Mathematics and Statistics 10(4): 884-893, 2022 DOI: 10.13189/ms.2022.100420 http://www.hrpub.org Evolution Equations of Pseudo Spherical Images for Timelike Curves in Minkowski 3-Space H. S. Abdel-Aziz 1 , H. Serry 2 , M. Khalifa Saad 1,3,* 1 Department of Mathematics, Faculty of Science, Sohag University, 82524 Sohag, EGYPT 2 Department of Mathematics, Faculty of Science, Suez Canal University, Ismailia, EGYPT 3 Department of Mathematics, Faculty of Science, Islamic University of Madinah, 170 Al-Madinah, KSA Received April 6, 2022; Revised June 21, 2022; Accepted July 11, 2022 Cite This Paper in the following Citation Styles (a): [1] H. S. Abdel-Aziz, H. Serry, M. Khalifa Saad, ”Evolution Equations of Pseudo Spherical Images for Timelike Curves in Minkowski 3-Space,” Mathematics and Statistics, Vol.10, No.4, pp. 884-893, 2022. DOI: 10.13189/ms.2022.100420 (b): H. S. Abdel-Aziz, H. Serry, M. Khalifa Saad, (2022). Evolution Equations of Pseudo Spherical Images for Timelike Curves in Minkowski 3-Space. Mathematics and Statistics, 10(4), 884-893. DOI: 10.13189/ms.2022.100420 Copyright ©2022 by authors, all rights reserved. Authors agree that this article remains permanently open access under the terms of the Creative Commons Attribution License 4.0 International License Abstract The pseudo spherical images of non-lightlike curves in Minkowski geometry are curves on the unit pseudo sphere, which are intimately related to the curvatures of the original ones. These images are obtained by means of Frenet-Serret frame vector fields associated with the curves. This classical topic is a well-known concept in Lorentzian geometry of curves. In this paper, we introduce the pseudo spherical images for a timelike curve in Minkowski 3-space. Our main purpose of the work is to obtain the time evolution equations of the orthonormal frame and curvatures of these images. The compatibility conditions for the evolutions are used. Finally, the theoretical results obtained through this study are given by some important theorems and explained in two computational examples with the corresponding graphs. Keywords Evolution Equations, Timelike Curves, Pseudo Spherical Images, Serret-Frenet Frame 1 Introduction The curves and their geometric properties play an important role in the field of differential geometry and in many branches of science such as mechanics and physics. They have some ap- plications such as computer aided geometric design (CAGD) and mathematical modeling [1]. Further, curves are usually studied as subsets of an ambient space with a notion of equiv- alence. For example, one may study curves in the plane, the usual three dimensional space, the Minkowski space, curves on a sphere, etc.[2, 3] Among these properties are the evolution equations for curves which is our concern, in which time is a fundamental element of the study. These equations have a clear effect in many physical and dynamic applications and also have a no- table affect in several fields as in image processing and how it is useful in images and shapes recognizing [4, 5]. Besides, how the evolution of a curve related to the level sets as well as the massive change of studying the generalization of the lo- cal maximum and minimum which are known as ridges and ravines [6, 7, 8]. In addition, one can see the effect of the evolution equations to the Fluid study such as studying the sur- faces evolution in Turbulence [9]. For studying these equations with respect to curves or surfaces in any space , there are some enormous ways such as equations of motion which describe the dynamical system of the curves via their frame field, via their velocity vector or even via their accelerations etc. For more details see [4, 7, 10]. In the Minkowski geometry, the pseudo spherical images of non-lightlike curves are curves on the unit pseudo sphere intimately related to the curvatures of the original curves. These images are obtained by means of Frenet-Serret frame vector fields associated to the meant curves, so this classical topic is a well-known concept in Lorentzian geometry of the curves, see [11, 12]. There are many studies of evolution equations that have been done in different spaces (see for instance, [7, 9, 10, 13]). Also, evolution equations for the elliptic partial differential equations and the magnetic geodesics equations have been obtained in [8]. As an extension of such studies, we interested here with the study of pseudo spherical images of a non-lightlike curve, es- pecially timelike curve and derive their evolution of time equa- tions attributed to the curvature and torsion of the considered