Articles
Kyungpook Mathematical Journal 2019; 59(1): 1322
Published online March 23, 2019 https://doi.org/10.5666/KMJ.2019.59.1.13
Copyright © Kyungpook Mathematical Journal.
On a Symbolic Method for Fully Inhomogeneous Boundary Value Problems
Srinivasarao Thota
Department of Applied Mathematics, School of Applied Natural Sciences, Adama Science and Technology University, Post Box No. 1888, Adama, Ethiopia
email: srinithota@ymail.com or srinivasarao.thota@astu.edu.et
Received: November 4, 2017; Revised: October 24, 2018; Accepted: October 25, 2018
Abstract
This paper presents a symbolic method for solving a boundary value problem with inhomogeneous Stieltjes boundary conditions over integrodifferential algebras. The proposed symbolic method includes computing the Green’s operator as well as the Green’s function of the given problem. Examples are presented to illustrate the proposed symbolic method.
Keywords: boundary value problems, Stieltjes conditions, Green's function, Green's operator, symbolic method.
1. Introduction
For the last five decades, many researchers and engineers have been actively developing applications of general boundary value problems (BVPs) of higher order ordinary differential equations with general boundary conditions. Symbolic analysis of boundary value problems, and the formulation of Green’s operator and Green’s function of semiinhomogeneous boundary value problems, were first attempted by Markus Rosenkranz et al. in 2004 [1], also see [3, 4, 5, 6, 7]. In this paper, we extend the idea of the symbolic method for semiinhomogeneous boundary value problems (inhomogeneous differential equation with homogeneous boundary conditions) to fully inhomogeneous boundary value problems (inhomogeneous differential equation with inhomogeneous boundary conditions) over integrodifferential algebras.
The paper is layed out as follows. In Section 1.1 we recall the algebra of integrodifferential operators, and in Section 1.2 we present the outline of the symbolic method for semiinhomogeneous boundary value problems. Section 2, describes the proposed symbolic method for fully inhomogeneous boundary value problems. Sample computations are provided in Section 3, using the proposed algorithm to illustrate the symbolic method.
1.1. Algebra of Integrodifferential Operators
To express the boundary value problem, Green’s operator and Green’s function in operator based notations, the basic concepts of integrodifferential algebras and the algebra of integrodifferential operators are recalled, see [1] or [3, 4, 5, 6, 7] for further details. Throughout this section denotes the field of characteristic zero and ℱ =
An algebraic structure (ℱ, D, A) is called an

D(A
f ) =f , 
D(
fg ) = (Df )g +f (Dg ), 
(AD
f )(ADg ) + AD(fg ) = (ADf )g +f (ADg ).
Here D: ℱ → ℱ and A: ℱ → ℱ are two maps defined by
For an ordinary integrodifferential algebra, the evaluation can be treated as a multiplicative linear functional E: , i.e., E(
Let (ℱ, D, A) be an ordinary integrodifferential algebra over and Φ ⊆ ℱ^{*}. The
For an integrodifferential algebra ℱ, a fully inhomogeneous BVP is given by a monic
The quantities {
The BVP with data {
where
In [1], Rosenkranz et al. presented a symbolic solution
1.2. Solution of Semiinhomogeneous Boundary Value Problems
The algorithm for computing the solution,
where
is regular.
The differential operator
where
The main steps [1] to determine the solution
Compute the fundamental right inverse
Compute the projector
Now the Green’s operator
2. Solution of Fullyinhomogeneous Boundary Value Problems
In this section, we present a method/algorithm to compute the solution
In Section 1.2, we computed the solution
Consider a semihomogeneous boundary value problem
Let
then one can observe that
and the solution of semiinhomogeneous BVP is computed as
Since
On simplification of (
where
Since
Lemma 2.1
From given data, we have conditions
is an interpolating function satisfying the given conditions, i.e.,
where
Since
It completes the proof.
Now the generalization of the above results is presented in the following theorem.
Theorem 2.2
Finally, we compute the solution of fully inhomogeneous BVP with data {
and the corresponding Green’s operator is
One can easily check that the Green’s operator
3. Sample Computations
Example 3.1
Consider the onedimensional problem of a thin rod occupying the interval (0,
for the temperature
The operator representation of the given BVP (
where the differential operator
The fundamental right inverse of
The right inverse
The Green’s operator of BVP (
By translating the symbols, the Green’s operator is
and integration can be performed in closed term using elementary integration techniques as
Now the solution of the given BVP (
where the Green’s function
For specific
Example 3.2
Consider the following BVP
Operator notation of the given BVP is
where
and
The Green’s operator of BVP (
and the complete solution is
where the Green’s function
If
Example 3.3
A particle of mass
Operator notation of (
The solution of (
where the Green’s function
Example 3.4
Consider a BVP of damped oscillator. Given a forcing function
Following the algorithm in Section 2 similar to Example 3.1, the solution is
where the Green’s function
Conclusion
In this paper, we discussed a symbolic method to solve BVPs with inhomogeneous Stieltjes boundary conditions over integrodifferential algebras. We extend the idea of the symbolic method for semiinhomogeneous BVPs to the fully inhomogeneous BVPs over algebras. Using the proposed method one can compute the Green’s operator and the Green’s function of the given problem. Sample computations are presented using the proposed method.
Acknowledgements
The author is thankful to the reviewer and editor for providing valuable inputs to improve the quality of manuscript.
Tables
Rewrite rules for integrodifferential operators
D  A  
D  A  
DA → 1  A 
References
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Green’s functions and boundary value problems ,, John Wiley & Sons, a WileyInterscience Publication, Pure and Applied Mathematics, 1979.  S. Thota, and S. D. Kumar.
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Symbolic method for polynomial interpolation with Stieltjes conditions , International Conference on Frontiers in Mathematics,(2015), 225228.  S. Thota, and S. D. Kumar.
On a mixed interpolation with integral conditions at arbitrary nodes . Cogent Mathematics.,3 (2016), 115161.  S. Thota, and S. D. Kumar.
Solving system of higherorder linear differential equations on the level of operators . Int. J. Pure Appl. Math.,106 (1)(2016), 1121.  S. Thota, and S. D. Kumar.
Symbolic algorithm for a system of differentialalgebraic equations . Kyungpook Math. J.,56 (4)(2016), 11411160.