This article is an open access article distributed under the terms and conditions of the Creative... more This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY
Journal of advances in mathematics and computer science, Nov 21, 2018
We present a special B-spline tight frame and use it to introduce our numerical approximation met... more We present a special B-spline tight frame and use it to introduce our numerical approximation method. We apply our method to investigate Gibbs effects and illustrate some features of the associated framelet expansion. It is shown that Gibbs effects occurs in the framelet expansion of a function with a jump discontinuity at 0 for certain classes of framelets. Numerical results are obtained regarding the behavior of the Gibbs effects. We present the results by expanding functions using the quasi-affine system. This system is generated by the B-spline tight framelets with a specific number of generators. We show numerically the existence of Gibbs effects in the truncated expansion of a given function by using some tight framelet representation.
Journal of Applied Mathematics and Computing, May 16, 2019
Integro-differential equations play an important role in may physical phenomena. For instance, it... more Integro-differential equations play an important role in may physical phenomena. For instance, it appears in fields like fluid dynamics, biological models and chemical kinetics. One of the most important physical applications is the heat transfer in heterogeneous materials, where physician are looking for efficient methods to solve their modeled equations. The difficulty of solving integro-differential equations analytically made mathematician to search about efficient methods to find an approximate solution. The present article is designed to supply numerical solution of a parabolic Volterra integro-differential equation under initial and boundary conditions. We have made an attempt to develop a numerical solution via the use of Sinc-Galerkin method, the convergence analysis via the use of fixed point theory has been discussed, and showed to be of exponential order. For comparison purposes, we approximate the solution of integro-differential equation using Adomian decomposition method. Sometimes, the Adomian decomposition method is a highly efficient technique used to approximate analytical solution of differential equations, applicability of Adomian decomposition method to partial integro-differential equations has not been studied in details previously in the literatures. In addition, we present numerical examples and comparisons to support the validity of these proposed methods.
In this paper, we present a robust algorithm to solve numerically a family of two-dimensional fra... more In this paper, we present a robust algorithm to solve numerically a family of two-dimensional fractional integro differential equations. The Haar wavelet method is upgraded to include in its construction the Laplace transform step. This modification has proven to reduce the accumulative errors that will be obtained in case of using the regular Haar wavelet technique. Different examples are discussed to serve two goals, the methodology and the accuracy of our new approach.
Representation theory of locally compact topological groups is a powerful tool to analyze Banach ... more Representation theory of locally compact topological groups is a powerful tool to analyze Banach spaces of functions and distributions. It provides a unified framework for constructing function spaces and to study several generalizations of the wavelet transform. Recently representation theory has been used to provide atomic decompositions for a large collection of classical Banach spaces. But in some natural situations, including Bergman spaces on bounded domains, representations are too restrictive. The proper tools are projective representations. In this paper we extend known techniques from representation theory to also include projective representations. This leads naturally to twisted convolution on groups avoiding the usual central extension of the group. As our main application we obtain atomic decompositions of Bergman spaces on the unit ball through the holomorphic discrete series for the group of isometries of the ball.
Mediterranean Journal of Mathematics, Oct 25, 2019
In this work, we study the L p estimates for a certain class of rough maximal functions with mixe... more In this work, we study the L p estimates for a certain class of rough maximal functions with mixed homogeneity associated with the surfaces of revolution. Using these estimates with an extrapolation argument, we obtain some new results that represent substantially improvements and extensions of many previously known results on maximal operators.
In this work, the generalized parametric Marcinkiewicz integral operators with mixed homogeneity ... more In this work, the generalized parametric Marcinkiewicz integral operators with mixed homogeneity related to surfaces of revolution are studied. Under some weak conditions on the kernels, the boundedness of such operators from Triebel-Lizorkin spaces to L p spaces are established. Our results, with the help of an extrapolation argument, improve and extend some previous known results.
