AMApplied Mathematics2152-7385Scientific Research Publishing10.4236/am.2014.513182AM-47624ArticlesCOMPUTER SCIENCE & COMMUNICATIONSENGINEERINGPHYSICS & MATHEMATICSStudy of Stability Analysis for a Class of Fourth Order Boundary Value ProblemsC.Bala Rama Krishna1*P.S. Rama Chandra Rao2*Department of Mathematics, Kakatiya Institute of Technology & Science, Warangal, IndiaDepartment of Mathematics, Chaitanya Degree College (Autonomous), Warangal, India* E-mail:cbrk2004@gmail.com(CBRK);patibanda20@yahoo.co.in(PSRCR);0707201405131887189316 April 201428 May 2014 8 June 2014© Copyright 2014 by authors and Scientific Research Publishing Inc. 2014This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

Fourth order differential equations are considered to develop the class of methods for the numerical solution of boundary value problems. In this paper, we have discussed the regions of absolute stability of fourth order boundary value problems. Methods proposed and derived in this paper are applied to solve a fourth-order boundary value problem. Numerical results are given to illustrate the efficiency of our methods and compared with exact solution.

Numerical Differentiation Initial Value Problem Boundary Value Problem Absolute Stability Multistep Methods
1. Introduction

The determination process for the numerical solution of initial value problems in ordinary differential equations can be classified into two categories-single step methods and multistep methods. Single step methods are those in which the approximation for the point involves information from only one of the previous points. Methods using the approximation at more than one previous points to determine the approximation at the next point are called multistep methods. Thus a k-step method requires information about the solution at k points to compute the solution at the point. Finite difference methods for boundary value problems are discussed in  . Linear multi step methods of second order differential equations are discussed in  . The methods based on numerical differentiation for first-order differential equations have been shown to be stiffly stable by Gear  . A detailed study of the single step and multistep methods has been carried out by Gear  , Gragg and Statter  and Henrici  . Gear  and Peter Henrici  have derived special multistep methods based on numerical integration and numerical differentiation for solving first-order differential equations. Jain  has considered high order stiffly stable methods. Further information can be had from  and  . Special multistep methods based on numerical differentiation for solving the initial value problem have been derived in Rama Chandra Rao  . The methods now to be discussed are based on replacing the function which is unknown, by an interpolating polynomial having the values on a set of points xn where yn has already been computed. The methods discussed in this paper are essentially based on the idea that the solution is best approximated by polynomials. The motivation for the work carried out in this paper arises from the methods based on numerical differentiation for the first-order differential equations, special multistep methods based on numerical integration for the solution of the special second-order differential equations by Henrici  and Special multistep methods based on numerical differentiation for solving the initial value problem by Rama Chandra Rao  . In Henrici  methods based on Numerical Integration have been derived by integrating twice and replacing the function by an interpolating polynomial. Special multistep methods have been derived by replacing on the left hand side of by an interpolating polynomial and differentiating it four times. We have investigated a class of implicit methods. It is found that the implicit methods have order. Some local truncation errors are provided. The regions of absolute stability of the methods are derived. Numerical tests of the performance of the methods are established by solving differential equation and compared with the exact solution. The numerical results reported show the validity of our methods.

2. General Linear Multistep Methods for Special Fourth-Order Differential Equations

The special fourth order differential equation

occurs frequently in many number of problems of science and engineering.

A general linear multistep method of step number k for the numerical solution of equation (1) is given by

where aj, bj are constants and “h” is the step length.

Introducing the polynomials

Equation (2) can be written as

In Equation (4), “E” is the shift operator defined by

Applying (4) to, we get the characteristic equation

The roots of the characteristic Equation (5) and are in general, complex and the region of absolute stability is defined to be the region of the complex -plane such that the roots of the characteristic Equation (5) lie within the unit circle whenever lies in the interior of the region. Denoting the region of absolute stability of R and its boundary by, the locus of is given by

3. Derivation of the Methods

Let be the backward difference interpolating polynomial of at abscissas. Then is given by

Differentiating (7) four times with respect to x, we get

Replacing by in Equation (1) and putting i.e., we get,

where

Taking r = 0 in (8), a class of methods can be attained which are given by

The coefficients are shown in Table 1.

Differences in (10) are expressed in terms of function values.

After simplification, the Equation (10) will turn out into the form

The coefficients are shown in Table 2.

