Using New Operational Matrix for Solving Nonlinear Fractional Integral Equations
محورهای موضوعی : مجله بین المللی ریاضیات صنعتی
1 - Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran.
2 - Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran.
کلید واژه: Fractional integral equations, Nonlinear integral equations, Operational matrix, Numerical methods, New basis function, Integral equations,
چکیده مقاله :
In this paper, a numerical method for solving nonlinear fractional integral equations (NFIE) is introduced. This method is based on the new basis functions (NFs) introduced in [M. Paripour and et al., Numerical solution of nonlinear Volterra Fredholm integral equations by using new basis functions, Communications in Numerical Analysis, (2013)]. Since the conventional operational matrices for fractional kernels are singular, the definition of these matrices is modified. In order to increase the accuracy of approximating integrals, the operational matrices are exactly calculated and parametrically presented. Then, the solution procedure is proposed and applied on NFIE. Furthermore, the error analysis is performed and rate of convergence is obtained. In addition, various numerical examples are provided for a wide range of fractional orders and nonlinearity of integral equations. Comparison of the results with the exact solutions and those reported in previous studies indicate the capability, salient accuracy, and superiority of the proposed method over similar ones.
در این مقاله یک روش عددی برای حل معادلات انتگرالی کسری غیر خطی (NFIE) بر اساس توابع پایه ی جدیدی که در مرجع ]16[ معرفی شده است، ارائه می گردد. ابتدا، ماتریس های عملیاتی تعمیم و بهبود داده شده تا بتواند مناسب انتگرال های کسری گردند. به کمک انتگرال گیری دقیق، ماتریس های مذکور به صورت پارامتری بدست می آیند. سپس، روش حل تشریح و بر روی معادلات انتگرالی غیر خطی اعمال می شوند. همچنین، تحلیل خطا صورت گرفته و مرتبه ی همگرایی بدست می آید. علاوه بر آن، مثال های عددی متعددی به ازای مقادیر بازه ی گسترده ای از مرتبه ی کسری بودن معادله و نیز توان جمله های غیر خطی ارائه می گردد. مقایسه ی نتایج با حل دقیق و نیز با نتایجی که در مطالعات پیشین گزارش شده اند توانایی، دقت قابل توجه و نیز برتری روش حاضر را نسبت به روش های مشابه نشان می دهد.
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