یک روش عددی برای حل معادله کابل با مشتق کسری بر پایه الگوی تفاضلات متناهی
محورهای موضوعی : آمارلیلا اعظمی 1 , امیر حسین رفاهی شیخانی 2 , هاشم صابری نجفی 3
1 - گروه ریاضی، واحد لاهیجان، دانشگاه آزاد اسلامی، لاهیجان، ایران
2 - گروه ریاضی، واحد لاهیجان، دانشگاه آزاد اسلامی، لاهیجان، ایران
3 - گروه ریاضی، واحد لاهیجان، دانشگاه آزاد اسلامی، لاهیجان، ایران
کلید واژه: Convergence, Stability, Caputo-Fabrizio derivative, Membrane potential, Fourier analysis,
چکیده مقاله :
در این مقاله، یک تقریب عددی برای حل نوعی از معادله کابل کسری بر پایه روش تفاضلات متناهی پیشنهاد نمودیم که می توان به آسانی به دیگر معادلات کابل کسری تعمیم داد. مشتق کسری بکار گرفته شده از نوع کاپوتو-فابریزیو است. با کمک فرمول تفاضلی پیشرو مشتق کسری را تقریب زده و سپس معادله کسری را همراه تقریب ذکر شده و طرح تفاضلی گسسته سازی می کنیم به منظور بررسی پایداری و همگرایی روش پیشنهادی، قضایایی ارائه نمودیم. در پایان با استفاده از دو مثال عددی از معادله کابل با شرایط مرزی و اولیه متفاوت کارآیی و دقت روش بکار گرفته شده، را مورد بررسی قرار دادیم. نتایج بدست آمده را با جوابهای دقیق معادله مقایسه نمودیم. نمودارهای اراﺋﻪ شده در هر دو مثال گویای سازگاری جوابهای عددی و جوابهای تحلیلی است که به وضوح دیده می شود. همچنین جدولها هم نمایشگر خطای کوچک بین دو جواب عددی و تحلیلی است که نشان میدهد روش پیشنهادی ما از دقت بالایی برخوردار است
In this paper, we proposed a numerical approximation to solve a type of fractional Cable equations based on the finite difference scheme that can be easily generalized to other fractional cable equations. The fractional derivatives are considered in the Caputo-Fabrizio type. By using the forward difference formula, the fractional derivative is approximated and then we can discretize the fractional equation with the mentioned approximation and difference scheme In order to demonstrate the stability and convergence, several Theorems and Lemmas are provided. Finally, two numerical examples of the cable equation with different boundary and initial conditions are used to demonstrate the effective and accuracy of the scheme. The obtained numerical results are compared with the analytical solution. Figures presented in both examples show consistency of numerical and analytical solutions that can be clearly seen. Tables also show a small error between the numerical and analytical solution which show that our proposed method is highly accurate.
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