Solving Volterra Integral Equations of the Second Kind with Convolution Kernel
محورهای موضوعی : مجله بین المللی ریاضیات صنعتی
M. S. Barikbin
1
(Department of Mathematics, Yadegar-e-Emam Khomeyni (RAH) Shahr-e-Rey Branch, Islamic Azad University, Tehran, Iran.)
A. R. Vahidi
2
(Department of Mathematics, Yadegar-e-Emam Khomeyni (RAH) Shahr-e-Rey Branch, Islamic Azad University, Tehran, Iran.)
T. ِDamercheli
3
(Department of Mathematics, Yadegar-e-Emam Khomeyni (RAH) Shahr-e-Rey Branch, Islamic Azad University, Tehran, Iran.)
کلید واژه: Ordinary differential equation, Volterra integral equation, Error analysis, Taylor expansion, Approximate solution,
چکیده مقاله :
In this paper, we present an approximate method to solve the solution of the second kind Volterra integral equations. This method is based on a previous scheme, applied by Maleknejad ‎et al., ‎‎[K. Maleknejad ‎and Aghazadeh, Numerical solution of Volterra integral equations of the second kind with convolution kernel by using Taylor-series expansion method, ‎Appl. Math. Comput.‎ (2005)]‎ to gain the approximate solution of the second kind Volterra integral equations with convolution kernel and Maleknejad ‎et al. ‎[K. Maleknejad ‎and‎ T. Damercheli, Improving the accuracy of solutions of the linear second kind volterra integral equations system by using the Taylor expansion method, ‎Indian J. Pure Appl. Math.‎ (2014)] ‎to gain the approximate solutions of systems of second kind Volterra integral equations with the help of Taylor expansion method. The Taylor expansion method transforms the integral equation into a linear ordinary differential equation (ODE) which, in this case, requires specified boundary conditions. Boundary conditions can be determined using the integration technique instead of differentiation technique. This method is more stable than derivative method and can be implemented to obtain an approximate solution of the Volterra integral equation with smooth and weakly singular kernels. An error analysis for the method is provided. A comparison between our obtained results and the previous results is made which shows that the suggested method is accurate enough and more ‎stable.‎
در این مقاله، یک روش تقریبی برای حل معادلات انتگرال ولترای نوع دوم ارائه می­دهیم. این روش بر مبنای روش بسط تیلوری است که مالک نژاد و آقازاده برای بدست آوردن جواب تقریبی معادلات انتگرال ولترای نوع دوم با هسته پیچشی و مالک نژاد و دمرچلی جهت یافتن جواب تقریبی دستگاه معادلات انتگرال ولترای نوع دوم به کار بسته­اند. روش بسط تیلور، معادله انتگرال را به یک دستگاه معادلات دیفرانسیل معمولی خطی تبدیل می­کند که در این حالت شرایط مرزی مشخص مورد نیاز است. شرایط مرزی می­تواند با استفاده از تکنیک انتگرالگیری به جای تکنیک مشتقگیری بدست آید. روش ارائه شده پایدارتر از روش مشتقگیری است و می­تواند جهت یافتن جواب تقریبی معادله انتگرال ولترا با هسته­های هموار و منفرد ضعیف استفاده شود. تحلیل خطای روش نیز ارائه شده است. مقایسه بین نتایج بدست آمده ما و نتایج قبلی نشان می­دهد که روش پیشنهادی دقیق­تر و پایدارتر است.
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