The local truncation error of the formula (11) is given by

Table 1. Coefficients of;.

Table 1

Table 1. Coefficients of ; .

M0123456789
000012

Table 2. Coefficients of;,.

Table 2

Table 2. Coefficients of ; , .

KJ
0123456789
41−46−41
53−1426−2411−2
6
7
8
9

It follows that the k-step method (14) has the order, which is absolutely stable for

For the method (13), we have

The regions of absolute stability of the method for k = 4, 5, 6, 7, 8 and 9 are shown in Figure 1 and Figure 2 (Taking real part on x-axis and imaginary part on y-axis). The region of absolute stability is the region lying outside the boundary.

The region of absolute stability of the method (13) for k = 4, 5 and 6

The region of absolute stability of the method (13) for k = 7, 8 and 9

4. Numerical Example

In this section, we have applied ND methods to solve the differential equation

in the interval with h = 0.01 and h = 0.02 and the results are shown in Table 3 and Table 4.

The fourth order numerical differentiation method derived in this paper for k = 6 is

5. Discussion and Conclusion

The methods based on numerical integration are found to be closed regions of absolute stability; the methods

Table 3

. Solution by fifth order ND with h = 0.01

XExact SolutionNumerical Solution by fifth order NDAbsolute Error
0.109.500016525794E−029.500016525794E−027.813194535800E−15
0.201.800052419090E−011.800052419090E−015.329070518201E−15
0.302.550394442682E−012.550394442681E−015.662137425588E−15
0.403.201646548346E−013.201646548346E−013.719247132494E−15
0.503.754975976986E−013.754975976986E−013.774758283726E−15
0.604.212257271813E−014.212257271813E−012.775557561563E−15
0.704.576217499812E−014.576217499812E−012.720046410332E−15
0.804.850567100313E−014.850567100313E−018.881784197001E−16
0.905.040115785530E−015.040115785530E−011.221245327088E−15
1.005.150873072263E−015.150873072263E−012.997602166488E−15
1.105.190133197868E−015.190133197868E−011.887379141863E−15
1.205.166544364783E−015.166544364783E−014.662936703426E−15
1.305.090162464214E−015.090162464214E−015.440092820663E−15
1.404.972489648573E−014.972489648573E−015.440092820663E−15
1.504.826498351587E−014.826498351587E−018.826273045770E−15
1.604.666641592316E−014.666641592316E−019.992007221626E−15
1.704.508850642096E−014.508850642096E−011.304512053935E−14
1.804.370521379547E−014.370521379547E−011.637578961322E−14
1.904.270490905788E−014.270490905789E−011.471045507628E−14
2.004.229006237963E−014.229006237963E−011.737499033538E−14
Table 4

. Solution by fifth order ND with h = 0.02

XExact SolutionNumerical Solution by fifth order NDAbsolute Error
0.109.500016525794E−029.500016525718E−027.650130529058E−13
0.201.800052419090E−011.800052419082E−017.430722703816E−13
0.302.550394442682E−012.550394442675E−016.964984144986E−13
0.403.201646548346E−013.201646548340E−016.305511668359E−13
0.503.754975976986E−013.754975976981E−015.463962615693E−13
0.604.212257271813E−014.212257271809E−014.425348976156E−13
0.704.576217499812E−014.576217499809E−013.217981436876E−13
0.804.850567100313E−014.850567100311E−011.886824030350E−13
0.905.040115785530E−015.040115785529E−014.274358644807E−14
1.005.150873072263E−015.150873072265E−011.179056852152E−13
1.105.190133197868E−015.190133197871E−012.902122986370E−13
1.205.166544364783E−015.166544364787E−014.607425552194E−13
1.305.090162464214E−015.090162464221E−016.451505996097E−13
1.404.972489648573E−014.972489648581E−018.351652702743E−13
1.504.826498351587E−014.826498351598E−011.023958695612E−12
1.604.666641592316E−014.666641592328E−011.214361944335E−12
1.704.508850642096E−014.508850642110E−011.404265592697E−12
1.804.370521379547E−014.370521379563E−011.590561016229E−12
1.904.270490905788E−014.270490905806E−011.771249813487E−12
2.004.229006237963E−014.229006237983E−011.950384298510E−12

based on numerical differentiation are found to be absolutely stable outside some closed boundaries. We have obtained the solution by numerical differentiation methods which are derived in this paper and are more accurate. The absolute errors are very small.

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