الفهرس 
Chapter 1 IntroductionOnly 14 pages are availabe for public view
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Abstract
This thesis investigates the numerical solution of mixed Fredholm-Volterra integral equations (MFVIEs). A type of mathematical problem characterized by its complexity. Even though MFVIEs are of major mathematical importance, it is frequently difficult to find their exact solutions. This study focuses on numerical solutions to overcome precisely this obstacle. In particular, the spotlight is on an approach called the two-grid iterative technique. Above all, it is important to highlight that the two-grid iterative method, which is thoughtfully carried out in this thesis, goes above traditional MFVIEs. It can also be used successfully in the more challenging domain of mixed Fredholm-Volterra integrodifferential equations (MFVIDEs). The convergence analysis showed that using this technique reduces computational costs by 85% compared with the direct method. Also, we introduce an algorithm to implement the proposed method and numerical simulations are presented to verify the accuracy rate of the theoretical analysis. The thesis’s investigation of this method is revealed in this summary: Chapter 1: Introduction: An introduction to integral equations is given in Chapter 1, highlighting the Fredholm and Volterra integral equations. The idea of mixed Fredholm-Volterra integral equations is then presented, along with an extension to mixed Fredholm-Volterra integrodifferential equations. To give a general overview of the approaches that have been used to solve these equations, a literature review is included. The chapter concludes with the statement of objectives for the research. Chapter 2: Numerical solution of Mixed Fredholm-Volterra Integral Equations: Chapter 2 is dedicated to addressing the solution of mixed Fredholm-Volterra integral equations. The chapter delves into the fundamental aspects of these equations, exploring the existence and uniqueness of solutions. The chapter introduces the two-grid iterative method as a powerful computational strategy for dealing with the numerical solution of MFVIEs.Furthermore, the chapter presents the proposed algorithm for implementing the two-grid iterative method, offering a step-by-step guide for its application. The algorithm serves as a practical tool for solving MFVIEs efficiently and accurately. In addition to the theoretical discussion, the chapter also addresses the practical considerations of computational costs associated with the method, providing valuable insights into the efficiency and feasibility of its implementation. To validate the effectiveness and accuracy of the two-grid iterative method, numerical examples are included in the chapter. These examples serve as concrete illustrations of the method’s capability to provide accurate solutions to MFVIEs. By showcasing the successful application of the approach in diverse scenarios, the chapter establishes its practical utility and robustness. Overall, Chapter 2 presents a comprehensive exploration of the solution of MFVIEs using the two-grid iterative method. Through theoretical analysis, algorithmic implementation, consideration of computational costs, and validation through numerical examples, the chapter provides a solid foundation for understanding and applying the two-grid iterative method in solving mixed Fredholm-Volterra integral equations. Chapter 3: Numerical solution of Mixed Fredholm-Volterra Integrodifferential Equations: In Chapter 3, the two-grid iterative method is extended to deal with mixed Fredholm-Volterra integrodifferential equation solutions. The chapter presents a detailed implementation algorithm for the method and discusses its adaptation specifically for MFVIDEs. Numerical examples demonstrating the accuracy of the solutions found are given to confirm the approach’s effectiveness and productivity.The chapter also examines the method’s computational costs, providing insight into the viability and effectiveness of the two-grid iterative approach to MFVIDE solution. The implementation technique, numerical examples, and computational cost analysis demonstrate the usefulness and benefits of using the two-grid iterative approach as a dependable and effective method for MFVIDE solution.Chapter 4: Conclusions and Future Work: As a conclusion, the last chapter summarizes the main findings derived from the study. It is expected to highlight how well the two-grid iterative method deals with MFVIEs as well as MFVIDEs. This chapter also points out possible shortcomings in the approach and suggests areas for additional research. The chapter concludes with a discussion on potential future research directions to further enhance and expand the applications of the two-grid iterative method in solving a broader range of integrodifferential equations. Overall, the application of the two-grid iterative method demonstrates its efficiency, accuracy, and potential for reducing computational costs in solving these types of equations. The findings of this research provide a valuable foundation for further advancements in the field and open up avenues for future investigations.