scholarly journals An Approximation Method for Solving Volterra Integrodifferential Equations with a Weighted Integral Condition

2015 ◽  
Vol 2015 ◽  
pp. 1-10 ◽  
Author(s):  
A. Guezane-Lakoud ◽  
N. Bendjazia ◽  
R. Khaldi
Keyword(s):  
Hilbert Space ◽  
Analytical Solution ◽  
Approximation Method ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Integral Condition ◽  
Approximate Solutions ◽  
Integrodifferential Equations ◽  
Numerical Examples ◽  
Weighted Integral

We apply the reproducing kernel Hilbert space (RKHS) method for getting analytical and approximate solutions for second-order hyperbolic integrodifferential equations with a weighted integral condition. The analytical solution is represented in the form of series; thus, then-terms approximate solutions are obtained. The results of the numerical examples are compared with the exact solutions to illustrate the accuracy and the effectivity of this method.

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 A Reproducing Kernel Hilbert Space Method for Solving Systems of Fractional Integrodifferential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Samia Bushnaq ◽  
Banan Maayah ◽  
Shaher Momani ◽  
Ahmed Alsaedi
Keyword(s):  
Hilbert Space ◽  
Fractional Calculus ◽  
Present Method ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Nonlinear Problems ◽  
Convergent Series ◽  
Integrodifferential Equations ◽  
Space Method

We present a new version of the reproducing kernel Hilbert space method (RKHSM) for the solution of systems of fractional integrodifferential equations. In this approach, the solution is obtained as a convergent series with easily computable components. Several illustrative examples are given to demonstrate the effectiveness of the present method. The method described in this paper is expected to be further employed to solve similar nonlinear problems in fractional calculus.

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 The Reproducing Kernel Hilbert Space Method for Solving System of Linear Weakly Singular Volterra Integral Equations

2018 ◽  
Vol 15 ◽  
pp. 8070-8080 ◽  
Author(s):  
Hameeda Oda Al-Humedi
Keyword(s):  
Hilbert Space ◽  
Exact Solutions ◽  
Integral Equations ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Approximate Solutions ◽  
Volterra Integral Equations ◽  
Reproducing Kernel Space ◽  
Linear Ordinary Differential Equations ◽  
Weakly Singular

The exact solutions of a system of linear weakly singular Volterra integral equations (VIE) have been a difficult to find.  The aim of this paper is to apply reproducing kernel Hilbert space (RKHS) method to find the approximate solutions to this type of systems. At first, we used Taylor's expansion to omit the singularity.  From an expansion the given system of linear weakly singular VIE is transform into a system of linear ordinary differential equations (LODEs).   The approximate solutions are represent in the form of series in the reproducing kernel space . By comparing with the exact solutions of two examples, we saw that RKHS is a powerful, easy to apply and full efficiency in scientific applications to build a solution without linearization and turbulence or discretization. 

scholarly journals Reproducing kernel functions for linear tenth-order boundary value problems

2018 ◽  
Vol 22 ◽  
pp. 01027
Author(s):  
Ali Akgül ◽  
Esra Karatas Akgül ◽  
Baris Orcan ◽  
Mustafa Inc
Keyword(s):  
Hilbert Space ◽  
Numerical Methods ◽  
Boundary Value Problems ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Boundary Value ◽  
Approximate Solutions ◽  
Kernel Functions ◽  
Space Method ◽  
Higher Order Differential Equations

Higher order differential equations have always been an onerous problem to investigate for the mathematicians and engineers. Different numerical methods were applied to get numerical approximations of such problems. This paper gives some reproducing kernel functions to find approximate solutions of the tenth-order boundary value problems (BVPs). These reproducing kernel functions are very important in the reproducing kernel Hilbert space method.

scholarly journals A Novel Method for Solutions of Fourth-Order Fractional Boundary Value Problems

2019 ◽  
Vol 3 (2) ◽  
pp. 33 ◽  
Author(s):  
Ali Akgül ◽  
Esra Karatas Akgül
Keyword(s):  
Hilbert Space ◽  
Boundary Value Problems ◽  
Fourth Order ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Boundary Value ◽  
Approximate Solutions ◽  
Novel Method ◽  
Fractional Boundary Value Problems ◽  
Space Method

In this paper, we find the solutions of fourth order fractional boundary value problems by using the reproducing kernel Hilbert space method. Firstly, the reproducing kernel Hilbert space method is introduced and then the method is applied to this kind problems. The experiments are discussed and the approximate solutions are obtained to be more correct compared to the other obtained results in the literature.

scholarly journals PICARD-REPRODUCING KERNEL HILBERT SPACE METHOD FOR SOLVING GENERALIZED SINGULAR NONLINEAR LANE-EMDEN TYPE EQUATIONS

2015 ◽  
Vol 20 (6) ◽  
pp. 754-767 ◽  
Author(s):  
Babak Azarnavid ◽  
Foroud Parvaneh ◽  
Saeid Abbasbandy
Keyword(s):  
Hilbert Space ◽  
Iterative Method ◽  
Error Estimates ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Picard Iteration ◽  
Combined Method ◽  
Numerical Examples ◽  
Space Method ◽  
Good Agreement

An iterative method is discussed with respect to its effectiveness and capability of solving singular nonlinear Lane-Emden type equations using reproducing kernel Hilbert space method combined with the Picard iteration. Some new error estimates for application of the method are established. We prove the convergence of the combined method. The numerical examples demonstrates a good agreement between numerical results and analytical predictions.

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