Reproducing Kernel Functions for Sard Spaces of Type B

1974 ◽  
Vol 11 (1) ◽  
pp. 37-44 ◽  
Author(s):  
Robert E. Barnhill ◽  
Gregory M. Nielson
Keyword(s):  
Reproducing Kernel ◽  
Kernel Functions ◽  
Type B

Numerical solutions for the Robin time-fractional partial differential equations of heat and fluid flows based on the reproducing kernel algorithm

2018 ◽  
Vol 28 (4) ◽  
pp. 828-856 ◽  
Author(s):  
Omar Abu Arqub
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Reproducing Kernel ◽  
Numerical Solutions ◽  
Fluid Flows ◽  
Kernel Functions ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Content Type

Purpose The purpose of this study is to introduce the reproducing kernel algorithm for treating classes of time-fractional partial differential equations subject to Robin boundary conditions with parameters derivative arising in fluid flows, fluid dynamics, groundwater hydrology, conservation of energy, heat conduction and electric circuit. Design/methodology/approach The method provides appropriate representation of the solutions in convergent series formula with accurately computable components. This representation is given in the W(Ω) and H(Ω) inner product spaces, while the computation of the required grid points relies on the R(y,s) (x, t) and r(y,s) (x, t) reproducing kernel functions. Findings Numerical simulation with different order derivatives degree is done including linear and nonlinear terms that are acquired by interrupting the n-term of the exact solutions. Computational results showed that the proposed algorithm is competitive in terms of the quality of the solutions found and is very valid for solving such time-fractional models. Research limitations/implications Future work includes the application of the reproducing kernel algorithm to highly nonlinear time-fractional partial differential equations such as those arising in single and multiphase flows. The results will be published in forthcoming papers. Practical implications The study included a description of fundamental reproducing kernel algorithm and the concepts of convergence, and error behavior for the reproducing kernel algorithm solvers. Results obtained by the proposed algorithm are found to outperform in terms of accuracy, generality and applicability. Social implications Developing analytical and numerical methods for the solutions of time-fractional partial differential equations is a very important task owing to their practical interest. Originality/value This study, for the first time, presents reproducing kernel algorithm for obtaining the numerical solutions of some certain classes of Robin time-fractional partial differential equations. An efficient construction is provided to obtain the numerical solutions for the equations, along with an existence proof of the exact solutions based upon the reproducing kernel theory.

scholarly journals Reproducing kernel functions for difference equations

2015 ◽  
Vol 8 (6) ◽  
pp. 1055-1064 ◽  
Author(s):  
Ali Akgül ◽  
◽  
Mustafa Inc ◽  
Esra Karatas ◽  
Keyword(s):  
Difference Equations ◽  
Reproducing Kernel ◽  
Kernel Functions

scholarly journals Reproducing kernel Hilbert space method for solutions of a coefficient inverse problem for the kinetic equation

2018 ◽  
Vol 8 (2) ◽  
pp. 145-151 ◽  
Author(s):  
Esra Karatas Akgül
Keyword(s):  
Hilbert Space ◽  
Inverse Problem ◽  
Kinetic Equation ◽  
Inverse Problems ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Approximate Solutions ◽  
Kernel Functions ◽  
Coefficient Inverse Problem ◽  
Space Method

On the basis of a reproducing kernel Hilbert space, reproducing kernel functions for solving the coefficient inverse problem for the kinetic equation are given in this paper. Reproducing kernel functions found in the reproducing kernel Hilbert space imply that they can be considered for solving such inverse problems. We obtain approximate solutions by reproducing kernel functions. We show our results by a table. We prove the eciency of the reproducing kernel Hilbert space method for solutions of a coefficient inverse problem for the kinetic equation.

scholarly journals Generalized coherent states, reproducing kernels, and quantum support vector machines

