Development and implemantation of various degree hermite b-spline based collocation schemes for heat transfer, EW and MEW equations

This doctoral thesis is structured into five chapters. In Chapter 1, the details of the starting point and fundamental objective of the study are presented. It presents a comprehensive literature review on the heat conduction equation, the equal-width wave equation, and the modified equal-width wave equation addressed in this thesis. In the second chapter, the methods and basis functions used in the thesis are introduced in detail. In this context, the finite difference methods, the finite elements method, variational methods, weighted residual methods, and the collocation method, a special type of these methods, are discussed. Furthermore, the structures, continuity properties, and application areas of the cubic, quintic and septic Hermite B-spline basis functions specifically used in this thesis together with the classical B-spline basis functions, trigonometric B-spline basis functions used in the literature are explained in detail. In the third chapter, the model problems examined in the thesis and the test problems handled with these model problems are defined in detail. In the fourth section, the numerical solutions of the test problems related to the heat conduction, EW, and MEW equations are obtained using the collocation finite elements method based on cubic, quintic, and septic Hermite B-spline basis functions. The calculated numerical results are presented in detail using tables and graphs, and the accuracy and effectiveness of the proposed methods are evaluated by comparing them with the results of some existing studies in the literature. In the fifth chapter, which is final the chapter of the thesis, the numerical results obtained from the method applied throughout the thesis are presented in comparative tables for the test problems of each model problem considered, and a general evaluation is made. Furthermore, considering the findings obtained from the study, the strengths and limitations of the method are discussed, and the thesis is completed by presenting suggestions for future work in this context.