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Clemens Markett - One of the best experts on this subject based on the ideXlab platform.
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properties of the solutions of the fourth order Bessel type Differential Equation
Journal of Mathematical Analysis and Applications, 2009Co-Authors: W N Everitt, Clemens Markett, L L LittlejohnAbstract:The structured Bessel-type functions of arbitrary even-order were introduced by Everitt and Markett in 1994; these functions satisfy linear ordinary Differential Equations of the same even-order. The Differential Equations have analytic coefficients and are defined on the whole complex plane with a regular singularity at the origin and an irregular singularity at the point of infinity. They are all natural extensions of the classical second-order Bessel Differential Equation. Further these Differential Equations have real-valued coefficients on the positive real half-line of the plane, and can be written in Lagrange symmetric (formally self-adjoint) form. In the fourth-order case, the Lagrange symmetric Differential expression generates self-adjoint unbounded operators in certain Hilbert function spaces. These results are recorded in many of the papers here given as references. It is shown in the original paper of 1994 that in this fourth-order case one solution exists which can be represented in terms of the classical Bessel functions of order 0 and 1. The existence of this solution, further aided by computer programs in Maple, led to the existence of a linearly independent basis of solutions of the Differential Equation. In this paper a new proof of the existence of this solution base is given, on using the advanced theory of special functions in the complex plane. The methods lead to the development of analytical properties of these solutions, in particular the series expansions of all solutions at the regular singularity at the origin of the complex plane.
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Fourier-Bessel Series for Second-Order and Fourth-Order Bessel Differential Equations
Computational Methods and Function Theory, 2008Co-Authors: W. Norrie Everitt, Clemens MarkettAbstract:In this paper we look at the Hilbert function space framework for Fourier-Bessel series, based on linear Differential operators generated by the second-order Bessel Differential Equation and the fourth-order Bessel-type Differential Equation. In the second-order case attention is restricted to the Differential Equation for Bessel functions of order zero $$-({xy^\prime}(x))^{\prime}=\lambda xy(x)\ \ \ \ \ {\rm for\ all}\ x\ \in\ (0,1\rbrack$$ , where λ ∈ ℂ, the complex plane, is the spectral parameter. In the fourth-order case we concentrate on the Bessel-type Differential Equation $$({xy^{\prime\prime}}(x))^{\prime\prime}-(({9x^{-1}}+{8M^{-1}}x)y^\prime(x))^\prime=\Lambda xy(x)\ \ \ {\rm for\ all}\ x\in\ (0,1\rbrack$$ , where Λ ∈ ℂ is the spectral parameter, and M > 0 is a given parameter. In both cases the analysis is concerned with the theory of unbounded linear operators, generated by the Differential Equation, in the Hilbert function space L ^2((0, 1); x ). The analysis depends on new results in special function theory to develop properties of the solutions of the fourth-order Bessel-type Differential Equation, in particular the series expansions of these solutions at the regular singularity at the origin of ℂ.
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the fourth order Bessel type Differential Equation
Applicable Analysis, 2004Co-Authors: Jyoti Das, W N Everitt, D B Hinton, L L Littlejohn, Clemens MarkettAbstract:The Bessel-type functions, structured as extensions of the classical Bessel functions, were defined by Everitt and Markett in 1994. These special functions are derived by linear combinations and limit processes from the classical orthogonal polynomials, classical Bessel functions and the Krall Jacobi-type and Laguerre-type orthogonal polynomials. These Bessel-type functions are solutions of higher-order linear Differential Equations, with a regular singularity at the origin and an irregular singularity at the point of infinity of the complex plane. There is a Bessel-type Differential Equation for each even-order integer; the Equation of order two is the classical Bessel Differential Equation. These even-order Bessel-type Equations are not formal powers of the classical Bessel Equation. When the independent variable of these Equations is restricted to the positive real axis of the plane they can be written in the Lagrange symmetric (formally self-adjoint) form of the Glazman–Naimark type, with real coeffic...
Heinzpeter Schlemmer - One of the best experts on this subject based on the ideXlab platform.
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the inverse laplace transform of the modified lommel functions
Integral Transforms and Special Functions, 2013Co-Authors: Christian H Ziener, Heinzpeter SchlemmerAbstract:Particular solutions of the inhomogeneous Bessel Differential Equation are the Lommel functions, usually denoted as s μ, ν(x), S μ, ν(x), and S μ, ν(x). The inhomogeneous modified Bessel Differential Equation has the particular solutions t μ, ν(x)=−i1−μ s μ, ν(ix), T μ, ν(x)=−i1−μ S μ, ν(ix), and T μ, ν(x)=−i1−μ S μ, ν(ix). For the modified Lommel functions T μ, ν(x) and T μ, ν(x) as well as T μ, ν(√x) and T μ, ν(√x), we give the inverse Laplace transform for μ=−1 and μ=0. For μ<−1, the inverse Laplace transform can be obtained from a recurrence relation.
