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Application of reproducing kernel method for solving a class of two-dimensional linear integral equations with weakly singular kernel | ||
Journal of New Researches in Mathematics | ||
مقاله 12، دوره 6، شماره 25، آذر و دی 2020، صفحه 159-166 اصل مقاله (254.88 K) | ||
نوع مقاله: research paper | ||
نویسندگان | ||
Mohammad Reza Eslahchi 1؛ Maryam Rezaeimirarkolaei2 | ||
1Associate Professor, Department of Applied Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, Tehran, Iran | ||
2Department of Applied Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, Tehran, Iran | ||
چکیده | ||
In this paper, we will present a new method for solving a class of two-dimensional linear Volterra integral equation of the second kind with weakly singular kernel from Abel type in the reproducing kernel space. The reproducing kernel function is discussed in detail. Weak singularity of problem is removed by applying integration by parts. Further, improper integral belongs to L_2 (Ω). In our method the exact solution ϕ(x,t) is represented in the form of series in the reproducing kernel space W(ω), and the approximate solution ϕ_n (x,t) is constructed via truncating the series to n terms. Convergence analysis of the method is proved in detail. Some numerical examples are also studied to demonstrate the efficiency and accuracy of the presented method. The obtained results show that the error of the approximate solution is monotone decreasing in the sense of the norm of W(ω), when increasing the number of the nodes. Also, that indicate the method is simple and effective. It turns out that this method is valid. | ||
کلیدواژهها | ||
Two-dimensional integral equation؛ Reproducing kernel Hilbert space؛ Volterra integral equation؛ Weakly singular kernel؛ Convergence analysis | ||
مراجع | ||
[1] A. M. Wazwaz. A first course in integral equations. World Scientific. Singapour (1997). [2] A. M. Wazwaz. Linear and nonlinear integral equation: methods and applications. Higher Education Press and Springer Verlage (2011). [3] M. H. Reihani, Z. Abadi. Rationalized Harr functions method for solving Fredholm and Volterra integral equations. Journal of Computational and Applied Mathematics 12-20 (2007). [4] J. Saberi-Nadjafi, M. Mehrabinezhad, T. Diogo. The Coiflet-Galerkin method for linear Volterra integral equations. Applied Mathematics and Computation 221:469-483 (2013). [5] J. Saberi-Nadjafi, M. Mehrabinezhad, H. Akbari. Solving Volterra integral equations of the second kind by WaveletGalerkin scheme. Computer and Mathematics with Applications 63:1536- 1547 (2012). [6] Miggen Cui, Yingzhen Lin.Nonlinear Numerical Analysis in the Reproducing Kernel Space.Nova Science Publishers, Inc (2008) | ||
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