On Solving Singular Multi Point Boundary Value Problems with Nonlocal Condition حول حل مسائل القيم الحدودية متعددة النقاط الشاذة مع شروط

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1 ISSN: GIF: On Solving Singular Multi Point Boundary Value Problems with Nonlocal Condition Heba. A. Abd Al-Razak* Department of Mathematics, College of Science for Women, Baghdad University, Baghdad, Iraq Abstract In this paper Hermite interpolation method is used for solving linear and nonlinear second order singular multi point boundary value problems with nonlocal condition. The approximate solution is found in the form of a rapidly convergent polynomial. We discuss behavior of the solution in the neighborhood of the singularity point which appears to perform satisfactorily for singular problems. The examples to demonstrate the applicability and efficiency of the method have been given. Keywords: Singular differential equation, Nonlocal condition, Hermite interpolation حول حل مسائل القيم الحدودية متعددة النقاط الشاذة مع شروط غير محلية هبة عواد عبد الر ازق* قسم الرياضيات, كلية العلوم للبنات, جامعة بغداد, بغداد, الع ارق الخالصة في هذا البحث استخدمت طريقة اند ارج هير مت لحل الرتبة الثانية الخطية وغير الخطية لمسائل القيم الحدودية الشاذة مع شروط غير محلية. ثم ايجاد حل تقريبي في شكل متعددة حدود متقاربة بسرعة.وناقشنا سلوك الحل في جوار نقطة منفردة حيث يظهر مرضيا للمسائل الشاذة.ثم اعطاء امثلة الظهار تطبيق وكفائة الطريقة. 1. Introduction In the study of non-linear phenomena in physics, engineering and other sciences, many mathematical models lead to singular multi point boundary value problems (SMPBVP) associated with non-linear ordinary differential equations (ODE) [1]. We will deal with no classical boundary value problems, the solution of singular multi point boundary value problems with nonlocal conditions specifications. The meaning of the problems is that the definite condition not only depends on the value of solution in the end of interval, but also depends on the value of solution in some points of the interior of interval.started to fairly late study this kind of problem, initialed by II'in and Moiseev [2] studied the existence of solutions for a linear multipoint BVP.Gupta studied certain three point boundary value problems for non-linear ordinary differential equation [3]. Since then, more general multi point boundary value problem have been studied [4-7]. Within the following ten years,the study on nonlocal boundary value problems for ordinary differential equations has been made great progress. However, it is not good enough. In this paper we introduce Hermite interpolation for construction polynomial solutions of singular boundary value problems with nonlocal condition. * heba.math@yahoo.com 2959

2 2. Singular Differential Equation with BC [8] The general form of the 2 nd order singular ordinary differential equations A(x) +B(x) +C(x)y=0 (1) How to solve a linear ODE. The first thing we do is, rewrite the ODE as: +P(x) +Q(x)y=0 (2) where, of course, P(x) = B(x) / A(x), and Q(x) = C(x) / A(x). There are two types of a point [0,1] : Ordinary point and singular point.also, there are two types of singular point : Regular and Irregular Points, A function y(x) is analytic at if it has a power series expansion at x 0 that converges to y(x) on an open interval containing. A point is an ordinary point of the ODE (2), if the functions P(x) and Q(x) are analytic at. Otherwise is a singular point of the ODE, if the functions P(x) and Q(x) are analytic at. i.e. P(x) = + (x- ) + (x- +.. = (3) Q(x) = + (x- ) + (x- +.. = (4) If A, B and C are polynomials then a point x0 such that A( ) 0 is an ordinary point. On the other hand if P(x) or Q(x) are not analytic at then is said to be a singular.a singular point of the ODE (3) is a regular singular point of the ODE if the functions xp(x) and x 2 Q(x) are analytic at. Otherwise is an irregular singular point of the ODE. Now, we state the following theorem without proof which gives us a useful way of testing if a singular point is regular[9]. Theorem 1: If the P(x) and Q(x) exist, are finite, and are not 0 then x = 0 is a regular singular point. If both limits are 0, then x = 0 may be a regular singular point or an ordinary point. If either limit fails to exists or is ± then x = 0 is an irregular singular point. There are four kinds of singularities: The first kind is the singularity at one of the ends of the interval [0, 1]. The second kind is the singularity at both ends of the interval [0, 1]. The third kind is the case of a singularity in the interior of the interval. The forth and final kind is simply treating the case of a regular differential equation on an infinite interval. In this paper, we focus of the first three kinds. 3. Hermite interpolation Hermite interpolation method used for construction polynomial solutions of singular multi point boundary value problems with nonlocal condition. Where the interpolation polynomial also matches the first derivatives,k=0,1,2, n [10].This interpolation technique is important since it has the property that gives high order of accuracy. Theorem 2 [10]: Suppose that f(x) [a,b], and that are distinct, then the unique polynomial of degree (at most) 2n + 1 denoted by Hermite interpolation, and such that ( ) ( ) ( ) is given by : [ ] (5) An error bound for Hermite interpolation is provided by the expression: E=,where f(x). 4. Solution for non-linear SMPBVP with Nonlocal Condition In this section, we apply the Hermite interpolation method and Taylor series to solve regular singular point. A general form order (SMPBVP) is: ( ) (6a) A associated with nonlocal condition of the form:, (6b) Where, εr and, ε j -2 Now, to solve the problem by suggested method doing the following steps: Step one: Evaluate Taylor series of y(x) about x = 0: 0692

