scholarly journals Numerical Simulation Characteristics of Logging Response in Water Injection Well by Reproducing Kernel Method

2015 ◽  
Vol 2015 ◽  
pp. 1-5
Author(s):  
Ming-Jing Du ◽  
Yu-Lan Wang ◽  
Temuer Chaolu
Keyword(s):  
Temperature Field ◽  
Water Temperature ◽  
Reproducing Kernel ◽  
Kernel Method ◽  
Reproducing Kernel Hilbert Space ◽  
Water Injection ◽  
Kernel Functions ◽  
Injection Well ◽  
Two Phase Flows ◽  
Two Phase

Reproducing kernel Hilbert space method (RKHSM) is an effective method. This paper, for the first time, uses the traditional RKHSM for solving the temperature field in two phase flows of multilayer water injection well. According to 2D oil-water temperature field mathematical model of two phase flows in cylindrical coordinates, selecting the properly initial and boundary conditions, by the process of Gram-Schmidt orthogonalization, the analytical solution was given by reproducing kernel functions in a series expansion form, and the approximate solution was expressed byn-term summation. The satisfied numerical results were carried out by Mathematica 7.0, showing that the larger the difference between injected water temperature and initial borehole temperature or water injection conditions, the more obvious the indication of water accepting zones. The numerical examples evidence the feasibility and effectiveness of the proposed method of the two phase flows in engineering.

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.

scholarly journals GENERALIZED JACOBI REPRODUCING KERNEL METHOD IN HILBERT SPACES FOR SOLVING THE BLACK-SCHOLES OPTION PRICING PROBLEM ARISING IN FINANCIAL MODELLING

2018 ◽  
Vol 23 (4) ◽  
pp. 538-553
Author(s):  
Mohammadreza Foroutan ◽  
Ali Ebadian ◽  
Hadi Rahmani Fazli
Keyword(s):  
Option Pricing ◽  
Reproducing Kernel ◽  
Initial Conditions ◽  
Kernel Method ◽  
Reproducing Kernel Hilbert Space ◽  
Kernel Functions ◽  
Dirichlet Boundary Conditions ◽  
Financial Modelling ◽  
Black Scholes ◽  
Reproducing Kernel Theory

Based on the reproducing kernel Hilbert space method, a new approach is proposed to approximate the solution of the Black-Scholes equation with Dirichlet boundary conditions and introduce the reproducing kernel properties in which the initial conditions of the problem are satisfied. Based on reproducing kernel theory, reproducing kernel functions with a polynomial form will be constructed in the reproducing kernel spaces spanned by the generalized Jacobi basis polynomials. Some new error estimates for application of the method are established. The convergence analysis is established theoretically. The proposed method is successfully used for solving an option pricing problem arising in financial modelling. The ideas and techniques presented in this paper will be useful for solving many other problems.

scholarly journals A Novel Method for Solving KdV Equation Based on Reproducing Kernel Hilbert Space Method

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Mustafa Inc ◽  
Ali Akgül ◽  
Adem Kiliçman
Keyword(s):  
Hilbert Space ◽  
Reproducing Kernel ◽  
Kernel Method ◽  
Kdv Equation ◽  
Reproducing Kernel Hilbert Space ◽  
Reproducing Kernel Method ◽  
Numerical Examples ◽  
The Kdv Equation ◽  
Novel Method ◽  
Reproducing Kernel Theory

We propose a reproducing kernel method for solving the KdV equation with initial condition based on the reproducing kernel theory. The exact solution is represented in the form of series in the reproducing kernel Hilbert space. Some numerical examples have also been studied to demonstrate the accuracy of the present method. Results of numerical examples show that the presented method is effective.

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 Solutions to Uncertain Volterra Integral Equations by Fitted Reproducing Kernel Hilbert Space Method

2016 ◽  
Vol 2016 ◽  
pp. 1-11 ◽  
Author(s):  
Ghaleb Gumah ◽  
Khaled Moaddy ◽  
Mohammed Al-Smadi ◽  
Ishak Hashim
Keyword(s):  
Hilbert Space ◽  
Integral Equations ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Convergent Series ◽  
Kernel Functions ◽  
Orthogonal Basis ◽  
Approximation Solution ◽  
Solution Methodology ◽  
Space Method

We present an efficient modern strategy for solving some well-known classes of uncertain integral equations arising in engineering and physics fields. The solution methodology is based on generating an orthogonal basis upon the obtained kernel function in the Hilbert spaceW21a,bin order to formulate the analytical solutions in a rapidly convergent series form in terms of theirα-cut representation. The approximation solution is expressed byn-term summation of reproducing kernel functions and it is convergent to the analytical solution. Our investigations indicate that there is excellent agreement between the numerical results and the RKHS method, which is applied to some computational experiments to demonstrate the validity, performance, and superiority of the method. The present work shows the potential of the RKHS technique in solving such uncertain integral equations.

