Analysis on existence of system of coupled multifractional nonlinear hybrid differential equations with coupled boundary conditions

This article dealt with a class of coupled hybrid fractional differential system. It consisted of a mixed type of Caputo and Hilfer fractional derivatives with respect to two different kernel functions, $\psi_{_1}$ and $\psi_{_2}$, respectively, in addition to coupled boundary conditions. The existence of the solution of the system was investigated using the Dhage fixed point theorem. Finally, an illustration was presented to validate our findings

Existence and attractivity results for ψ -Hilfer hybrid fractional differential equations

In this work, we present some results on the existence of attractive solutions of fractional differential equations of the ψ-Hilfer hybrid type. The results on the existence of solutions are a consequence of the Schauder fixed point theorem. Next, we prove that all solutions are uniformly locally attractive

Existence results by Mönch's fixed point theorem for a tripled system of sequential fractional differential equations

In this paper, we study the existence of the solutions for a tripled system of Caputo sequential fractional differential equations. The main results are established with the aid of Mönch's fixed point theorem. The stability of the tripled system is also investigated via the Ulam-Hyer technique. In addition, an applied example with graphs of the behaviour of the system solutions with different fractional orders are provided to support the theoretical results obtained in this study

Piecewise implicit coupled system under ABC fractional differential equations with variable order

This research paper presented a novel investigation into an implicit coupled system of fractional variable order, which has not been previously studied in the existing literature. The study focused on establishing and developing sufficient conditions for the existence and uniqueness of solutions, as well as the Ulam-Hyers stability, for the proposed coupled system without using semigroup property. By extending the existing conclusions examined for the Atangana-Baleanu-Caputo (ABC) operator, we contributed to advancing the understanding of variable-order fractional differential equations. The paper provided a solid theoretical foundation for further analysis, numerical simulations, and practical applications. The obtained results have implications for designing and controlling systems modeled using fractional variable order equations and serve as a basis for addressing a wide range of dynamical problems. The transformation techniques, qualitative analysis, and illustrative examples presented in this work highlight its unique contributions and potential to serve as a foundation for future research

Fractional Calculus Operators and the Mittag-Leffler Function

This book focuses on applications of the theory of fractional calculus in numerical analysis and various fields of physics and engineering. Inequalities involving fractional calculus operators containing the Mittag–Leffler function in their kernels are of particular interest. Special attention is given to dynamical models, magnetization, hypergeometric series, initial and boundary value problems, and fractional differential equations, among others

Improved modeling for fluid flow through porous media

Petroleum production is one of the most important technological challenges in the current world. Modeling and simulation of porous media flow is crucial to overcome this challenge. Recent years have seen interest in investigation of the effects of history of rock, fluid, and flow properties on flow through porous media. This study concentrates on the development of numerical models using a ‘memory’ based diffusivity equation to investigate the effects of history on porous media flow. In addition, this study focusses on developing a generalized model for fluid flow in packed beds and porous media. The first part of the thesis solves a memory-based fractional diffusion equation numerically using the Caputo, Riemann-Liouville (RL), and Grünwald-Letnikov (GL) definitions for fractional-order derivatives on uniform meshes in both space and time. To validate the numerical models, the equation is solved analytically using the Caputo, and Riemann-Liouville definitions, for Dirichlet boundary conditions and a given initial condition. Numerical and analytical solutions are compared, and it is found that the discretization method used in the numerical model is consistent, but less than first order accurate in time. The effect of the fractional order on the resulting error is significant. Numerical solutions found using the Caputo, Riemann-Liouville, and Grünwald-Letnikov definitions are compared in the second part. It is found that the largest pressure values are found from Caputo definition and the lowest from Riemann-Liouville definition. It is also found that differences among the solutions increase with increasing fractional order

A finite volume approach for the numerical analysis and solution of the Buckley-Leverett equation including capillary pressure

