چكيده فارسي :
ر اين پايان نامه به بررسي برخي از روش هاي عددي براي تقريب مشتق كپوتو مرتبه αام، (1, 0 ∈ (α
مي پردازيم. سپس به كمك الگوريتم هاي عددي مرتبه بالا براي تقريب مشتق كپوتو، يك طرح تفاضل متناهي براي معادله
انتشار-وزش نوع كپوتو را بررسي خواهيم كرد. علاوه بر اين تقريب هاي عددي مرتبه بالا براي مشتق كپوتو با استفاده از
درونيابي و گسسته سازي مورد بررسي قرار گرفته و به آناليز پايداري و همگرايي روش ها پرداخته خواهد شد
چكيده انگليسي :
In recent years, fractional calculus has been a fruitful field of research in science and engineering. In fact, many
scientific areas are currently paying attention to the fractional calculus concepts, and we can refer to its adoption in viscoelasticity and damping, diffusion and wave propagation, electromagnetism, chaos and fractals,
heat transfer, biology, electronics, and signal processing. The concept of fractional calculus can be summarized in the derivative and integral of non-integer order. Fractional differential equations are generalizations of
classical differential equations of integer order that have recently proved to be valuable tools for the modeling
of many physical phenomena. This topic has been the focus of many studies due to its frequent appearances in
various applications, such as physics, biology, finance, fractional dynamics, engineering, signal processing,
and control theory. However, because of the nonlocal properties of fractional operators, it is difficult, even
impossible to obtain the analytical solutions of fractional differential equations. So, deriving numerical methods for fractional differential equations is one of the core issues in the studies of numerical fractional calculus.
Hence, the development of methods is useful and reliable for solving fractional differential equations. Up to
now, there exist several numerical methods to approximate Caputo derivatives. Constructing a series of new
high-order numerical approximations to αth (0 < α < 1) order Caputo derivative by using rth (r4 is a
positive integer) degree interpolation approximation for the integral function refers to some works of Li et al.
They apply a high-order finite difference scheme for Caputo-type advection-diffusion equation with Dirichlet
boundary conditions by using the derived numerical approximation. Besides, practical numerical stability is
also shown for the case r = 4. The other cases (r4 for positive integer r) can be similarly investigated.
In this thesis, we aim to investigate some recent works by Li et al. related to high-order approximations
for Caputo derivatives and applying new numerical approximations for time-fractional advection-diffusion
equations. For this purpose first, some preliminaries which will be used in the sequel are prepared and the
introduction of fractional calculus is mentioned. Some important preliminaries are some useful definitions
such as Grunwald-Letnikov derivative, Riemann-Liouville derivative, Caputo fractional derivative, and Riesz
derivative. Then, numerical methods such as L1, L2, and L2c are represented. After that, by using the Taylor
expansion of Lagrange polynomials, high-order approximations for Caputo derivatives are obtained. In the
continuation of the text, fractional derivative approximation by a central finite difference scheme is investigated and discussed with initial boundary conditions to check the stability and the amount of local truncation
error. Finally, two different examples are solved by considering r = 4, 5 to demonstrate the efficiency of
these schemes and confirm the theoretical results. For this purpose, several suitable MATLAB codes are prepared. Numerical results confirm theoretical results and show that the numerical order of convergence in time
and space are approximately r + 1 − α and two, respectively