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The inverse Laplace transform of the modified Lommel functions
Integral Transforms and Special Functions, 2013Co-Authors: Christian H Ziener, Heinzpeter SchlemmerAbstract:Particular solutions of the inhomogeneous Bessel Differential Equation are the Lommel functions, usually denoted as s μ, ν(x), S μ, ν(x), and S μ, ν(x). The inhomogeneous modified Bessel Differential Equation has the particular solutions t μ, ν(x)=−i1−μ s μ, ν(ix), T μ, ν(x)=−i1−μ S μ, ν(ix), and T μ, ν(x)=−i1−μ S μ, ν(ix). For the modified Lommel functions T μ, ν(x) and T μ, ν(x) as well as T μ, ν(√x) and T μ, ν(√x), we give the inverse Laplace transform for μ=−1 and μ=0. For μ
W N Everitt - One of the best experts on this subject based on the ideXlab platform.
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properties of the solutions of the fourth order Bessel type Differential Equation
Journal of Mathematical Analysis and Applications, 2009Co-Authors: W N Everitt, Clemens Markett, L L LittlejohnAbstract:The structured Bessel-type functions of arbitrary even-order were introduced by Everitt and Markett in 1994; these functions satisfy linear ordinary Differential Equations of the same even-order. The Differential Equations have analytic coefficients and are defined on the whole complex plane with a regular singularity at the origin and an irregular singularity at the point of infinity. They are all natural extensions of the classical second-order Bessel Differential Equation. Further these Differential Equations have real-valued coefficients on the positive real half-line of the plane, and can be written in Lagrange symmetric (formally self-adjoint) form. In the fourth-order case, the Lagrange symmetric Differential expression generates self-adjoint unbounded operators in certain Hilbert function spaces. These results are recorded in many of the papers here given as references. It is shown in the original paper of 1994 that in this fourth-order case one solution exists which can be represented in terms of the classical Bessel functions of order 0 and 1. The existence of this solution, further aided by computer programs in Maple, led to the existence of a linearly independent basis of solutions of the Differential Equation. In this paper a new proof of the existence of this solution base is given, on using the advanced theory of special functions in the complex plane. The methods lead to the development of analytical properties of these solutions, in particular the series expansions of all solutions at the regular singularity at the origin of the complex plane.
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the Bessel Differential Equation and the hankel transform
Journal of Computational and Applied Mathematics, 2007Co-Authors: W N Everitt, H KalfAbstract:This paper studies the classical second-order Bessel Differential Equation in Liouville form: Here, the parameter ν represents the order of the associated Bessel functions and λ is the complex spectral parameter involved in considering properties of the Equation in the Hilbert function space L2(0,∞). Properties of the Equation are considered when the order νe[0,1), in this case the singular end-point 0 is in the limit-circle non-oscillatory classification in the space L2(0,∞); the Equation is in the strong limit-point and Dirichlet condition at the end-point +∞ Applying the generalised initial value theorem at the singular end-point 0 allows of the definition of a single Titchmarsh-Weyl m-coefficient for the whole interval (0,∞). In turn this information yields a proof of the Hankel transform as an eigenfunction expansion for the case when νe[0,1), a result which is not available in the existing literature. The application of the principal solution, from the end-point 0 of the Bessel Equation, as a boundary condition function yields the Friedrichs self-adjoint extension in L2(0,∞); the domain of this extension has many special known properties, of which new proofs are presented.
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the fourth order Bessel type Differential Equation
Applicable Analysis, 2004Co-Authors: Jyoti Das, W N Everitt, D B Hinton, L L Littlejohn, Clemens MarkettAbstract:The Bessel-type functions, structured as extensions of the classical Bessel functions, were defined by Everitt and Markett in 1994. These special functions are derived by linear combinations and limit processes from the classical orthogonal polynomials, classical Bessel functions and the Krall Jacobi-type and Laguerre-type orthogonal polynomials. These Bessel-type functions are solutions of higher-order linear Differential Equations, with a regular singularity at the origin and an irregular singularity at the point of infinity of the complex plane. There is a Bessel-type Differential Equation for each even-order integer; the Equation of order two is the classical Bessel Differential Equation. These even-order Bessel-type Equations are not formal powers of the classical Bessel Equation. When the independent variable of these Equations is restricted to the positive real axis of the plane they can be written in the Lagrange symmetric (formally self-adjoint) form of the Glazman–Naimark type, with real coeffic...