2017 ◽  
Vol 17 (15&16) ◽  
pp. 1292-1306 ◽  
Author(s):  
Rupak Chatterjee ◽  
Ting Yu
Keyword(s):  
Hilbert Space ◽  
Coherent States ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Feature Space ◽  
Reproducing Kernels ◽  
Kernel Functions ◽  
Support Vector ◽  
Machine Learning Classification ◽  
Generalized Coherent States

The support vector machine (SVM) is a popular machine learning classification method which produces a nonlinear decision boundary in a feature space by constructing linear boundaries in a transformed Hilbert space. It is well known that these algorithms when executed on a classical computer do not scale well with the size of the feature space both in terms of data points and dimensionality. One of the most significant limitations of classical algorithms using non-linear kernels is that the kernel function has to be evaluated for all pairs of input feature vectors which themselves may be of substantially high dimension. This can lead to computationally excessive times during training and during the prediction process for a new data point. Here, we propose using both canonical and generalized coherent states to calculate specific nonlinear kernel functions. The key link will be the reproducing kernel Hilbert space (RKHS) property for SVMs that naturally arise from canonical and generalized coherent states. Specifically, we discuss the evaluation of radial kernels through a positive operator valued measure (POVM) on a quantum optical system based on canonical coherent states. A similar procedure may also lead to calculations of kernels not usually used in classical algorithms such as those arising from generalized coherent states.

scholarly journals Numerical Solution of Fractional Order Burgers’ Equation with Dirichlet and Neumann Boundary Conditions by Reproducing Kernel Method

2020 ◽  
Vol 4 (2) ◽  
pp. 27 ◽  
Author(s):  
Onur Saldır ◽  
Mehmet Giyas Sakar ◽  
Fevzi Erdogan
Keyword(s):  
Fractional Order ◽  
Reproducing Kernel ◽  
Burgers Equation ◽  
Kernel Method ◽  
Reproducing Kernel Hilbert Space ◽  
Neumann Boundary ◽  
Kernel Functions ◽  
Iterative Approach ◽  
Reproducing Kernel Method ◽  
Reproducing Kernel Spaces

In this research, obtaining of approximate solution for fractional-order Burgers’ equation will be presented in reproducing kernel Hilbert space (RKHS). Some special reproducing kernel spaces are identified according to inner products and norms. Then an iterative approach is constructed by using kernel functions. The convergence of this approach and its error estimates are given. The numerical algorithm of the method is presented. Furthermore, numerical outcomes are shown with tables and graphics for some examples. These outcomes demonstrate that the proposed method is convenient and effective.

THE ELEMENT-FREE kp-RITZ METHOD FOR VIBRATION OF LAMINATED ROTATING CYLINDRICAL PANELS

2002 ◽  
Vol 02 (04) ◽  
pp. 523-558 ◽  
Author(s):  
K. M. LIEW ◽  
X. ZHAO ◽  
T. Y. NG
Keyword(s):  
Reproducing Kernel ◽  
Ritz Method ◽  
Free Vibration Analysis ◽  
Kernel Functions ◽  
Structural Vibration ◽  
Moderate Amount ◽  
Significant Development ◽  
Methodology Development ◽  
Cylindrical Panels ◽  
Element Free

In this paper, the element-free kernel particle Ritz method or in short the kp-Ritz method, is developed and implemented for the free vibration analysis of rotating cylindrical panels. Over the years, researchers have employed various trial or admissible functions, such as harmonic and hyperbolic functions as well as polynomials, in the Ritz approximate approach for structural vibration analyses. The present work represents a significant development whereby 2-D reproducing kernel functions are employed to approximate the displacement field. The resulting kp-Ritz method represents the methodology development aspect of this paper. Further the rotating panel problem selected for demonstrating the accuracy and efficiency of this new methodology is also an important contribution of this paper. Although there is a moderate amount of published information on the vibration of rotating cylindrical shells, corresponding studies on rotating panels are conspicuously lacking. This is the first instance in which results have been obtained for rotating cylindrical panels whereby the essential boundary conditions have been fully enforced.

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