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Kyungpook Mathematical Journal 2019; 59(1): 13-22

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
e-mail: srinithota@ymail.com or srinivasarao.thota@astu.edu.et

Received: November 4, 2017; Revised: October 24, 2018; Accepted: October 25, 2018

This paper presents a symbolic method for solving a boundary value problem with inhomogeneous Stieltjes boundary conditions over integro-differential 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.

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 semi-inhomogeneous 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 semi-inhomogeneous boundary value problems (inhomogeneous differential equation with homogeneous boundary conditions) to fully inhomogeneous boundary value problems (inhomogeneous differential equation with inhomogeneous boundary conditions) over integro-differential algebras.

The paper is layed out as follows. In Section 1.1 we recall the algebra of integro-differential operators, and in Section 1.2 we present the outline of the symbolic method for semi-inhomogeneous 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 Integro-differential Operators

To express the boundary value problem, Green’s operator and Green’s function in operator based notations, the basic concepts of integro-differential algebras and the algebra of integro-differential 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 = C[a, b] for simplicity.

Definition 1.1.([1, 7])

An algebraic structure (ℱ, D, A) is called an integro-differential algebra over if ℱ is a commutative -algebra with -linear operators D and A such that the following conditions are satisfied

  • D(Af) = f,

  • D(fg) = (Df)g + f(Dg),

  • (ADf)(ADg) + AD(fg) = (ADf)g + f(ADg).

Here D: ℱ → ℱ and A: ℱ → ℱ are two maps defined by D=ddx, a derivation, and A=axdx, a -linear right inverse of D, i.e. D∘A = 1 (the identity map). The map A is called an integral for D and A∘D = 1–E, where E is called the evaluation operator of ℱ defined as E: ff(a), evaluates at initial point a. An integro-differential algebra over is called ordinary if .

For an ordinary integro-differential algebra, the evaluation can be treated as a multiplicative linear functional E: , i.e., E(fg) = (Ef)(Eg), for all f, g ∈ ℱ. Let Φ ⊆ ℱ* be a set of all multiplicative linear functionals including E. To specify a BVP, we also need a collection of “point evaluations” as new generators. For example, the boundary conditions u(2) = 1, u′(1) = 5, 02u   dx=0 on a function u ∈ ℱ = C[a, b] gives rise to the functional E2u = 1, E1Du = 5, E2Au = 0 ∈ ℱ*.

Definition 1.2.([1, 7])

Let (ℱ, D, A) be an ordinary integro-differential algebra over and Φ ⊆ ℱ*. The integro-differential operators ℱ[D, A] are defined as the -algebra generated by the symbols D and A, the functions f ∈ ℱ and the characters (functionals) Ec, φ, χ ∈ Φ, modulo the Noetherian and confluent rewrite system given in Table 1.

For an integro-differential algebra ℱ, a fully inhomogeneous BVP is given by a monic differential operator L = Dn + an−1Dn−1 + · · · + a1D + a0 and the boundary conditions b1, …, bn ∈ ℱ[D, A] with boundary data α1, …, αn ∈ ℝ. Given a forcing function f ∈ ℱ and a set of boundary data α1, …, αn ∈ ℝ, we want to find u ∈ ℱ such that

Lu=f,b1u=α1,,bnu=αn

The quantities {f; α1, …, αn} are known collectively as the data for the BVP. We are not only interested to solve the BVP (1.1) for a specific data but also finding a suitable form of the solution that will exhibit its dependence on the data. The feature of (1.1) that enables us to achieve this goal is its linearity. If u1 is the solution of the data {f1; α11, …, αn1} and u2 is the solution of the data {f2; α12, …, αn2}, then λ1u1+λ2u2 is the solution of the data {λ1f1+λ2f2; λ1α11+λ2α12, …, λ1αn1+ λ2αn2}. Hence one can decompose the data as

{f;α1,,αn}={f;0,,0}+{0;α1,,αn}.

The BVP with data {f; 0, …, 0} is an inhomogeneous differential equation with homogeneous boundary conditions; the BVP with data {0; α1, …, αn} is a homogeneous differential equation with inhomogeneous boundary conditions. Symbolically, we can write the solution u of (1.1) as

u=F(f,α1,,αn),

where F is a linear operator that transforms the data into the solution. Hence we regard F as the inverse operator of L.

