Operational matrix based on orthogonal polynomials and artificial neural networks methods for solving fractal-fractional differential equations

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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