In this paper we introduce fractional-and bi-calculi using Riemann-Liouville approach and Caputo ... more In this paper we introduce fractional-and bi-calculi using Riemann-Liouville approach and Caputo approach as well. An effort is put into explaining the basic principles of these calculi since they are not as common as classical calculus. This was also done for tanh-, bi-tanh-multiplicative, and bigeometric calculi and in the general case as well. Generalizations are also investigated where the homeomorphisms , are arbitrary.
In this paper we introduce fractional φ- and biφ-calculi using Riemann-Liouville approach and Cap... more In this paper we introduce fractional φ- and biφ-calculi using Riemann-Liouville approach and Caputo approach as well. An effort is put into explaining the basic principles of these calculi since they are not as common as classical calculus. This was also done for tanh-, bi-tanh- multiplicative, and bigeometric calculi and in the general case as well. Generalizations are also investigated where the homeomorphisms φ, η are arbitrary.
Banach spaces of functions, or more generally, of distributions are one of the main topics in ana... more Banach spaces of functions, or more generally, of distributions are one of the main topics in analysis. In this thesis, we present an abstract framework for construction of invariant Banach function spaces from projective group representations. Coorbit theory gives a unified method to construct invariant Banach function spaces via representations of Lie groups. This theory was introduced by Feichtinger and Gröchenig in [23, 24, 25, 26] and then extended in [9]. We generalize this concept by constructing coorbit spaces using projective representation which is first studied by O. Christensen in [10]. This allows us to describe wider classes of function spaces as coorbits, in order to construct frames and atomic decompositions for these spaces. As in the general coorbit theory, we construct atomic decompositions and Banach frames for coorbit spaces under certain smoothness conditions. By this modification, we can discretize the Bergman spaces A p α (B n) via the family of projective representations {ρ s } of the group SU(n, 1), for any real parameter s > n. v
Representation theory of locally compact topological groups is a powerful tool to analyze Banach ... more Representation theory of locally compact topological groups is a powerful tool to analyze Banach spaces of functions and distributions. It provides a unified framework for constructing function spaces and to study several generalizations of the wavelet transform. Recently representation theory has been used to provide atomic decompositions for a large collection of classical Banach spaces. But in some natural situations, including Bergman spaces on bounded domains, representations are too restrictive. The proper tools are projective representations. In this paper we extend known techniques from representation theory to also include projective representations. This leads naturally to twisted convolution on groups avoiding the usual central extension of the group. As our main application we obtain atomic decompositions of Bergman spaces on the unit ball through the holomorphic discrete series for the group of isometries of the ball.
This work deals with a new modified version of the Adomian-Rach decomposition method (MDM). The M... more This work deals with a new modified version of the Adomian-Rach decomposition method (MDM). The MDM is based on combining a series solution and decomposition method for solving nonlinear differential equations with Adomian polynomials for nonlinearities. With application to a class of nonlinear oscillators known as the Lienard-type equations, convergence and error analysis are discussed. Several physical problems modeled by Lienard-type equations are considered to illustrate the effectiveness, performance and reliability of the method. In comparison to the 4th Runge-Kutta method (RK4), highly accurate solutions on a large domain are obtained.
In this paper, we present a robust algorithm to solve numerically a family of two-dimensional fra... more In this paper, we present a robust algorithm to solve numerically a family of two-dimensional fractional integro differential equations. The Haar wavelet method is upgraded to include in its construction the Laplace transform step. This modification has proven to reduce the accumulative errors that will be obtained in case of using the regular Haar wavelet technique. Different examples are discussed to serve two goals, the methodology and the accuracy of our new approach.
In this work, the generalized parametric Marcinkiewicz integral operators with mixed homogeneity ... more In this work, the generalized parametric Marcinkiewicz integral operators with mixed homogeneity related to surfaces of revolution are studied. Under some weak conditions on the kernels, the boundedness of such operators from Triebel–Lizorkin spaces to L p spaces are established. Our results, with the help of an extrapolation argument, improve and extend some previous known results.