In [1], Rosenkranz et al. presented a symbolic solution F(f, 0, …, 0) of a BVP with data {f; 0, …, 0}. In this paper we find the solution of fully inhomogeneous BVP with data {f; α1, …, αn}. To motivate the solution, we briefly recall the symbolic solution F(f, 0, …, 0) in Section 1.2. In Section 2, we give the solution F(f,α1, …, αn) of fully inhomogeneous BVP.

1.2. Solution of Semi-inhomogeneous Boundary Value Problems

The algorithm for computing the solution, F(f, 0, …, 0), of a semi-inhomogeneous BVP is described in [1] with details, and also [6]. Consider a semi-inhomogeneous BVP

Lu=f,b1u=0,,bnu=0,

where L is a surjective linear map and B = {b1, …, bn} ⊆ ℱ* is a closed subspace of the dual space. We call F(f, 0, …, 0) ∈ ℱ a solution of (1.3) for a given forcing function f ∈ ℱ, if LF(f, 0, …, 0) = f and F(f, 0, …, 0) ∈ B. In operator notations LF = 1 and BF = 0, and the operator F is called Green’s operator. The Green’s operator maps each f to its unique solution F(f, 0, …, 0). The BVP (1.3) is called regular if and only if B is complement of Ker(L) so that ℱ = Ker(L)⊕B as a direct sum. The regularity of a BVP can be tested algorithmically [1, p. 30] as follows: If u1, …, un is a basis for Ker(L) and {b1, …, bn} is a basis for B, then the BVP is regular if and only if the evaluation matrix

b(u)=(b1(u1)b1(un)bn(u1)bn(un))

is regular.

The differential operator L is always surjective and the dim Ker(L) = n < ∞. Moreover, we can find the right inverse of L using variation of parameters as follows: Let (ℱ, D, A) be an ordinary integro-differential algebra and let L ∈ ℱ[D] be monic with regular fundamental system u1, …, un. Then the fundamental right inverse of L is [1, Corollary 29] given by

M=i=1nuiAd-1di[D,A],

where d is the determinant of Wronskian matrix W for u1, …, un and di the determinant of Wi obtained from W by replacing the i-th column by the n-th unit vector. If {b1, …, bn} and {u1, …, un} are bases for B and Ker(L) respectively with bi biorthogonal to ui, then the projector operator P is [1, p. 26] determined by

P=i=1nuibi.

The main steps [1] to determine the solution F(f, 0, …, 0) of a semi-inhomogeneous BVP and the corresponding Green’s operator F are:

  • Compute the fundamental right inverse M ∈ ℱ[D, A] from a given fundamental system as in (1.5).

  • Compute the projector P ∈ ℱ[D, A] onto Ker(L) along B as in (1.6).

  • Now the Green’s operator F is computed as F = MPM, and the solution is u = F(f, 0, …, 0) = (MPM)f.

  • In this section, we present a method/algorithm to compute the solution u = F(f,α1, …, αn) and the corresponding Green’s operator F for a BVP with the data {f; α1, …, αn}. From equation (1.2), one can decompose the solution as

    F(f,α1,,αn)=F(f,0,,0)+F(0,α1,,αn).

    In Section 1.2, we computed the solution F(f, 0, …, 0). In this section, we present a method for computing the solution F(0, α1, …, αn) and then F(f,α1, …, αn) as a composition of two solutions F(f, 0, …, 0) and F(0, α1, …, αn).

    Consider a semi-homogeneous boundary value problem

    Lu=0,b1u=α1,,bnu=αn.

    Let H be any function (not necessarily satisfying the differential operator L) such that biH = αi, for i = 1, …, n. Set

    u=H+v,

    then one can observe that v satisfies the semi-inhomogeneous BVP

    Lv=-LHb1v=0,,bnv=0.

    and the solution of semi-inhomogeneous BVP is computed as

    v=F(-LH,0,,0)=(M-PM)(-LH).

