resistivity structures were reconstructed. The synthetic examples show that the smoothness-constrained regularized inversion method is effective and can be reasonable to reconstruct three-dimensional subsurface resistivity structures.
文摘3D inversion of borehole-surface electrical data for complex geo-electrical models is still a challenging problem in geophysical exploration. We have developed a program for 3D inversion to borehole-surface electrical data based on the quasi-analytical approximation （QA） and re-weighted regularized conjugate gradient method （RRCG） algorithms using Visual Fortran 6.5. Application of the QA approximation to forward modeling and Frechet derivative computations speeds up the calculation dramatically. The trial calculation for synthetic data of theoretical model showed that the program is fast and highly precise.
基金supported by the National Natural Science Foundation of China (Grant No. 10871124)the Innovation Program of the Shanghai Municipal Education Commission,China (Grant No. 09ZZ99)
文摘This paper presents a meshless method for the nonlinear generalized regularized long wave （GRLW） equation based on the moving least-squares approximation. The nonlinear discrete scheme of the GRLW equation is obtained and is solved using the iteration method. A theorem on the convergence of the iterative process is presented and proved using theorems of the infinity norm. Compared with numerical methods based on mesh, the meshless method for the GRLW equation only requires the.scattered nodes instead of meshing the domain of the problem. Some examples, such as the propagation of single soliton and the interaction of two solitary waves, are given to show the effectiveness of the meshless method.
基金Supported by National Natural Science Foundation of China (No60373061)Joint Programof National Natural Science Foundation of ChinaGeneral Administration of Civil Aviation of China (No60672168)
文摘An intensity-based non-rigid registration algorithm is discussed, which uses Gaussian smoothing to constrain the transformation to be smooth, and thus preserves the topology of images. In view of the insufficiency of the uniform Gaussian filtering of the deformation field, an automatic and accurate non-rigid image registration method based on B-splines approximation is proposed. The regularization strategy is adopted by using multi-level B-splines approximation to regularize the displacement fields in a coarse-to-fine manner. Moreover, it assigns the different weights to the estimated displacements according to their reliabilities. In this way, the level of regularity can be adapted locally. Experiments were performed on both synthetic and real medical images of brain, and the results show that the proposed method improves the registration accuracy and robustness.
基金The National Basic Research Program (973)of China (No 2005CB724303)
文摘Approximate entropy (ApEn), a measure quantifying regularity and complexity, is believed to be an effective analyzing method of diverse settings that include both deterministic chaotic and stochastic processes, particularly operative in the analysis of physiological signals that involve relatively small amount of data. However, the similarity definition of vectors based on Heaviside function, of which the boundary is discontinuous and hard, may cause some problems in the validity and accuracy of ApEn. To overcome these problems, a modified ApEn based on fuzzy similarity (mApEn) was proposed. The performance on the MIX stochastic model, as well as those on the Logistic map and the Hennon map with noise, shows that the fuzzy similarity-based ApEn gets more satisfying results than the standard ApEn when characterizing systems with different regularities.
文摘For a subset K of a metric space(X,d)and x∈X,Px(x)={y∈K:d(x,y)=d(x,K)≡inf{d(x,k):k∈K}}is called the set of best K-approximant to x.An element go E K is said to be a best simulta-neous approximation of the pair y1,y2 E∈if max{d(y1,go),d(y2,go)}=inf g∈K max{d(y1,g),d(y2,g)}.In this paper,some results on the existence of common fixed points for Banach operator pairs in the framework of convex metric spaces have been proved.For self mappings T and S on K,results are proved on both T-and S-invariant points for a set of best simultaneous approximation.Some results on best K-approximant are also deduced.The results proved generalize and extend some results of I.Beg and M.Abbas^[1],S.Chandok and T.D.Narang^[2],T.D.Narang and S.Chandok^[11],S.A.Sahab,M.S.Khan and S.Sessa^[14],P.Vijayaraju^[20]and P.Vijayaraju and M.Marudai^[21].
文摘A new method for approximating the inerse Laplace transform is presented. We first change our Laplace transform equation into a convolution type integral equation, where Tikhonov regularization techniques and the Fourier transformation are easily applied. We finally obtain a regularized approximation to the inverse Laplace transform as finite sum
基金NNSF of China(10531020)the Program of 985 Innovation Engineering on Information in Xiamen University(2004-2007)
文摘This article is concerned with the partial regularity for the weak solutions of stationary Navier-Stokes system under the controllable growth condition.By A-harmonic approximation technique,the optimal regularity is obtained.
