480 AMERICAN Journal of Public Diplomacy and International Studies www. grnjournal.us AMERICAN Journal of Public Diplomacy and International Studies Volume 01, Issue 10, 2023 ISSN (E): 2993-2157 MAXIMUM PRINCIPLE AND ERROR ANALYSIS Behruz Ulug‘bek o‘g‘li Quvvatov teacher of the "General Technical Sciences" department of the International University of Asia ulughbekovich.bekhruz@mail.ru Abstract: The maximum principle is a fundamental concept in mathematics and engineering, providing crucial insights into the behavior of various mathematical and physical systems. This principle has wide-ranging applications in fields such as differential equations, optimal control, and numerical analysis. Error analysis, on the other hand, plays a critical role in assessing the accuracy and reliability of numerical methods and computational algorithms. This article discusses the maximum principle and its significance in error analysis, highlighting its applications and implications in various disciplines. Keywords: maximum principle, error analysis, differential equations, numerical methods, optimal control Introduction. The purpose of this article is to introduce finite difference and finite element methods for solving ordinary and partial differential equations of boundary value problems. Emphasis is placed on understanding the important details of finite difference and finite element methods and their implementation with sound mathematical theory . We begin with a comprehensive discussion of one-dimensional problems before considering two or higher dimensions. We also list some useful references for those who want to learn more about related fields. First, let's consider the concept of the maximum principle.   ,yx,for ,0b ,2 2 2 22 2 2           ac y c yx h x aL we consider the elliptic differential operator and 0 ,0  ca assume without loss of generality that The maximum principle is given in the following theorem. Theorem 1. If in a     3, Cyxu bounded domain   0, yxLu satisfies , then .  yxu , will have a maximum value at the boundary of the field. Proof: If the theorem is not true, then   00 , yx has a local point,    yxuyxu ,, 00  all   yx, for, a local extremum  00 , yx A prerequisite for this is: 481 AMERICAN Journal of Public Diplomacy and International Studies www. grnjournal.us     0, ,0, 0000       yx x u yx x u Now  00 , yx is not on the boundary of the domain and since u(x, y) is continuous, in the domain  00 , yx Q There exists a neighborhood of , where we have a Taylor expansion () with superscript of              ,, 2 2 1 ,, 33 02002 0000 yxO uyuyxuxyxuyyxxu yyxxxx   means that functions are evaluated.  00 , yx at , that is  002 2 0 , yx x u uxx    in  00 , yx ifo at evaluated and so on.    0000 ,, yxuyyxxu  for all sufficiently small x and , y      02 2 1 02002  yyxxxx uyuyxux (1) On the other hand, from the given condition ,02 0000000  yyxxxx ucubuaLu (2) Here  00 0 , yxaa  and others, we rewrite the above inequality to obtain the opposition to the Taylor expansion as follows.   ,0 2 0 20 0 0 0 2 0 0 0 0 00 0 0                              a b c M u u Ma b u Ma b M a u M a yy yyxyxx (3) Here M > 0 is a constant. M 's position was small enough x and y is to choose . Now if we look at the following expression, 482 AMERICAN Journal of Public Diplomacy and International Studies www. grnjournal.us Ma b y M a 0 00 ,x  (4) From expression (1), we know the following. (5) The result is the following. (6) and again from (1), (7) Thus, the left-hand side of ( 4 ) and ( 5 ) of ( 3 ) must not be positive, that is (8) contradicts the condition and the proof is completed with this. On the other hand, if Lu < 0, the minimum value of u is m will be within the limit. The maximum principle for general elliptic equations is as follows. (9) be here is a limited area. Then from theorem 1 u(x, y) is an expression  of the local part cannot have a positive local maximum or a negative m local minimum. The maximum principle and error analysis form an inseparable pair in the study of numerical methods and computational algorithms for solving differential equations and control problems. 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