The reproducing kernel algorithm for handling differential algebraic systems of ordinary differential equations
The aim of the present analysis is to implement a relatively recent computational method, reproducing kernel Hilbert space, for obtaining the solutions of differential algebraic systems for ordinary differential equations. The reproducing kernel Hilbert space ⊕j=1mW22a,b⊕⊕j=m+1nW_21a,b is constructe...
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| Published in: | Mathematical methods in the applied sciences Vol. 39; no. 15; pp. 4549 - 4562 |
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| Format: | Journal Article |
| Language: | English |
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Freiburg
Blackwell Publishing Ltd
01.10.2016
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| ISSN: | 0170-4214, 1099-1476 |
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| Abstract | The aim of the present analysis is to implement a relatively recent computational method, reproducing kernel Hilbert space, for obtaining the solutions of differential algebraic systems for ordinary differential equations. The reproducing kernel Hilbert space
⊕j=1mW22a,b⊕⊕j=m+1nW_21a,b is constructed in which the initial conditions of the systems are satisfied. While, two smooth kernel functions are used throughout the evolution of the algorithm in order to obtain the required grid points. An efficient construction is given to obtain the numerical solutions for the systems together with an existence proof of the exact solutions based upon the reproducing kernel theory. In this approach, computational results of some numerical examples are presented to illustrate the viability, simplicity, and applicability of the algorithm developed. Finally, the utilized results show that the present algorithm and simulated annealing provide a good scheduling methodology to such systems. Copyright © 2016 John Wiley & Sons, Ltd. |
|---|---|
| AbstractList | The aim of the present analysis is to implement a relatively recent computational method, reproducing kernel Hilbert space, for obtaining the solutions of differential algebraic systems for ordinary differential equations. The reproducing kernel Hilbert space
⊕j=1mW22a,b⊕⊕j=m+1nW_21a,b is constructed in which the initial conditions of the systems are satisfied. While, two smooth kernel functions are used throughout the evolution of the algorithm in order to obtain the required grid points. An efficient construction is given to obtain the numerical solutions for the systems together with an existence proof of the exact solutions based upon the reproducing kernel theory. In this approach, computational results of some numerical examples are presented to illustrate the viability, simplicity, and applicability of the algorithm developed. Finally, the utilized results show that the present algorithm and simulated annealing provide a good scheduling methodology to such systems. Copyright © 2016 John Wiley & Sons, Ltd. The aim of the present analysis is to implement a relatively recent computational method, reproducing kernel Hilbert space, for obtaining the solutions of differential algebraic systems for ordinary differential equations. The reproducing kernel Hilbert space [Formulaomitted] is constructed in which the initial conditions of the systems are satisfied. While, two smooth kernel functions are used throughout the evolution of the algorithm in order to obtain the required grid points. An efficient construction is given to obtain the numerical solutions for the systems together with an existence proof of the exact solutions based upon the reproducing kernel theory. In this approach, computational results of some numerical examples are presented to illustrate the viability, simplicity, and applicability of the algorithm developed. Finally, the utilized results show that the present algorithm and simulated annealing provide a good scheduling methodology to such systems. The aim of the present analysis is to implement a relatively recent computational method, reproducing kernel Hilbert space, for obtaining the solutions of differential algebraic systems for ordinary differential equations. The reproducing kernel Hilbert space is constructed in which the initial conditions of the systems are satisfied. While, two smooth kernel functions are used throughout the evolution of the algorithm in order to obtain the required grid points. An efficient construction is given to obtain the numerical solutions for the systems together with an existence proof of the exact solutions based upon the reproducing kernel theory. In this approach, computational results of some numerical examples are presented to illustrate the viability, simplicity, and applicability of the algorithm developed. Finally, the utilized results show that the present algorithm and simulated annealing provide a good scheduling methodology to such systems. Copyright © 2016 John Wiley & Sons, Ltd. The aim of the present analysis is to implement a relatively recent computational method, reproducing kernel Hilbert space, for obtaining the solutions of differential algebraic systems for ordinary differential equations. The reproducing kernel Hilbert space j =1 m W 2 2 a ,b j =m +1 n W _ 2 1 a ,b is constructed in which the initial conditions of the systems are satisfied. While, two smooth kernel functions are used throughout the evolution of the algorithm in order to obtain the required grid points. An efficient construction is given to obtain the numerical solutions for the systems together with an existence proof of the exact solutions based upon the reproducing kernel theory. In this approach, computational results of some numerical examples are presented to illustrate the viability, simplicity, and applicability of the algorithm developed. Finally, the utilized results show that the present algorithm and simulated annealing provide a good scheduling methodology to such systems. Copyright © 2016 John Wiley & Sons, Ltd. |
| Author | Arqub, Omar Abu |
| Author_xml | – sequence: 1 givenname: Omar Abu surname: Arqub fullname: Arqub, Omar Abu email: Correspondence to: Omar Abu Arqub, Department of Mathematics, Faculty of Science, Al-Balqa Applied University, Al-Salt 19117, Jordan,, o.abuarqub@bau.edu.jo organization: Department of Mathematics, Faculty of Science, Al-Balqa Applied University, 19117, Al-Salt, Jordan |
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| References_xml | – reference: Yang LH, Lin Y. Reproducing kernel methods for solving linear initial-boundary-value problems. Electronic Journal of Differential Equations 2008; 2008:1-11. – reference: Celik E, Bayram M. The numerical solution of physical problems modeled as a systems of differential-algebraic equations (DAEs). Journal of the Franklin Institute 2005; 342:1-6. – reference: Cui M, Lin Y. Nonlinear Numerical Analysis in the Reproducing Kernel Space. Nova Science: New York, NY, USA, 2009. – reference: Geng F, Cui M. A reproducing kernel method for solving nonlocal fractional boundary value problems. Applied Mathematics Letters 2012; 25:818-823. – reference: Moaddy K, AL-Smadi M, Hashim I. A novel representation of the exact solution for differential algebraic equations system using residual power-series method. Discrete Dynamics in Nature and Society 2015; 2015:12, doi:10.1155/2015/205207.Article ID 205207. – reference: Celik E. On the numerical solution of chemical differential-algebraic equations by Pade series. Applied Mathematics and Computation 2004; 153:13-17. – reference: Jiang W, Chen Z. A collocation method based on reproducing kernel for a modified anomalous subdiffusion equation. Numerical Methods for Partial Differential Equations 2014; 30:289-300. – reference: Berlinet A, Agnan CT. Reproducing Kernel Hilbert Space in Probability and Statistics. Kluwer Academic Publishers: Boston, Mass, USA, 2004. – reference: Abu Arqub O. An iterative method for solving fourth-order boundary value problems of mixed type integro-differential equations. Journal of Computational Analysis and Applications 2015; 18:857-874. – reference: Daniel A. Reproducing Kernel Spaces and Applications. Springer: Basel, Switzerland, 2003. – reference: Abu Arqub O. Adaptation of reproducing kernel algorithm for solving fuzzy Fredholm-Volterra integrodifferential equations. 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| Title | The reproducing kernel algorithm for handling differential algebraic systems of ordinary differential equations |
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