    Since u = H + v, we have

    u=F(0,α1,,αn)=(M-PM)(-LH)+H,

    On simplification of (2.1), we have

    u=F(0,α1,,αn)=PH,

    where P ∈ ℱ[D, A] is the projector onto Ker(L) along B as given in equation (1.6) and H ∈ ℱ is an irrespective operator of L such that biH = αi, for i = 1, …, n. We call H as a right inverse of B such that biH = αi and it is computed as in the following lemma.

    Since H is depending only on the boundary data, this amounts to an interpolation problem with Stieltjes boundary conditions, presented in the next lemma. The rest of paper, right inverse of B means the right inverse of each element of B such that biH = αi.

    Lemma 2.1

    Let {u1, …, un} ⊂ ℱ and {b1, …, bn} ⊂ ℱ*be bases for Ker(L) and B respectively, and {α1, …, αn} ⊂ ℝ be boundary data. Then there exists a unique right inverse H of B such that biH = αi is given by

    H=uvTb(u)-1αv,

    where uv = (u1, …, un) and αv = (α1, …, αn) are column vectors and b(u) is evaluation matrix as in equation (1.4).

    Proof

    From given data, we have conditions b1u = α1, …, bnu = αn. Suppose

    H=c1u1++cnun

    is an interpolating function satisfying the given conditions, i.e., biH = αi, where ci are the unknown coefficients to be determined, for i = 1, …, n. From the given boundary conditions {b1, …, bn} with {α1, …, αn}, one can write equation (2.3) as

    b(u)(c1,,cn)T=(α1,,αn)T,

    where b(u) is the evaluation matrix. Since the evaluation matrix is regular, there exists inverse of b(u), and we have

    (c1cn)=b(u)-1(α1αn).

    Since H is the linear combination of u1, …, un with coefficients c1, …, cn, we have

    H=uvTb(u)-1αv.

    It completes the proof.

    Now the generalization of the above results is presented in the following theorem.

    Theorem 2.2

    Let (ℱ, D, A) be an ordinary integro-differential algebra and B = {b1, …, bn} ⊂ ℱ*. Given a boundary data {α1, …, αn} ⊂ ℝ, the BVP

    Lu=0,b1u=α1,,bnu=αn.

    has the unique solution

    u=F(0,α1,,αn)=PH,

    where P ∈ ℱ[D, A] is a projector onto Ker(L) along Bas in equation (1.6) and H ∈ ℱ such that biH = αi as in equation (2.2).

    Finally, we compute the solution of fully inhomogeneous BVP with data {f; α1, …, αn} as composition of two solutions

    F(f,α1,,αn)=(M-PM)(f)+PH

    and the corresponding Green’s operator is

    F=(M-PM)+PH.

    One can easily check that the Green’s operator F satisfies LF = 1 and BF = α, and the solution F(f,α1, …, αn) satisfies LF(f,α1, …, αn) = f and BF(f,α1, …, αn) = α.

    Example 3.1

    Consider the one-dimensional problem of a thin rod occupying the interval (0, a) on the x-axis. This is one of the classical examples of the ordinary linear BVPs [2]. We solve

    d2udx2=f,         0<x<1;         u(0)=α,         u(1)=β,

    for the temperature uC[0, 1], where fC[0, 1] is the prescribed source density (per unit length of the rod) of heat and α, β are the prescribed end temperatures.

    The operator representation of the given BVP (3.1) is

    Lu=fE0u=α,E1u=β,

    where the differential operator L = D2 with Ker(L) = {1, x}, and the set of boundary operators is B = {E0, E1} with boundary data {α, β}. The null space projector P is computed as in equation (1.6), and it is given by

    P=(1-x)E0+xE1

    The fundamental right inverse of L, computed as described in equation (1.5), and it is given by

    M=xA-Ax.

    The right inverse H of B is computed as follows: For a given fundamental system {1, x}, boundary operators {E0, E1} with boundary data {α, β}, the operator H calculated as

    H=(1,x)   (10-11)   (αβ)=α(1-x)+βx.

    The Green’s operator of BVP (3.1) can be computed using the proposed algorithm as

    F=(1-P)M+PH=xA-Ax-xE1A+xE1Ax+α(1-x)+βx.