This article is an open access article distributed under the terms and conditions of the Creative... more This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY
Journal of advances in mathematics and computer science, Nov 21, 2018
We present a special B-spline tight frame and use it to introduce our numerical approximation met... more We present a special B-spline tight frame and use it to introduce our numerical approximation method. We apply our method to investigate Gibbs effects and illustrate some features of the associated framelet expansion. It is shown that Gibbs effects occurs in the framelet expansion of a function with a jump discontinuity at 0 for certain classes of framelets. Numerical results are obtained regarding the behavior of the Gibbs effects. We present the results by expanding functions using the quasi-affine system. This system is generated by the B-spline tight framelets with a specific number of generators. We show numerically the existence of Gibbs effects in the truncated expansion of a given function by using some tight framelet representation.
Journal of Applied Mathematics and Computing, May 16, 2019
Integro-differential equations play an important role in may physical phenomena. For instance, it... more Integro-differential equations play an important role in may physical phenomena. For instance, it appears in fields like fluid dynamics, biological models and chemical kinetics. One of the most important physical applications is the heat transfer in heterogeneous materials, where physician are looking for efficient methods to solve their modeled equations. The difficulty of solving integro-differential equations analytically made mathematician to search about efficient methods to find an approximate solution. The present article is designed to supply numerical solution of a parabolic Volterra integro-differential equation under initial and boundary conditions. We have made an attempt to develop a numerical solution via the use of Sinc-Galerkin method, the convergence analysis via the use of fixed point theory has been discussed, and showed to be of exponential order. For comparison purposes, we approximate the solution of integro-differential equation using Adomian decomposition method. Sometimes, the Adomian decomposition method is a highly efficient technique used to approximate analytical solution of differential equations, applicability of Adomian decomposition method to partial integro-differential equations has not been studied in details previously in the literatures. In addition, we present numerical examples and comparisons to support the validity of these proposed methods.
In this paper, we present a robust algorithm to solve numerically a family of two-dimensional fra... more In this paper, we present a robust algorithm to solve numerically a family of two-dimensional fractional integro differential equations. The Haar wavelet method is upgraded to include in its construction the Laplace transform step. This modification has proven to reduce the accumulative errors that will be obtained in case of using the regular Haar wavelet technique. Different examples are discussed to serve two goals, the methodology and the accuracy of our new approach.
Representation theory of locally compact topological groups is a powerful tool to analyze Banach ... more Representation theory of locally compact topological groups is a powerful tool to analyze Banach spaces of functions and distributions. It provides a unified framework for constructing function spaces and to study several generalizations of the wavelet transform. Recently representation theory has been used to provide atomic decompositions for a large collection of classical Banach spaces. But in some natural situations, including Bergman spaces on bounded domains, representations are too restrictive. The proper tools are projective representations. In this paper we extend known techniques from representation theory to also include projective representations. This leads naturally to twisted convolution on groups avoiding the usual central extension of the group. As our main application we obtain atomic decompositions of Bergman spaces on the unit ball through the holomorphic discrete series for the group of isometries of the ball.
Mediterranean Journal of Mathematics, Oct 25, 2019
In this work, we study the L p estimates for a certain class of rough maximal functions with mixe... more In this work, we study the L p estimates for a certain class of rough maximal functions with mixed homogeneity associated with the surfaces of revolution. Using these estimates with an extrapolation argument, we obtain some new results that represent substantially improvements and extensions of many previously known results on maximal operators.
In this work, the generalized parametric Marcinkiewicz integral operators with mixed homogeneity ... more In this work, the generalized parametric Marcinkiewicz integral operators with mixed homogeneity related to surfaces of revolution are studied. Under some weak conditions on the kernels, the boundedness of such operators from Triebel-Lizorkin spaces to L p spaces are established. Our results, with the help of an extrapolation argument, improve and extend some previous known results.