Priya Kumari - One of the best experts on this subject based on the ideXlab platform.
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wavelet approximation of functions belonging to generalized lipschitz class by extended haar wavelet series and its application in the solution of Bessel Differential Equation using haar operational matrix
International Journal of Applied and Computational Mathematics, 2019Co-Authors: Shyam Lal, Priya KumariAbstract:In this paper, a class \(Lip_\alpha ^{(s)}[0,1]\) is introduced. This class of functions is a generalization of known Lipschitz class, i.e., \(Lip_\alpha [0,1], 0<\alpha \le 1\) of functions. Four new estimators \(E_N^{(1)}(f), E_{J,N}^{(2)}(f), E_N^{(3)}(f)\) and \(E_N^{(4)}(f)\) of functions of \(Lip_\alpha [0,1]\) and \(Lip_\alpha ^{(s)}[0,1]\) classes have been obtained. These estimators are sharper and best possible in the approximation of functions by wavelet methods. The developed estimators, the solution of Bessel Differential Equation of order zero by Haar wavelet operational matrix and its comparison with the exact solution are the main significant achievements of this research paper.
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Wavelet Approximation of Functions Belonging to Generalized Lipschitz Class by Extended Haar Wavelet Series and its Application in the Solution of Bessel Differential Equation Using Haar Operational Matrix
International Journal of Applied and Computational Mathematics, 2019Co-Authors: Shyam Lal, Priya KumariAbstract:In this paper, a class $$Lip_\alpha ^{(s)}[0,1]$$ L i p α ( s ) [ 0 , 1 ] is introduced. This class of functions is a generalization of known Lipschitz class, i.e., $$Lip_\alpha [0,1], 0
L L Littlejohn - One of the best experts on this subject based on the ideXlab platform.
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properties of the solutions of the fourth order Bessel type Differential Equation
Journal of Mathematical Analysis and Applications, 2009Co-Authors: W N Everitt, Clemens Markett, L L LittlejohnAbstract:The structured Bessel-type functions of arbitrary even-order were introduced by Everitt and Markett in 1994; these functions satisfy linear ordinary Differential Equations of the same even-order. The Differential Equations have analytic coefficients and are defined on the whole complex plane with a regular singularity at the origin and an irregular singularity at the point of infinity. They are all natural extensions of the classical second-order Bessel Differential Equation. Further these Differential Equations have real-valued coefficients on the positive real half-line of the plane, and can be written in Lagrange symmetric (formally self-adjoint) form. In the fourth-order case, the Lagrange symmetric Differential expression generates self-adjoint unbounded operators in certain Hilbert function spaces. These results are recorded in many of the papers here given as references. It is shown in the original paper of 1994 that in this fourth-order case one solution exists which can be represented in terms of the classical Bessel functions of order 0 and 1. The existence of this solution, further aided by computer programs in Maple, led to the existence of a linearly independent basis of solutions of the Differential Equation. In this paper a new proof of the existence of this solution base is given, on using the advanced theory of special functions in the complex plane. The methods lead to the development of analytical properties of these solutions, in particular the series expansions of all solutions at the regular singularity at the origin of the complex plane.
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the fourth order Bessel type Differential Equation
Applicable Analysis, 2004Co-Authors: Jyoti Das, W N Everitt, D B Hinton, L L Littlejohn, Clemens MarkettAbstract:The Bessel-type functions, structured as extensions of the classical Bessel functions, were defined by Everitt and Markett in 1994. These special functions are derived by linear combinations and limit processes from the classical orthogonal polynomials, classical Bessel functions and the Krall Jacobi-type and Laguerre-type orthogonal polynomials. These Bessel-type functions are solutions of higher-order linear Differential Equations, with a regular singularity at the origin and an irregular singularity at the point of infinity of the complex plane. There is a Bessel-type Differential Equation for each even-order integer; the Equation of order two is the classical Bessel Differential Equation. These even-order Bessel-type Equations are not formal powers of the classical Bessel Equation. When the independent variable of these Equations is restricted to the positive real axis of the plane they can be written in the Lagrange symmetric (formally self-adjoint) form of the Glazman–Naimark type, with real coeffic...