LAPLACE AND ELZAKI TRANSFORMS COLLOCATION METHOD FOR TELEGRAPH EQUATIONS DEFINED BY CARGO DERIVATIVE

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LAPLACE AND ELZAKI TRANSFORMS COLLOCATION METHOD FOR TELEGRAPH EQUATIONS DEFINED BY CARGO DERIVATIVE

CHAPTER ONE

INTRODUCTION

Background

Fractional differential equations (FDEs) have appeared as a new branch of applied mathematics and have been utilized in several mathematical systems in applied science. In fact, FDEs are an alternative type to non-linear equations. Various forms play an essential role and techniques, not only in mathematics but also in mechanics, process control, complex schemes and technology, to produce mathematical modelling of several natural processes. These calculations, of course, need to be overcome. A number of experiments on fractional and FDEs involving various operators, such as Erdelyi-Kober, Riemann-Liouville, Caputo, Weyl Riesz and Grunwald-Letnikov operators, have emerged over the past three centuries with implementations in other areas [1,2,3,4,5].

The communication process plays a critical role in the global community in this modern world. High-frequency communication technologies continue to profit from important industrial attention, triggered by a host of microwave communication and radio frequency schemes. Certainly, all transmission media have a signal loss. Signal losses need to be determined to the transmission media. Telegraph equations are used for electrical signal propagation in the signal analysis, wave propagation, transmission line cable, random walk, and so forth. Heaviside has created a transmission line. This transmission can be classified into two categories, unguided and guided. In the guided medium, the signal is transmitted via the transmission system or copper wire. These guided media convey the higher frequencies current and voltage waves. While in unguided media, electromagnetic fields carry the signal over part or all communication channels through microwave communication and radiofrequency systems. Such electromagnetic waves are broadcast and processed by the antenna. Specifically, cable transmission mediums are investigated in controlled transmission media to resolve effective telegraph transmission. A link transmission medium can be delegated a guided transmission medium and speaks to a physical framework that legitimately proliferates the data between at least two areas. To improve the controlled communications system, it is necessary to calculate or predict the power and signal losses in the system, as all systems have these losses. Different analytical and numerical methods have been implemented to solve time-fractional telegraph equations, such as the Homotopy perturbation transform technique [6], the q-Homotopy analysis transform technique [7], the Adomian decomposition technique, the Reduced differential transform technique, the Reproducing Kernel technique, the Variational iteration technique, Haar wavelet and the Sinc-collocation technique

The study of partial differential equations (PDEs) has been a cornerstone of mathematical modeling in various scientific and engineering disciplines. PDEs describe the dynamics of physical phenomena by taking into account spatial and temporal variations. The Telegraph equation is a specific type of PDE that arises in the modeling of wave propagation and signal transmission in telecommunication systems, electrical circuits, and transmission lines. It plays a critical role in the analysis and design of communication networks, ensuring efficient data transmission and signal processing.

To solve PDEs like the Telegraph equation effectively, mathematicians and scientists have developed various mathematical tools and techniques. Laplace and Elzaki transforms are essential methods that can transform a PDE into an algebraic equation, making it more accessible for analysis and solution. The application of these transforms, combined with numerical methods like collocation, has been instrumental in solving complex PDEs in various domains.

This research project delves into the Laplace and Elzaki transforms and their combined application with the collocation method for solving Telegraph equations. It explores the theoretical foundations, computational aspects, and practical implications of this approach, with a particular focus on problems defined by cargo derivatives. The study aims to contribute to the understanding and application of advanced mathematical techniques in solving PDEs in the field of telecommunications and signal processing.

1.2 Research Objectives

The primary objectives of this research project are as follows:

To introduce and analyze the Laplace and Elzaki transforms and their application in solving Telegraph equations.

To explore the collocation method as a numerical approach to solving PDEs and its compatibility with Laplace and Elzaki transforms.

To investigate the specific application of these mathematical techniques in addressing Telegraph equations defined by cargo derivatives.

To provide practical insights into the efficient solution of Telegraph equations in the context of telecommunication systems and signal processing.

1.3 Research Questions

This research project will address the following research questions:

What are the theoretical foundations and mathematical principles of the Laplace and Elzaki transforms?

How can the Laplace and Elzaki transforms be applied in solving Telegraph equations?

What is the collocation method, and how does it complement the Laplace and Elzaki transforms in solving PDEs?

What are the challenges and complexities in addressing Telegraph equations defined by cargo derivatives, and how can they be effectively overcome?

1.4 Significance of the Study

The study of Laplace and Elzaki transforms in combination with the collocation method for solving Telegraph equations defined by cargo derivatives holds significant importance for the following reasons:

Advancing Mathematical Techniques: The research contributes to the advancement of mathematical methods in solving complex PDEs, with potential applications beyond the scope of this study.

Practical Relevance: The findings have practical implications for improving the efficiency and accuracy of signal transmission and processing in telecommunication systems, benefiting various industries and technologies.

Mathematical Education: The research enhances the understanding of mathematical techniques among students, researchers, and practitioners in the field of applied mathematics and engineering.

Interdisciplinary Applications: The mathematical methods studied can be applied in diverse interdisciplinary domains, including physics, electrical engineering, and data science.

1.5 Scope of the Study

This research project focuses on the Laplace and Elzaki transforms in conjunction with the collocation method for solving Telegraph equations defined by cargo derivatives. The study encompasses both theoretical and computational aspects of these mathematical techniques. While the primary application context is telecommunication systems, the findings may have broader relevance in other domains involving wave propagation and signal transmission.

1.6 Organization of the Study

This thesis is structured into multiple chapters to provide a comprehensive analysis of the Laplace and Elzaki transforms and their application in solving Telegraph equations defined by cargo derivatives:

Chapter 2 will review the theoretical foundations of the Laplace and Elzaki transforms, including their mathematical properties and principles.

Chapter 3 will discuss the collocation method as a numerical approach for solving PDEs and its compatibility with the transforms.

Chapter 4 will explore the specific application of these mathematical techniques in addressing Telegraph equations, with a focus on cargo derivatives.

Chapter 5 will provide practical insights, including computational examples and case studies, to demonstrate the effectiveness of the approach in real-world scenarios.

1.7 Conclusion

This introductory chapter has set the stage for the research project, highlighting the importance of studying the Laplace and Elzaki transforms in combination with the collocation method for solving Telegraph equations defined by cargo derivatives. The subsequent chapters will delve into the theoretical and practical aspects of these mathematical techniques, offering valuable insights and contributions to the field of applied mathematics and engineering.

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