3 (7) Where y(0)=,y'(0)=, =,,,i=3,4, Evaluate Taylor series of (x) about x =, : (8) Where ( )=, ( )=, =,,,i=3,4, and j=1,2,,m-2 Evaluate Taylor series of (x) about x =, : (9) Where ( )=, ( )=, =,,,i=3,4, and j=1,2,,m-2 And evaluate Taylor series of y(x) about x = 1: (10) Where y(1)=,y'(1)=, =,,,i=3,4, Step two: Insert the series form (7) into equation (6) and put x = 0, then equate the coefficients of powers of x to obtain.then derive equation(6) with respect to x,then insert the series form (7)in to equation derived and put x=0 equate the coefficients of power of x to obtain.iterate the process many times to obtain then and so on. Step three: Make up x= in the case j=1 of the series form (7) to obtain ( )=,to find ( )= derive the series (7) and requite x=,then insert the series (8) into equation (6) and put x=,then equate the coefficients of power of (x- ) to obtain.to find derive equation(6) with respect to x, Then insert the series form (8)in to equation derived and put x= and equate the coefficient of power of (x- ). Iterate the process many times to obtain then and so on. In the same way, we get, and,i=2,3,. And j=2,3,,m-2. Step four: Make up x= in the case j=1 of the series form (8) to obtain ( )=,to find ( )= derive the series (8) and requite x=,then insert the series (9) into equation (6) and put x=,then equate the coefficients of power of (x- ) to obtain.to find derive equation(6) with respect to x, Then insert the series form (9)in to equation derived and put x= and equate the coefficient of power of (x- ). Iterate the process many times to obtain then and so on. In the same way, we get, and, i=2,3,. And j=2,3,,m-2. Step five: Insert the series form (10) into equation (6) and put x = 1, then equate the coefficients of powers of( x-1) to obtain,to find derive equation(6) with respect to x, Then insert the series form (10)in to equation derived and put x=1 and equate the coefficient of power of (x-1).. Iterate the process many times to obtain then and so on. Step six: The notation implies that the coefficients depend only on the indicated unknowns ( ) ( ), then substitute these date ( ) in Hermite interpolation polynomial equation(5). To find the unknowns coefficients by reduction order equation and use as a replacement of y(x) and substitute the nonlocal conditions, we have (m- 2) unknown coefficients, we can find this for any (m-2) by solving this system of algebraic equations using MATLAB,so insert the value of the unknown coefficients into equation(5),thus equation(5) represent the solution of the equation(6). To illustrate the effectiveness of the presented method,we will consider the following example of nonlinear(smpbvp) with nonlocal conditions. Example1: Consider the second-order non-linear (SMPBVP)[11]:, 0 x 1 With nonlocal conditions ( ) ( ), ( ) ( ) The exact solution y ( ) =sinh( ). Now, It is clear that x = 0 and 1, are regular singular point and it is singularity of second kind. We solve this example using Taylor polynomials, with order 12. We get the values nonlocal conditions y(0)= ,y(2/3)= ,y(4/5)= , y(1)= , y'(0)= ,y'(2/3)= , y'(4/5)= , y'(1)=

