scholarly journals Tensor Product and Certain Solutions of Fractional Wave Type Equation

2021 ◽  
Vol 14 (3) ◽  
pp. 942-948
Author(s):  
Ibtisam Benkemache ◽  
Mohammad Al-horani ◽  
Roshdi Rashid Khalil
Keyword(s):  
Fourier Series ◽  
Tensor Product ◽  
Banach Spaces ◽  
Separation Of Variables ◽  
Type Equation ◽  
Wave Type

In this paper we Önd certain solutions of some fractional partial di§erential equations. Tensor product of Banach spaces is used to Önd some solutions where separation of variables does not work. We solve the fractional wave type equation using fractional Fourier Series

Solutions of Certain Fractional Partial Differential Equations

2021 ◽  
Vol 20 ◽  
pp. 504-507
Author(s):  
Alsauodi Maha ◽  
Alhorani Mohammed ◽  
Khalil Roshdi
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Tensor Product ◽  
Banach Spaces ◽  
Separation Of Variables ◽  
Fractional Partial Differential Equations ◽  
Partial Differential

In this paper we find certain solutions of some fractional partial differential equations. Tensor product of Banach spaces is used where separation of variables does not work.

scholarly journals The dual of the space of bounded operators on a Banach space

2021 ◽  
Vol 8 (1) ◽  
pp. 48-59
Author(s):  
Fernanda Botelho ◽  
Richard J. Fleming
Keyword(s):  
Banach Space ◽  
Tensor Product ◽  
Banach Spaces ◽  
Dual Space ◽  
Tensor Products ◽  
Bounded Operators ◽  
Projective Tensor Product ◽  
Projective Tensor

Abstract Given Banach spaces X and Y, we ask about the dual space of the 𝒧(X, Y). This paper surveys results on tensor products of Banach spaces with the main objective of describing the dual of spaces of bounded operators. In several cases and under a variety of assumptions on X and Y, the answer can best be given as the projective tensor product of X ** and Y *.

scholarly journals SUBPROJECTIVITY OF PROJECTIVE TENSOR PRODUCTS OF BANACH SPACES OF CONTINUOUS FUNCTIONS

2021 ◽  
pp. 1-14
Author(s):  
R.M. CAUSEY
Keyword(s):  
Tensor Product ◽  
Banach Spaces ◽  
Tensor Products ◽  
Continuous Functions ◽  
Hausdorff Spaces ◽  
Projective Tensor Product ◽  
Projective Tensor ◽  
Spaces Of Continuous Functions

Abstract Galego and Samuel showed that if K, L are metrizable, compact, Hausdorff spaces, then $C(K)\widehat{\otimes}_\pi C(L)$ is c0-saturated if and only if it is subprojective if and only if K and L are both scattered. We remove the hypothesis of metrizability from their result and extend it from the case of the twofold projective tensor product to the general n-fold projective tensor product to show that for any $n\in\mathbb{N}$ and compact, Hausdorff spaces K1, …, K n , $\widehat{\otimes}_{\pi, i=1}^n C(K_i)$ is c0-saturated if and only if it is subprojective if and only if each K i is scattered.

An analytical solution of a two-dimensional temperature field in a hollow cylinder under a time periodic boundary condition using Fourier series

2009 ◽  
Vol 223 (8) ◽  
pp. 1889-1901 ◽  
Author(s):  
G Atefi ◽  
M A Abdous ◽  
A Ganjehkaviri ◽  
N Moalemi
Keyword(s):  
Boundary Condition ◽  
Temperature Field ◽  
Fourier Series ◽  
Temperature Distribution ◽  
Analytical Solution ◽  
Hollow Cylinder ◽  
Periodic Boundary Condition ◽  
Periodic Boundary ◽  
Separation Of Variables ◽  
Two Dimensional

The objective of this article is to derive an analytical solution for a two-dimensional temperature field in a hollow cylinder, which is subjected to a periodic boundary condition at the outer surface, while the inner surface is insulated. The material is assumed to be homogeneous and isotropic with time-independent thermal properties. Because of the time-dependent term in the boundary condition, Duhamel's theorem is used to solve the problem for a periodic boundary condition. The periodic boundary condition is decomposed using the Fourier series. This condition is simulated with harmonic oscillation; however, there are some differences with the real situation. To solve this problem, first of all the boundary condition is assumed to be steady. By applying the method of separation of variables, the temperature distribution in a hollow cylinder can be obtained. Then, the boundary condition is assumed to be transient. In both these cases, the solutions are separately calculated. By using Duhamel's theorem, the temperature distribution field in a hollow cylinder is obtained. The final result is plotted with respect to the Biot and Fourier numbers. There is good agreement between the results of the proposed method and those reported by others for this geometry under a simple harmonic boundary condition.

