# تقریب تابع مجهول وابسته به زمان در مسئله معکوس تلگراف یک بعدی

نوع مقاله : مقاله پژوهشی

نویسندگان

گروه ریاضی، دانشکده علوم، دانشگاه علم و فناوری مازندران، بهشهر، ایران

چکیده

دراین مقاله، مسئله معکوس خطی جهت تقریب تابع وابسته به زمان در متن معادله هذلولوی مرتبه دوم را

مورد مطالعه قرار می دهیم. برای حل مسئله، معلوماتی نظیر شرایط مرزی نیومان بهمراه یک شرط انتگرالی و شرایط

اولیه در لحظه ی آغازین و لحظه نهایی در اختیار ما قرار داده شده است. در گام نخست نشان می دهیم که این مسئله

دارای جواب یکتا می باشد. سپس مسئله اولیه را با استفاده از تبدیلاتی مناسب به یک مسئله جدید تبدیل می کنیم و

سپس تقریب طیفی بر اساس روش هم مکانی-ریتز را برای بازیابی توابع مجهول ارایه می دهیم. گسسته سازی مسئله

با استفاده از تکنیک ارایه شده به یک دستگاه معادلات خطی منتهی می شود که از روش منظم سازی تیخونوف جهت

حل آن استفاده می شود. نتیجه شبیه سازی عددی موید دقت بالا و پایداری جواب تقریبی می باشند. نتایج حاصل از

به کارگیری روش ارائه شده در این مقاله، برای چند مثال عددی نمایش داده شده است

کلیدواژه‌ها

موضوعات

عنوان مقاله [English]

### Approximation of a timewise dependent function in the inverse one-dimensional telegraph equation

نویسندگان [English]

• Kamal Rashedi
• Ali Tahmasbi
Department of Mathematics, University of Science and Technology of Mazandaran, Behshahr, Iran
چکیده [English]

In this article, we study the linear inverse problem for approximating a timewise-dependent

function in the second-order hyperbolic equation. To solve the problem, information such as Neumann

boundary conditions along with an integral condition and initial conditions at the initial moment and the

final instant have been provided. In the first step, we show that this problem has a unique solution. Then,

we change the main problem into a new one and then we present the spectral approximation based on

the Ritz-collocation method to recover the unknown functions. Discretization of the problem by using

the presented technique leads to a linear system of algebraic equations, which Tikhonov’s regularization

method is used to obtain stable solutions. The results of the numerical simulation confirm the high accuracy

and stability of the approximate solution.

کلیدواژه‌ها [English]

• Inverse telegraph equation
• Existence and Uniqueness of solution
• Tikhonov regularization
• Spectral method
• Discretization

#### مراجع

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### سابقه مقاله

• تاریخ دریافت: 21 اسفند 1401
• تاریخ بازنگری: 17 خرداد 1402
• تاریخ پذیرش: 29 خرداد 1402
• تاریخ اولین انتشار: 01 تیر 1402