scholarly journals Reproducing kernel functions for the generalized Kuramoto-Sivashinsky equation

2018 ◽  
Vol 22 ◽  
pp. 01028
Author(s):  
Ali Akgül ◽  
Esra Karatas Akgül ◽  
Sahin Korhan ◽  
Mustafa Inc
Keyword(s):  
Hilbert Space ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Kernel Functions ◽  
Sivashinsky Equation ◽  
Space Method

Reproducing kernel functions are obtained for the solution of generalized Kuramoto–Sivashinsky (GKS) equation in this paper. These reproducing kernel functions are valuable in the reproducing kernel Hilbert space method. They will be useful for interested researchers.

scholarly journals A Two-Phase Flow Model for Pressure Transient Analysis of Water Injection Well considering Water Imbibition in Natural Fractured Reservoirs

2018 ◽  
Vol 2018 ◽  
pp. 1-12
Author(s):  
Mengmeng Li ◽  
Qi Li ◽  
Gang Bi ◽  
Jiaen Lin
Keyword(s):  
Transient Analysis ◽  
Two Phase Flow ◽  
Fractured Reservoirs ◽  
Water Injection ◽  
Injection Well ◽  
Phase Flow ◽  
Two Phase ◽  
Pressure Transient Analysis ◽  
Pressure Transient ◽  
Water Imbibition

The pressure injection falloff test for water injection well has the advantages of briefness and convenience, with no effect on the oil production. It has been widely used in the oil field. Tremendous attention has been focused on oil-water two-phase flow model based on the Perrine-Martin theory. However, the saturation gradient is not considered in the Perrine-Martin method, which may result in errors in computation. Moreover, water imbibition is important for water flooding in natural fractured reservoirs, while the pressure transient analysis model has rarely considered water imbibition. In this paper, we proposed a semianalytical oil-water two-phase flow imbibition model for pressure transient analysis of a water injection well in natural fractured reservoirs. The parameters in this model, including total compressibility coefficient, interporosity flow coefficient, and total mobility, change with water saturation. The model was solved by Laplace transform finite-difference (LTFD) method coupled with the quasi-stationary method. Based on the solution, the model was verified by the analytical method and a field water injection test. The features of typical curves and the influences of the parameters on the typical curves were analyzed. Results show that the shape of pressure curves for single phase flow resembles two-phase flow, but the position of the two-phase flow curves is on the upper right of the single phase flow curves. The skin factor and wellbore storage coefficient mainly influence the peak value of the pressure derivatives and the straight line of the early period. The shape factor has a major effect on the position of the “dip” of pressure derivatives. The imbibition rate coefficient mainly influences the whole system radial flow period of the curves. This work provides valuable information in the design and evaluation of stimulation treatments in natural fractured reservoirs.

scholarly journals New Numerical Method for Solving Tenth Order Boundary Value Problems

Mathematics ◽  
2018 ◽  
Vol 6 (11) ◽  
pp. 245 ◽  
Author(s):  
Ali Akgül ◽  
Esra Karatas Akgül ◽  
Dumitru Baleanu ◽  
Mustafa Inc
Keyword(s):  
Hilbert Space ◽  
Boundary Value Problems ◽  
Numerical Method ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Boundary Value ◽  
Approximate Solutions ◽  
Kernel Functions ◽  
Numerical Results ◽  
Space Method

In this paper, we implement reproducing kernel Hilbert space method to tenth order boundary value problems. These problems are important for mathematicians. Different techniques were applied to get approximate solutions of such problems. We obtain some useful reproducing kernel functions to get approximate solutions. We obtain very efficient results by this method. We show our numerical results by tables.

scholarly journals A REPRODUCING KERNEL METHOD FOR SOLVING A CLASS OF NONLINEAR SYSTEMS OF PDES

2014 ◽  
Vol 19 (2) ◽  
pp. 180-198 ◽  
Author(s):  
Maryam Mohammadi ◽  
Reza Mokhtari
Keyword(s):  
Hilbert Space ◽  
Nonlinear Systems ◽  
System Performance ◽  
Reproducing Kernel ◽  
Kernel Method ◽  
Reproducing Kernel Hilbert Space ◽  
Coupled System ◽  
Reproducing Kernel Method ◽  
Nonlinear Hyperbolic System ◽  
Systems Of Pdes

This paper is concerned with a technique for solving a class of nonlinear systems of partial differential equations (PDEs) in the reproducing kernel Hilbert space. The analytical solution is represented in the form of series. An iterative method is given to obtain the approximate solution. The convergence analysis is established theoretically. The proposed method is successfully used for solving a coupled system of viscous Burgers’ equations and a nonlinear hyperbolic system. Performance of the method is tested in terms of various error norms. In the case of non-availability of exact solution, it is compared with the existing methods.

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