In this paper we introduce fractional-and bi-calculi using Riemann-Liouville approach and Caputo ... more In this paper we introduce fractional-and bi-calculi using Riemann-Liouville approach and Caputo approach as well. An effort is put into explaining the basic principles of these calculi since they are not as common as classical calculus. This was also done for tanh-, bi-tanh-multiplicative, and bigeometric calculi and in the general case as well. Generalizations are also investigated where the homeomorphisms , are arbitrary.
In this paper we introduce fractional φ- and biφ-calculi using Riemann-Liouville approach and Cap... more In this paper we introduce fractional φ- and biφ-calculi using Riemann-Liouville approach and Caputo approach as well. An effort is put into explaining the basic principles of these calculi since they are not as common as classical calculus. This was also done for tanh-, bi-tanh- multiplicative, and bigeometric calculi and in the general case as well. Generalizations are also investigated where the homeomorphisms φ, η are arbitrary.
Banach spaces of functions, or more generally, of distributions are one of the main topics in ana... more Banach spaces of functions, or more generally, of distributions are one of the main topics in analysis. In this thesis, we present an abstract framework for construction of invariant Banach function spaces from projective group representations. Coorbit theory gives a unified method to construct invariant Banach function spaces via representations of Lie groups. This theory was introduced by Feichtinger and Gröchenig in [23, 24, 25, 26] and then extended in [9]. We generalize this concept by constructing coorbit spaces using projective representation which is first studied by O. Christensen in [10]. This allows us to describe wider classes of function spaces as coorbits, in order to construct frames and atomic decompositions for these spaces. As in the general coorbit theory, we construct atomic decompositions and Banach frames for coorbit spaces under certain smoothness conditions. By this modification, we can discretize the Bergman spaces A p α (B n) via the family of projective representations {ρ s } of the group SU(n, 1), for any real parameter s > n. v
Representation theory of locally compact topological groups is a powerful tool to analyze Banach ... more Representation theory of locally compact topological groups is a powerful tool to analyze Banach spaces of functions and distributions. It provides a unified framework for constructing function spaces and to study several generalizations of the wavelet transform. Recently representation theory has been used to provide atomic decompositions for a large collection of classical Banach spaces. But in some natural situations, including Bergman spaces on bounded domains, representations are too restrictive. The proper tools are projective representations. In this paper we extend known techniques from representation theory to also include projective representations. This leads naturally to twisted convolution on groups avoiding the usual central extension of the group. As our main application we obtain atomic decompositions of Bergman spaces on the unit ball through the holomorphic discrete series for the group of isometries of the ball.
This work deals with a new modified version of the Adomian-Rach decomposition method (MDM). The M... more This work deals with a new modified version of the Adomian-Rach decomposition method (MDM). The MDM is based on combining a series solution and decomposition method for solving nonlinear differential equations with Adomian polynomials for nonlinearities. With application to a class of nonlinear oscillators known as the Lienard-type equations, convergence and error analysis are discussed. Several physical problems modeled by Lienard-type equations are considered to illustrate the effectiveness, performance and reliability of the method. In comparison to the 4th Runge-Kutta method (RK4), highly accurate solutions on a large domain are obtained.
In this paper, we present a robust algorithm to solve numerically a family of two-dimensional fra... more In this paper, we present a robust algorithm to solve numerically a family of two-dimensional fractional integro differential equations. The Haar wavelet method is upgraded to include in its construction the Laplace transform step. This modification has proven to reduce the accumulative errors that will be obtained in case of using the regular Haar wavelet technique. Different examples are discussed to serve two goals, the methodology and the accuracy of our new approach.
In this work, the generalized parametric Marcinkiewicz integral operators with mixed homogeneity ... more In this work, the generalized parametric Marcinkiewicz integral operators with mixed homogeneity related to surfaces of revolution are studied. Under some weak conditions on the kernels, the boundedness of such operators from Triebel–Lizorkin spaces to L p spaces are established. Our results, with the help of an extrapolation argument, improve and extend some previous known results.
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