    By translating the symbols, the Green’s operator is

    F=x0x-0xξ-x01+x01ξ+α(1-x)+βx,

    and integration can be performed in closed term using elementary integration techniques as

    F=(1-x)0xξ+xx1(1-ξ)+α(1-x)+βx.

    Now the solution of the given BVP (3.1) is

    u=F(f,α,β)=01g(x,ξ)f(ξ)dξ+α(1-x)+βx,

    where the Green’s function g(x, ξ) is

    g(x,ξ)={(x-1)ξif0ξx1,x(ξ-1)if0xξ1.

    For specific f(x), the integration in (3.2) can be expressed as

    u=(1-x)0xξf(ξ)   dξ+xx1(1-ξ)f(ξ)   dξ+α(1-x)+βx.

    Example 3.2

    Consider the following BVP

    d3udx3-3d2udx2+3dudx-u=x2ex,u(0)=0,u(0)=0,u(1)=0.

    Operator notation of the given BVP is

    Lu=fE0u=α1,E0Du=α2,E1u=α3,

    where L = D3 − 3D2 + 3D − 1, E0u = u(0), E0Du = u′(0), E1u = u(1) and α1 = α2 = α3 = 0. Fundamental right inverse of L is

    M=12exAe-xx2-exxAe-xx+12exx2Ae-x,

    and H is computed as

    H=(exxexx2ex)(100110eee)         (000)=0.

    The Green’s operator of BVP (3.3) is computed as follows

    F=(1-P)M+PH=12exAe-xx2-xexAxe-x+12x2exAe-x-12x2exE1Ae-xx2+x2exE1Axe-x-12x2exE1Ae-x,

    and the complete solution is

    u=F(f,α,β)=01g(x,ξ)f(ξ)   dξ+0

    where the Green’s function g(x, ξ) is

    g(x,ξ)={-12ex-ξξ(-ξ+2x+x2ξ-2x2)if 0ξand ξxand x1,-12ex-ξx2(ξ2-2ξ+1)                        if 0xand xξand ξ1.

    If f(x) = x2ex, then the exact solution of the given BVP (3.3) is

    u=160x5ex-160x2ex.

    Example 3.3

    A particle of mass m moves along the u axis under the influence of a force f(t) directed along the axis. The motion of the particle is determined by Newton’s law with initial conditions

    md2udt2=f(t),t>0;         u(0)=α,dudt(0)=β.

    Operator notation of (3.4) is

    mD2u=f(t),E0u=α,E0Du=β.

    The solution of (3.4) computed similar to Example 3.1, and it is given by

    u=F(f,α,β)=0g(t,ξ)f(ξ)   dξ+α+βt

    where the Green’s function g(t, ξ) is

    g(t,ξ)={0if0tξ<,t-ξmif0ξt<.

    Equation (3.5) can be written now as

    u=1m0t(t-ξ)f(ξ)   dξ+α+βt.

    Example 3.4

    Consider a BVP of damped oscillator. Given a forcing function fC[0, π], we find uC[0, π] such that

    d2udx2+2dudx+u=f,         u(0)=α,         u(π)=β.

    Following the algorithm in Section 2 similar to Example 3.1, the solution is

    u=F(f,α,β)=0πg(x,ξ)f(ξ)   dξ+(βeπ-απ)xe-x+αe-x,

    where the Green’s function g(x, ξ) is given by

    g(x,ξ)={1π(π-x)ξeξ-xif0ξxπ,1π(π-ξ)xeξ-xif0xξπ.

    In this paper, we discussed a symbolic method to solve BVPs with inhomogeneous Stieltjes boundary conditions over integro-differential algebras. We extend the idea of the symbolic method for semi-inhomogeneous 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.

    The author is thankful to the reviewer and editor for providing valuable inputs to improve the quality of manuscript.

    Rewrite rules for integro-differential operators

    fgf · gDff D + f′AfA → (Af)A − A(Af)
    χϕϕDϕ → 0AfD → f − Af′ − (Ef)E
    ϕf → (ϕf)ϕDA → 1Afϕ → (Af)ϕ
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    5. S. Thota, and S. D. Kumar. On a mixed interpolation with integral conditions at arbitrary nodes. Cogent Mathematics., 3(2016), 115-161.
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