4 Then using these data to find and when k=0,1,2,3. Every time we get polynomial of degree seven then compensate in equation (5), we get: Table-1, gives the result y( ) exact solution and results also the error between them. Table- 2 gives the maximum absolute error for Hermite interpolation method with third-kind Chebyshev wavelets Collocation method (3CWCM) and fourth-kind Chebyshev wavelets collocation method (4CWCM)[11]. Figure-1 we give a comparison between the exact solutions with suggested solution. Table 1- The result of the method for of example 1 x i Exact Solution y( ) Hermite interpolation Error= y( ) e e e e e e e e e e e-11 Table 2- The result maximum absolute error of example 1 Method Hermite interpolation M.E CWCM CWCM Figure 1- Comparison between the exact and suggested solution 5. Solution for linear SMPBVP In this section, we apply the Hermite interpolation method and Taylor series to solve regular singular point. A general form order (SMPBVP) is: (11a) A associated with nonlocal condition of the form:, (11b) Where, εr and, ε j

5 {, b(x)= { (11c) And a(x) have singularity α(0,1) because. Now, to solve the problem by suggested method defines on 1 st sub domain then define on 2 nd sub domain. Therefore, the construct,to do this we repeat the same steps the previous in solution second order non-linear (SMPBVP) on 1 st sub domain [0, and 2 nd sub domain (.To obtain two polynomials represent the solution of equation (11)on sub domain [0,,and ( Finally, we sum the result of the two solutions, we get a solution to the equation (11) on domain [0,1]. To illustrate the effectiveness of the presented method, we will consider the following example of linear (SMPBVP). Example2: Consider the second-order singular linear MBVP[11]:, With nonlocal conditions ( ), ( ) Where {, { The exact solution is: y(x)=x(1-x) It is clear that x = 1/2 is regular singular point and it is singularity of third kind. Method used when the 1 st sub domain [0, 1/2] and got the following results using Taylor polynomials, with order 12 y(0)= , ( )= , y( )= ,y'(0)= , ( )= ,y'( )= Then using these data to find and when k=0,1,2 Every time we get polynomial of degree five then compensate in equation (5), we get:: = Method used again when the 2 nd sub domain (1/2,1] and got the following results usingtaylor polynomials, with order 12 ( )= , y(1)= ( )= , y'(1)= Then using these data to find and when k=0,1,2. Every time we get polynomial of degree five then compensate in equation (5), we get: = x Details about the solution to the equation on the domain [0,1] as following, Table-3, gives the result y( ) exact solution and results also the error between them. Table-4, gives the maximum absolute error for Hermite interpolation with third-kind Chebyshev wavelets Collocation method (3CWCM) and fourth-kind Chebyshev wavelets collocation method (4CWCM) [11]. Figure 2 we give a comparison between the exact solutions with suggested solution. Table 3- The result of the Hermitei interpolation and exact solution for example 2 x i Exact Solution y( ) Hermite interpolation e e-06 Error= y( ) e e e e e e e e e e e

6 Table 4- The result maximum absolute error of example 2 Method Hermite interpolation M.E CWCM 7.1 4CWCM 2.8 Figure 2- Comparison between the exact and suggested solution 6. Behavior of the solution in the neighborhood of the singularity x= 0 [12] Our main concern in this section will be the study of the behavior of the solution in the neighborhood of singular point. Consider the following singular initial value problem: y (x) + ((N 1) / x ) y (x) = f(y), N 1, 0 < x < 1, (12) y(0) =, = 0,, (13) Where f(y) is continuous function.as the same manner in [12], let us look for a solution of this problem in the form: y(x)= y'(x)=,y''(x)= ( ) (14) where C is a positive constant and k > 1. If we substitute (14) in (12) we obtain : C= (1/ k) (f( ) /N, (15) In order to improve representation (14) we perform the variable substitution: y(x) = C (1 + g(x)), (16) we easily obtain the following result which is similar to the results in [12]. Theorem 3[12]: For each > 0, problem (12), (13) has, in the neighborhood of x = 0, a unique solution that can be represented by: y(x, ) = C (1 + g + o( ) ), where k, C and g are given by (15) and (16), respectively.we see that these results are in good agreement with the ones obtained by the method in [12], they are also consistent with the results presented in [13]. In order to estimate the convergence order of the suggested method at x = 0, we have carried out several experiments with different values of n and used the formula: = ( / ), (17) Where is the approximate value of obtained with ni, ni= 1,2, 3, 4, 7. Conclusion In this paper, we presented Hermite interpolation method to solve the singular multi point boundary value problem with nonlocal condition. It is clearly seen that present method is a powerful and accurate seen method for finding singular differential equation in the form of polynomial and presents a rapid convergence for the solutions. The numerical results showed that the Hermite interpolation method can solve the problem effectively and the comparison shows that the present method is in good agreement with the existing results in the literature. References 1. Robert, L.B., and Courtney, S. C Differential Equations A Modeling perspective, Preliminary Edition, United States of America. 069.

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