The study of petroleum recovery is significant for reservoir engineers. Mathematical models of the immiscible displacement process contain various assumptions and parameters, resulting in nonlinear governing equations which are tough to solve. The Buckley-Leverett equation is one such model, where controlling forces like gravity and capillary forces directly act on saturation profiles. These saturation profiles have important features during oil recovery. In this thesis, the Buckley-Leverett equation is solved through a finite volume scheme, and capillary forces are considered during this calculation. The detailed derivation and calculation are also illustrated here. First, the method of characteristics is used to calculate the shock speed and characteristics curve behaviour of the Buckley-Leverett equation without capillary forces. After that, the local Lax-Friedrichs finite-volume scheme is applied to the governing equation (assuming there are no capillary and gravity forces). This mathematical formulation is used for the next calculation, where the cell-centred finite volume scheme is applied to the Buckley- Leverett equation including capillary forces. All calculations are performed in MATLAB. The fidelity is also checked when the finite-volume scheme is computed in the case where an analytical solution is known. Without capillary pressure, all numerical solutions are calculated using explicit methods and smaller time steps are used for stability. Later, the fixed-point iteration method is followed to enable the stability of the local Lax-Friedrichs and Cell-centred finite volume schemes using an implicit formulation. Here, we capture the number of iterations per time-steps (including maximum and average iterations per time-step) to get the solution of water saturation for a new time-step and obtain the saturation profile. The cumulative oil production is calculated for this study and illustrates capillary effects. The influence of viscosity ratio and permeability in capillary effects is also tested in this study. Finally, we run a case study with valid field data and check every calculation to highlight that our proposed numerical schemes can capture capillary pressure effects by generating shock waves and providing single-valued saturation at each position. These saturation profiles help find the amount of water needed in an injection well to displace oil through a production well and obtains good recovery using the water flooding technique

Nonlinear Dynamics

This volume covers a diverse collection of topics dealing with some of the fundamental concepts and applications embodied in the study of nonlinear dynamics. Each of the 15 chapters contained in this compendium generally fit into one of five topical areas: physics applications, nonlinear oscillators, electrical and mechanical systems, biological and behavioral applications or random processes. The authors of these chapters have contributed a stimulating cross section of new results, which provide a fertile spectrum of ideas that will inspire both seasoned researches and students

Wavelet Theory

The wavelet is a powerful mathematical tool that plays an important role in science and technology. This book looks at some of the most creative and popular applications of wavelets including biomedical signal processing, image processing, communication signal processing, Internet of Things (IoT), acoustical signal processing, financial market data analysis, energy and power management, and COVID-19 pandemic measurements and calculations. The editor’s personal interest is the application of wavelet transform to identify time domain changes on signals and corresponding frequency components and in improving power amplifier behavior

Special Functions: Fractional Calculus and the Pathway for Entropy

Historically, the notion of entropy emerged in conceptually very distinct contexts. This book deals with the connection between entropy, probability, and fractional dynamics as they appeared, for example, in solar neutrino astrophysics since the 1970's (Mathai and Rathie 1975, Mathai and Pederzoli 1977, Mathai and Saxena 1978, Mathai, Saxena, and Haubold 2010). The original solar neutrino problem, experimentally and theoretically, was resolved through the discovery of neutrino oscillations and was recently enriched by neutrino entanglement entropy. To reconsider possible new physics of solar neutrinos, diffusion entropy analysis, utilizing Boltzmann entropy, and standard deviation analysis was undertaken with Super-Kamiokande solar neutrino data. This analysis revealed a non-Gaussian signal with harmonic content. The Hurst exponent is different from the scaling exponent of the probability density function and both Hurst exponent and scaling exponent of the Super-Kamiokande data deviate considerably from the value of ½, which indicates that the statistics of the underlying phenomenon is anomalous. Here experiment may provide guidance about the generalization of theory of Boltzmann statistical mechanics. Arguments in the so-called Boltzmann-Planck-Einstein discussion related to Planck's discovery of the black-body radiation law are recapitulated mathematically and statistically and emphasize from this discussion is pursued that a meaningful implementation of the complex ‘entropy-probability-dynamics’ may offer two ways for explaining the results of diffusion entropy analysis and standard deviation analysis. One way is to consider an anomalous diffusion process that needs to use the fractional space-time diffusion equation (Gorenflo and Mainardi) and the other way is to consider a generalized Boltzmann entropy by assuming a power law probability density function. Here new mathematical framework, invented by sheer thought, may provide guidance for the generalization of Boltzmann statistical mechanics. In this book Boltzmann entropy, generalized by Tsallis and Mathai, is considered. The second one contains a varying parameter that is used to construct an entropic pathway covering generalized type-1 beta, type-2 beta, and gamma families of densities. Similarly, pathways for respective distributions and differential equations can be developed. Mathai's entropy is optimized under various conditions reproducing the well-known Boltzmann distribution, Raleigh distribution, and other distributions used in physics. Properties of the entropy measure for the generalized entropy are examined. In this process the role of special functions of mathematical physics, particularly the H-function, is highlighted