A General Solution of Reynolds’ Equation for a Full Finite Journal Bearing

1967 ◽  
Vol 89 (2) ◽  
pp. 203-210 ◽  
Author(s):  
R. R. Donaldson
Keyword(s):  
Fourier Series ◽  
Series Solution ◽  
Reynolds Equation ◽  
Journal Bearing ◽  
Load Capacity ◽  
Separation Of Variables ◽  
Hill Equation ◽  
Eigenvalues And Eigenvectors ◽  
General Series ◽  
Bearing Load

Reynolds’ equation for a full finite journal bearing lubricated by an incompressible fluid is solved by separation of variables to yield a general series solution. A resulting Hill equation is solved by Fourier series methods, and accurate eigenvalues and eigenvectors are calculated with a digital computer. The finite Sommerfeld problem is solved as an example, and precise values for the bearing load capacity are presented. Comparisons are made with the methods and numerical results of other authors.

scholarly journals Types of Radon-Nikodym properties for the projective tensor product of Banach spaces

2003 ◽  
Vol 47 (4) ◽  
pp. 1303-1326 ◽  
Author(s):  
Qingying Bu ◽  
Joe Diestel ◽  
Patrick Dowling ◽  
Eve Oja
Keyword(s):  
Tensor Product ◽  
Banach Spaces ◽  
Projective Tensor Product ◽  
Projective Tensor

On a Theorem of Beurling and Livingston

1965 ◽  
Vol 17 ◽  
pp. 367-372 ◽  
Author(s):  
Felix E. Browder
Keyword(s):  
Banach Space ◽  
Fourier Series ◽  
Banach Spaces ◽  
General Result ◽  
Functional Equations ◽  
Linear Functional ◽  
Monotone Mappings ◽  
Conjugate Space ◽  
Non Linear ◽  
Duality Mappings

In their paper (1), Beurling and Livingston established a generalization of the Riesz-Fischer theorem for Fourier series in Lp using a theorem on duality mappings of a Banach space B into its conjugate space B*. It is our purpose in the present paper to give another proof of this theorem by deriving it from a more general result concerning monotone mappings related to recent results on non-linear functional equations in Banach spaces obtained by the writer (2, 3, 4, 5) and G. J. Minty (6).

scholarly journals Quantization of a 3D Nonstationary Harmonic plus an Inverse Harmonic Potential System

2016 ◽  
Vol 2016 ◽  
pp. 1-6 ◽  
Author(s):  
Salim Medjber ◽  
Hacene Bekkar ◽  
Salah Menouar ◽  
Jeong Ryeol Choi
Keyword(s):  
Separation Of Variables ◽  
Type Equation ◽  
Invariant Operator ◽  
Wave Functions ◽  
Time Dependent ◽  
Harmonic Potential ◽  
Uncertainty Relations ◽  
Mathematical Methods ◽  
Potential System ◽  
Uvarov Method

The Schrödinger solutions for a three-dimensional central potential system whose Hamiltonian is composed of a time-dependent harmonic plus an inverse harmonic potential are investigated. Because of the time-dependence of parameters, we cannot solve the Schrödinger solutions relying only on the conventional method of separation of variables. To overcome this difficulty, special mathematical methods, which are the invariant operator method, the unitary transformation method, and the Nikiforov-Uvarov method, are used when we derive solutions of the Schrödinger equation for the system. In particular, the Nikiforov-Uvarov method with an appropriate coordinate transformation enabled us to reduce the eigenvalue equation of the invariant operator, which is a second-order differential equation, to a hypergeometric-type equation that is convenient to treat. Through this procedure, we derived exact Schrödinger solutions (wave functions) of the system. It is confirmed that the wave functions are represented in terms of time-dependent radial functions, spherical harmonics, and general time-varying global phases. Such wave functions are useful for studying various quantum properties of the system. As an example, the uncertainty relations for position and momentum are derived by taking advantage of the wave functions.

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