Solution and comparison of differential equations with some approximate solutions methods

In the world of science and technology, second-order ordinary differential equations hold significant importance in solving scientific problems. Many scientists engaged in applied sciences work on such differential equations. One of these differential equations studied is the Lane-Emden type of differential equation. This equation bears the names of astrophysicists Jonathan Homer Lane and Robert Emden. In astrophysics, the Lane-Emden equation is a dimensionless form of Poisson's equation for the gravitational potential of a Newtonian self-gravitating, spherically symmetric, polytrophic fluid. The Lane-Emden equation was initially utilized to describe the internal structure of a self-gravitating polytropic body. These equations constitute a system of differential equations used to calculate the radius and mass of a star, define its internal structure, analyze the distribution of variables such as pressure, density, and temperature, and examine the hydrostatic equilibrium state. In cases where temperature effects are included, such as when studying isothermal gas spheres, the astrophysical phenomenon is described by a Lane-Emden equation. In the latter half of the 20th century, interesting applications of the isothermal solution (singular isothermal sphere) and its non-singular modifications were utilized in the structures of collisionless systems like globular clusters and early-type galaxies. These equations are particularly useful when studying certain types of stars, such as low-mass stars, where we aim to understand their mass and structural properties. A second-order ordinary differential equation emerging in the study of stellar interiors is also known as a polytropic differential equation. Furthermore, these equations have broader applications beyond astrophysics. They serve as fundamental models in the study of fluid dynamics, specifically in compressible fluid spheres, which find relevance in various engineering fields, including chemical engineering and aerodynamics. The Lane-Emden equations' mathematical elegance and utility make them an integral part of theoretical astrophysics and applied mathematics. Their versatility allows mathematicians and scientists to explore and comprehend complex physical systems, providing valuable insights into the behavior of celestial objects and contributing to advancements in mathematical modeling across diverse scientific disciplines. Solving and analyzing Lane-Emden type differential equations has been challenging, especially due to singularities occurring at certain points, like the origin. While solutions can be found through computational means, there's a recognized need for analytical solutions to enhance physical understanding. This thesis aims to emphasize the significance of Lane-Emden type differential equations by introducing various methods used in solving these equations and evaluating the impact of these methods on applications. Based on the results of the applications, an opinion will be stated regarding which solution method proves to be more practical, thereby contributing to further research in the relevant field. This work comprises four sections structured as follows: In the first chapter, the formulated representation of the Lane-Emden type differential equations introduced in the abstract, its basic elements and examples of equations are given. Then, the aim of the thesis and the methods developed in the literature for solving Lane-Emden type equations are mentioned. The second section delves into the theoretical aspect, introducing fundamental concepts and the solution methods used to solve problems: the HE's variational iteration method, finite differences method, homotopy perturbation method, homotopy analysis method, and Adomian decomposition method. In the third chapter of the thesis, we solved two Lane-Emden type differential equations under specified boundary conditions using several methods. Firstly, solutions were obtained using the HE variational iteration method. For this purpose, Lagrange multiplier and initial approximation function were determined. Subsequently, functional solutions were computed through iterative processes up to a certain level. Secondly, we employed the Adomian decomposition method for the solution. Here, we initially identified Adomian polynomials as initial approximation function and then proceeded with iterative procedures, achieving functional solutions up to a specified level. Thirdly, for the solution using the homotopy perturbation method, we established the homotopy structure to calculate approximation functions. Within the homotopy analysis method, we formulated the zeroth-order deformation equation, followed by higher-order deformation equations, and conducted iterative procedures to solve them. Additionally, solutions were obtained in Matlab using the finite differences method based on the chosen value of 'n'." The final section interprets the results obtained from Lane-Emden type differential equations using the HE variational iteration method, Adomian decomposition method, homotopy perturbation method, homotopy analysis method, and finite differences method. it is seen that closer results can be obtained by calculating higher-order approximations for all methods used. Adomian decomposition method can be easily applied to differential equations. Approximate solutions obtained using this method are nearly identical to the analytical solutions of nonlinear equations. Adomian polynomials can be separately determined for different types of Lane-Emden equations, leading to approximate solutions. Homotopy analysis method is applicable to nonlinear differential equations. The approximate solutions obtained through this method are almost the same as the analytical solutions of the Lane-Emden (Poisson) equation. Writing approximate solutions using the homotopy analysis method can yield close results to the exact solutions. Convergence in the homotopy perturbation method depends on the initial approximation chosen and is also related to the homotopy path. Results close to the exact solutions have been observed. Moreover, this method requires more computational effort compared to others. The cost of the finite differences method is also higher compared to others, but selecting a large value can yield close results. Due to the high computational load, computer assistance might be necessary. The values used in the solution methods in this thesis have been arbitrarily chosen, without any specific reason. There is no particular justification for their selection. Changing the values for a more detailed comparison of solutions could be the subject of another study.