| सारांश: | This study provided some new methods to solve initial value problems (IVPs) and boundary
value problems (BVPs) of fractal-fractional differential equations (FFDEs) using operational
matrix (OM) and artificial neural networks (ANNs). This research is centered on deriving two
methods and formulating two novel definitions of fractal-fractional differential and integral
operators. The first part of this thesis presents a new definition of the generalized Caputo
differential and integral operators with fractional order and fractal dimension. Utilizing
the OM based on orthogonal polynomials (Legendre and Jacobi), a numerical method for
addressing various types of FFDEs is provided. This thesis emphasizes the existence theory
and numerical solutions of multi-order boundary and initial value FFDEs. In these chapters,
we explore convergence, existence, and uniqueness of solutions to FFDEs, aiming to determine
the existence and uniqueness of at least one solution. Additionally, an error-bound analysis
is conducted to confirm the validity and convergence of the method. The OM simplifies
FFDEs into algebraic systems, resulting in straightforward and easily solvable problems.
Subsequently, the performance of the proposed technique in addressing real-world problems
is demonstrated. In the second part of the thesis, we developed the Hilfer fractal-fractional
derivative definition. Similarly, the OM with the tau method for Hilfer fractal-fractional
differentiability is generalized for solving FFDEs based on orthogonal polynomials. Numerical
results suggest that the proposed method is quite accurate compared to other existing methods.
The Jacobi polynomial, with its two parameters, ξ and ϑ, leads to distinct collections of
orthogonal polynomials. Adjusting these parameters generates different types of orthogonal
polynomials, each with unique characteristics. We also investigated numerical illustrations
by varying the values of fractional and fractal parameters as well as the number of terms
from truncated shifted Legendre polynomials (SLPs) and shifted Jacobi polynomials (SJPs).
Our OM techniques based on SLPs and SJPs require only a few terms to obtain an accurate
solution. In the third part, ANNs based on a generalized power series method in the generalized
Caputo fractal-fractional derivative (GCFFD) are derived to approximate solutions of linear
and non-linear FFDEs. Finally, ANNs employing a combination of power series methods
in the GCFFD are developed to approximate solutions of higher-order linear FFDEs with
both constant and variable coefficients. Initially, the algorithm utilized a truncated series.
The values of the unknown coefficients in this truncated power series were then determined
using an optimization technique to minimize the criterion function. This discovery indicates
convergence toward optimal model coefficients as the learning process advances. Compared
to other traditional methods, the suggested approach has proven to be more accurate. The
definitions and techniques provided surpass traditional methods in accuracy, representing a
significant advancement in the field.
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