基金Supported by NSF of China (10531020)the Education Department of Fujian Province(JK2009045)the Program of 985 Innovation Engieering on Information in Xiamen University(2004-2007)
文摘In this article, we consider nonlinear elliptic systems of divergence type with Dini continuous coefficients. The authors use a new method introduced by Duzaar and Grotowski, to prove partial regularity for weak solutions, based on a generalization of the technique of harmonic approximation and directly establish the optimal HSlder exponent for the derivative of a weak solution on its regular set.
基金supported by the National Natural Science Foundation of China(11271305,11531010)
文摘In this article, we consider the partial regularity of stationary Navier-Stokes system under the natural growth condition. Applying the method of A-harmonic approximation,we obtain some results about the partial regularity and establish the optimal Holder exponent for the derivative of a weak solution on its regular set.
文摘Two new regularization algorithms for solving the first-kind Volterra integral equation, which describes the pressure-rate deconvolution problem in well test data interpretation, are developed in this paper. The main features of the problem are the strong nonuniform scale of the solution and large errors (up to 15%) in the input data. In both algorithms, the solution is represented as decomposition on special basic functions, which satisfy given a priori information on solution, and this idea allow us significantly to improve the quality of approximate solution and simplify solving the minimization problem. The theoretical details of the algorithms, as well as the results of numerical experiments for proving robustness of the algorithms, are presented.
文摘This paper is devoted to the long time behavior of the solution to the initial boundary value problems for a class of the Kirchhoff wave equations with nonlinear strongly damped terms: . Firstly, in order to prove the smoothing effect of the solution, we make efficient use of the analytic property of the semigroup generated by the principal operator of the equation in the phase space. Then we obtain the regularity of the global attractor and construct the approximate inertial manifold of the equation. Finally, we prove that arbitrary trajectory of the Kirchhoff wave equations goes into a small neighbourhood of the approximate inertial manifold after large time.
基金Supported by National Natural Science Foundation of China (10976026)the Education Department of Fujian Province (JK2009045)
文摘In this article,we consider interior regularity for weak solutions to nonlinear elliptic systems of divergence type with Dini continuous coefficients under natural growth condition for the case 1〈m〈 2.All estimates in the case of m≥2 is no longer suitable,and we can’t obtain the Caccioppoli’s second inequality by using these techniques developed in the case of m≥2.But the Caccioppoli’s second inequality is the key to use A-harmonic approximation method.Thus,we adopt another technique introduced by Acerbi and Fcsco to overcome the difficulty and we also overcome those difficulties due to Dini condition.And then we apply the A-harmonic approximation method to prove partial regularity of weak solutions.
基金supported by the NSFC(Nos.11771054,12072042,91852207)the Sino-German Research Group Project(No.GZ1465)the National Key Project GJXM92579.
文摘Hyperbolic conservation laws arise in the context of continuum physics,and are mathematically presented in differential form and understood in the distributional(weak)sense.The formal application of the Gauss-Green theorem results in integral balance laws,in which the concept of flux plays a central role.This paper addresses the spacetime viewpoint of the flux regularity,providing a rigorous treatment of integral balance laws.The established Lipschitz regularity of fluxes(over time intervals)leads to a consistent flux approximation.Thus,fully discrete finite volume schemes of high order may be consistently justified with reference to the spacetime integral balance laws.
基金This work was Supported by the Natural Science Foundation of Guangdong Province (Grant Nos.06105648,05008289,032038)the Doctoral Foundation of Guangdong Province (Grant No.04300917)
文摘The concept of two-direction refinable functions and two-direction wavelets is introduced.We investigate the existence of distributional(or L2-stable) solutions of the two-direction refinement equation: φ(x)=∑p+kφ(mx-k)+∑p-kφ(k-mx) where m ≥ 2 is an integer. Based on the positive mask {pk+} and negative mask {p-k}, the conditions that guarantee the above equation has compactly distributional solutions or L2-stable solutions are established. Furthermore, the condition that the L2-stable solution of the above equation can generate a two-direction MRA is given. The support interval of φ(x) is discussed amply. The definition of orthogonal two-direction refinable function and orthogonal two-direction wavelets is presented, and the orthogonality criteria for two-direction refinable functions are established. An algorithm for constructing orthogonal two-direction refinable functions and their two-direction wavelets is presented. Another construction algorithm for two-direction L2-refinable functions, which have nonnegative symbol masks and possess high approximation order and regularity, is presented. Finally, two construction examples are given.
文摘Abstract In the present paper, we construct two approximate inertial manifolds for the generalized symmetric regularized long wave equations with damping term. The orders of approximations of these manifolds to the global attractor are derived.
