# کاربرد روش موجک لژاندر همراه با قانون مربعی گاوس در حل معادلات انتگرال - دیفرانسیل کسری

نوع مقاله : مقاله پژوهشی

نویسنده

گروه ریاضی، مجتمع آموزش عالی سراوان، سراوان، ایران

چکیده

در این مقاله، روشی جدید برای حل معادلات انتگرال - دیفرانسیل ولترا - فردهلم کسری غیرخطی پیشنهاد شده است. این روش تابع مجهول مسئله را توسط موجک‌های لژاندر تقریب می‌زند. برای انجام این کار، موجک‌های لژاندر به‌همراه قانون مربعی گاوس برای تبدیل مسئله به یک سیستم خطی یا غیر‌خطی از معادلات جبری استفاده می‌شود که این سیستم به‌سادگی با استفاده از فن‌های برنامه‌نویسی ریاضی قابل‌حل است. علاوه بر این، وجود و یکتایی راه‌حل ارائه‌شده با استفاده از برخی قضایا و لم‌ها اثبات می‌شود. همچنین تخمین خطا و تحلیل همگرایی این روش نشان داده خواهد شد. علاوه بر این، به‌منظور نشان دادن قابلیت و دقت این روش، چند مثال عددی بیان‌شده است و نتایج به‌دست آمده از این مثال‌ها با نتایج به‌دست آمده از روش‌های موجک چبیشف، روش توابع کلاهی توسعه‌یافته و نیستروم و نیوتن - کانتورویچ مقایسه شده‌اند.

کلیدواژه‌ها

موضوعات

عنوان مقاله [English]

### Application of Legendre wavelet method coupled with the Gauss quadrature rule for solving fractional integro-differential equations

نویسنده [English]

• Mohsen Riahi Beni
Department of Mathematics, Higher Education Complex of Saravan, Saravan, Iran
چکیده [English]

In this work, we propose a novel technique for solving the nonlinear fractional Volterra-Fredholm integro-differential equations (FVFIDEs). This method approximates the unknown function with the Legendre wavelets. To do this, the Legendre wavelets are used in conjunction with the quadrature rule for converting the problem into a linear or nonlinear system of algebraic equations which can be easily solved by applying the mathematical programming techniques. Furthermore, the existence and uniqueness of the solution are proved by preparing some theorems and lemmas. Also, the error estimate and convergence analysis of the method will be shown. Moreover, some examples are presented and their results are compared to the results of Chebyshev wavelet, modiﬁcation of hat functions, NystrÖm and Newton-Kantorovitch methods to show the capability and accuracy of this scheme.

کلیدواژه‌ها [English]

• Legendre wavelet
• Collocation method
• fractional Volterra-Fredholm integro-differential equations

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### سابقه مقاله

• تاریخ دریافت: 20 اسفند 1399
• تاریخ بازنگری: 29 خرداد 1400
• تاریخ پذیرش: 01 مرداد 1400