الفهرس 
Chapter 1Only 14 pages are availabe for public view
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Abstract
This thesis mainly sheds light on five different famous partial differential equations of second order, namely, the one-dimensional nonlinear Schrödinger equation, the one-dimensional nonlinear Schrödinger equation with detuning term, the complex Ginzburg-Landau equation, the two-dimensional nonlinear Schrödinger equation, and the two-dimensional linear time-dependent Schrödinger equation. At the same time, trying to find the approximate optimal solution for each of them, which results in the lowest possible error and the highest accuracy.In this regard, four effective numerical approaches are applied, and their results are compared to reach the most reliable method of solution. These methods are the split-step Fourier transform, the pseudo-spectral Fourier method, the Hopscotch method, and the Crank-Nicolson method. Specifically, the first two methods are members of the pseudo-spectral family, while the last two are finite difference methods.The thesis is organized into approximately six chapters: Introductory chapter: gives a brief introduction to five powerful equations relevant to modeling some eminent electromagnetic problems, along with their significance and potential methods of solutions. The objectives of this thesis are also exhibited.Chapter one: is dedicated to studying the analytical and approximate solutions of the one-dimensional nonlinear Schrödinger equation, which is used in many basic applications such as modeling wave propagation in lossless optical fibers. The results obtained in this chapter are published in:Farag, N. G., Eltanboly, A. H., EL-Azab, M. S., & Obayya, S. S. A. (2021). On the Analytical and Numerical Solutions of the One-Dimensional Nonlinear Schrödinger Equation. athematical Problems in Engineering, 2021. [Q3]Chapter two: deals with the search for approximate solutions to the one-dimensional nonlinear Schrödinger equation with detuning term, which is a prominent equation in optical fiber modeling with losses, and at the same time, is not supported by an accurate analytical solution. The results obtained in this chapter are published in:Farag, N. G., Eltanboly, A. H., El-Azab, M. S., & Obayya, S. S. A. (2022). Extended Split-Step Fourier Transform Approach for Accurate characterization of Soliton Propagation in a Lossy Optical Fiber. Journal of Function Spaces, 2022. [Q2]Chapter three: turns our attention to a more complex nonlinear partial differential equation called the complex Ginzburg-Landau equation, showing six different types of nonlinearities that accompany it, namely Kerr law, power law, quadratic-cubic, cubic-quintic, dual power law, and polynomial law nonlinearity, in addition to presenting and developing a pseudo-spectral technique to deal with these different models. The results obtained in this chapter are published in:Farag, N. G., Eltanboly, A. H., El-Azab, M. S., & Obayya, S. S. A. (2022). Pseudo-spectral approach for extracting optical solitons of the complex Ginzburg Landau equation with six nonlinearity forms. Optik, 254, 168662. [Q2]Chapter four: introduces the two-dimensional model. Two versions of the equation are explained, namely the two-dimensional nonlinear Schrödinger equation and the two-dimensional linear time-dependent Schrödinger equation. The two-dimensional nonlinear Schrödinger equation is briefly introduced and examined using three numerical methods. Accordingly, new modifications to these approaches have been developed to deal with the numerical solution of the two-dimensional linear time-dependent Schrödinger equation, which is an important equation in modeling electron wave propagation through a quantum wire. In addition, analytical solutions were shown for the purposes of comparison and validation of the approximate solutions presented. The results obtained in this chapter are published in:Farag, N. G., Eltanboly, A. H., El-Azab, M. S., & Obayya, S. S. (2023). Numerical Solutions of the (2+ 1)-Dimensional Nonlinear and Linear Time-Dependent Schrödinger Equations Using Three Efficient Approximate Schemes. Fractal and Fractional, 7(2), 188. [Q1]