ON THE FORMULATION OF BOUNDARY VALUE PROBLEMS FOR ONE SECOND-ORDER EQUATION

Аннотация

This paper addresses the formulation of boundary value problems (BVPs) for second-order differential equations. Boundary value problems are essential in various scientific and engineering applications where solutions must satisfy specific conditions at the boundaries of the domain. The study outlines a systematic approach to defining the differential equation, determining the domain, and specifying the appropriate boundary conditions. The discussion includes different types of boundary conditions such as Dirichlet, Neumann, and mixed conditions. An example is provided to illustrate the formulation process, demonstrating how to combine the differential equation with boundary conditions to define a complete BVP. Methods for solving these problems, including analytical and numerical techniques, are also reviewed, highlighting their importance in obtaining accurate solutions for complex systems. KEYWORDS Boundary Value Problems (BVPs), Second-order Differential Equations, Dirichlet Boundary Condition, Neumann Boundary Condition, Mixed Boundary Condition, Analytical Methods. INTRODUCTIO Consider the equation: Volume 04 Issue 06-2024 58 American Journal Of Applied Science And Technology (ISSN – 2771-2745) VOLUME 04 ISSUE 06 Pages: 58-63 OCLC – 1121105677 Publisher: Oscar Publishing Services Servi LU =K(x;y)U +U +(x;y)U +b(x;y)U +mu U = f(x;y) (1) yy xx y where К(х,у)-continuously differentiable function, and The region D-which consists at у>0 of a К(х,у) 0 at у0,К(х,у)0 at rectangle with vertices at points А(0;0), В(1;0), А 1 (0;1), В(1;1), and at 𝑦 0 Let W1(D)denote the space of functions Definition 1. The functions (x;y)W1(D)) are 2 2 from W1(D)that satisfy the boundary conditions (2)- called the generalized solution of the problem (1)-(3), if 2 the integral identity holds. (3) −U (KV) −U V +a(x;y)U V +bUY +mU  UVdD =  fVdD  y Y x x y  d D 0 1 of functions from the space W1(D) possessing the for any function V from W 2 (D). 2 property that all elements of (x,y) are linearly n The existence of a generalized solution to the independent, and their linear combinations are dense boundary value problem (1)-(3) will be established in this space. Such a set, as known from [1],[6] exists. using the Galerkin method. Let {(x,y)}– be the set n Volume 04 Issue 06-2024 59 American Journal Of Applied Science And Technology (ISSN – 2771-2745) VOLUME 04 ISSUE 06 Pages: 58-63 OCLC – 1121105677 Publisher: Oscar Publishing Services Servi Let's consider an auxiliary problem l W = e −  y W ( x ; y ) =  ( x (;5)y ) n ny n W (x;1)=(x)W (x;y) (6) n n S The solution to the problem (5)-(6) y 1 W (x, y) = e(x;)dt + et(x;t)dt n n −1 n s S It is clear that W (х,у) is linearly independent. n N Indeed, if C W =0 is for any set of W, W , …, W , n n 1 2 n n=1 then by applying the operator L to this sum, we have N C(x;y)=0C =0,n n n n n=1 It is clear that W (x;y)W1(D) is easily obtainable an estimation n 2 p p W n L  m  n L (D) p(D) p Moreover, W (x,y) satisfies the conditions (6) for any n 𝑛. We will seek an approximate solution to the problem (1)-(3) in the form of N U N (x;y) = C n W n (x; y) n=1 where С are constants determined from a system of n nonlinear algebraic equations in the form (LUNU ) =(fU ) ,n=1,N (7) , n 0 , n 0 The solvability of this system of algebraic equations approximate solutions and Lemma « acute angle» from follows from the a priori estimates obtained for the [8] Volume 04 Issue 06-2024 60 American Journal Of Applied Science And Technology (ISSN – 2771-2745) VOLUME 04 ISSUE 06 Pages: 58-63 OCLC – 1121105677 Publisher: Oscar Publishing Services Servi Lemma 1. Suppose the conditions К(х;1)0 are satisfied and the inequalities 2а(х;у)−К (х;у)−К(х;у)0 y Then the estimate holds true 2 p UN + UN  K (8) W1(D) L (D) 2 2 p 𝐾 does not depend on n. 2 Proof. Multiplying (7) by C and summing over n from 1 to N, we obtain the identity n eyU y NLUNdD= eyU y N fdD (9) D D Integrating the left-hand side of equation (9) by parts, we obtain (U y N)2+(2a−K−Ky)(U x N)2++(UN)2+ 2 p m UN dD− − e  1 (UN)2dx+ e  1 K(x;1)(UN)2dx− e  1 (UN)2dx+ 1 ey ((UN)2−K(UN)2+mUN p +(UN)2)n −2(UN)(UN)n  ds 2 x 2 y 2 2  y y 1 x y 2 0 0 0 S where n=(n;n )– is the unit vector of the inward normal to 𝜕D. Using conditions (3) and the conditions of the lemma, 1 2 we obtain inequality (8). Let's return to the question of the solvability of the system of equations (7). If we write it in F (C) = 0 (F (C),C) the form , where C =(C .....C ) then as we have just seen multiplying we get the m 1 m