خواص ساختاری ضرب خارجی اعداد فازی و کاربردهای آن
محورهای موضوعی : آمار
رباب علیخانی
1
(ریاضی کاربردی-دانشکده علوم ریاضی-دانشگاه تبریز-تبریز)
کلید واژه: Triangular fuzzy number, Generalized differentiability, Cross product, Fuzzy variable coefficients, Linear fuzzy differential equations,
چکیده مقاله :
در حساب اعداد فازی، عمل ضرب و جمع بر اساس اصل توسیع زاده بنا نهاده شده است. این ضرب از دیدگاه نظری و عملی دارای چندین خاصیت غیرطبیعی است. برای غلبه بر چنین معایبی اخیراً یک عمل ضرب جدید با عنوان ضرب خارجی ارائه شده است. مزیت اصلی این ضرب این است که شکل اعداد فازی مثلثی و ذوزنقه‌ای تحت ضرب خارجی حفظ می‌شود و از دیدگاه محاسباتی خیلی کاربردی تر از ضرب معمولی است. بنابراین ضرب خارجی دو عدد فازی می‌تواند یک انتخاب دیگر به جای ضرب معمولی بدست آمده از اصل توسیع زاده، در مسائل کاربردی باشد. هدف این مقاله، ارائه‌ی فرمولی صریح برای ضرب خارجی اعداد فازی مثلثی بر اساس ضرب اسکالر اعداد فازی و سپس با استفاده از آن فرمولی برای طول ضرب خارجی دو عدد فازی مثلثی و مشتق ضرب خارجی دو تابع فازی مثلثی است. همچنین در این مقاله رابطه‌ی بین هسته ضرب خارجی و معمولی اعداد فازی بیان شده است. در نهایت، به عنوان یک کاربرد، مفهوم ضرب خارجی در معادلات دیفرانسیل خطی مرتبه‌ی اول با ضرایب متغییر فازی بکار برده شده و جواب‌های مثلثی آن تحت مشتق‌پذیری تعمیم یافته بدست آورده می‌شود. چندین مثال برای بیان کارایی نتایج نظری و مقایسه با روش‎های پیشین آورده می‌شود.
In the fuzzy arithmetic, the definitions of addition and multiplication of fuzzy numbers are based on Zadeh’s extension principle. From theoretical and practical points of view, this multiplication of fuzzy numbers owns several unnatural properties. Recently, to avoid this shortcoming, a new multiplicative operation of product type is introduced, the so-called cross-product of fuzzy numbers. The main advantage is that this product preserves the shape of triangular or trapezoidal fuzzy numbers under multiplication and from computational point of view the cross product is more applicable than the usual product. The above mentioned properties motivate us to use the cross product in applications as a possible alternative of the product obtained by Zadeh's extension Principle. The aim of the present paper is to give an explicit formula for the cross product of triangular fuzzy numbers based on the scalar product of fuzzy numbers and then, explicit formulas for the length of cross product of triangular fuzzy numbers and fuzzy derivative of cross product of triangular fuzzy functions. As an application, we apply the cross product concept for the first order linear fuzzy differential equations with fuzzy variable coefficients and obtain its triangular solutions under generalized differentiability. Finally, some examples are given to illustrate the theoretical results.
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