# حل عددی معادلات دیفرانسیل جزئی کسری با استفاده از ترکیب روش تبدیل دیفرانسیل با روش‌های چند گامی خطی کسری

نوع مقاله : مقاله پژوهشی

نویسندگان

گروه ریاضی کاربردی، دانشکده ریاضی، آمار و علوم کامپیوتر، دانشگاه تبریز، تبریز، ایران

چکیده

در این مقاله یک روش تکراری برای به‌دست آوردن جواب‌های عددی معادلات دیفرانسیل کسری جزئی

%\LTRfootnote{Fractional partial differential equations}

% معرفی شده است. این روش براساس ترکیب روش تبدیل دیفرانسیل \LTRfootnote{Differential transform method} با روش‌های چند گامی خطی کسری \LTRfootnote{Fractional linear multi-step methods} ،(FLMM) بنا شده است. روش پیشنهاد شده دارای هزینه محاسباتی بسیار کم است که با استفاده از آن معادلات دیفرانسیل کسری جزئی به یک دستگاه از معادلات دیفرانسیل معمولی تبدیل می‌شوند. سپس معادلات حاصل با استفاده از اعمال روش‌های چند گامی خطی کسری همانند اویلر کسری با دقت بالا حل می‌شوند. سری جواب به‌دست آمده در روش تبدیل دیفرانسیل در ناحیه‌های بزرگ سرعت همگرایی کندی دارد. در این مقاله با ترکیب روش یادشده با روش‌های چند گامی خطی کسری این نقیصه برطرف می‌شود. نتایج عددی نشان می‌دهند که جواب‌های به‌دست آمده با جواب دقیق معادله دیفرانسیل کسری مطابقت خوبی دارند. نتایج حاصل شده پایداری و دقت اثبات شده روش را تایید می‌کنند.

کلیدواژه‌ها

موضوعات

عنوان مقاله [English]

### Solving Fractional Partial Differential Equations Using Differential Transform Method Combined with Fractional Linear Multi-step Methods

نویسندگان [English]

• Hamidreza Marasi
• Abdolbaghi Soltani
Department of Mathematics, Faculty of Mathematics, Statistics and Computer Science, University of Tabriz„ Tabriz, Iran
چکیده [English]

In this paper, an iterative method for obtaining the numerical solutions of fractional partial

differential equations (FPDEs) is introduced. This method is based on the combination of the differential

transformation method (DTM) with fractional linear multistep methods (FLMM). The proposed method

has a very low computational cost, with the help of which partial fractional differential equations are

converted into a system of ordinary differential equations. Then the resulting equations are solved with high

accuracy by applying fractional linear multi-step methods such as fractional Euler The series of solutions

obtained in the differential transformation method has a slow convergence speed in large regions. In this

study, by combining the mentioned method with linear multi-step methods, this shortcoming is solved.

Numerical results show that the obtained solutions. They are in good agreement with the exact solution of

the fractional differential equation. The obtained results confirm the proven stability and accuracy of the

method.

کلیدواژه‌ها [English]

• Fractional Differential Equation
• Numerical solution
• Stability
• Differtential Transform method
• Fractional linear multi-step method

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### سابقه مقاله

• تاریخ دریافت: 27 تیر 1401
• تاریخ بازنگری: 16 دی 1401
• تاریخ پذیرش: 09 بهمن 1401
• تاریخ اولین انتشار: 25 بهمن 1401