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Circuit analysis as a discrete boundaryvalue problemRaghib Abu-Saris a & Wajdi Ahmad ba Department of Basic Sciences , University of Sharjah , P. O. Box27272, Sharjah, U.A.Eb Department of Electrical and Electronics Engineering ,University of Sharjah , P. O. Box 27272, Sharjah, U.A.EPublished online: 02 Nov 2009.

To cite this article: Raghib Abu-Saris & Wajdi Ahmad (2007) Circuit analysis as a discrete boundaryvalue problem, International Journal of Mathematical Education in Science and Technology, 38:6,832-839, DOI: 10.1080/00207390701240877

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Circuit analysis as a discrete boundary value problem

RAGHIB ABU-SARIS*y and WAJDI AHMADz

Department of Basic Sciences, University of Sharjah,P. O. Box 27272, Sharjah, U.A.E.

Department of Electrical and Electronics Engineering,University of Sharjah, P. O. Box 27272, Sharjah, U.A.E.

(Received 30 March 2006; accepted 19 October 2006)

In this note, it is shown that mesh and nodal techniques for electric circuitanalysis can be cast as non-autonomous discrete boundary value problems.An elegant necessary and sufficient condition for the existence of a uniquesolution for such problems is described. The developed condition iscomputationally tractable, as it only involves checking a determinant that isspecially constructed from the coefficients of the underlying difference equation.Illustrative examples are presented to demonstrate the validity of the result.

1. Motivation and problem formulation

The discrete boundary value problem (DBVP) addresses difference equations subjectto boundary conditions. Such a problem may result from discretizing a continuousboundary value problem (CBVP) with an underlying ordinary differential equation.Alternatively, it can be a result of modelling a discrete (dynamical) process that hasto satisfy a set of prescribed conditions or targets. Typically, part of these conditionsis known at the initial or present time, while the rest are described at future orlater time(s). Such problems appear in a wide variety of science and engineeringapplications. The following example, in particular, is drawn from the field ofelectrical circuit analysis, where it is shown that a circuit analysis problem can becast as a non-autonomous DBVP.

Nodal and Mesh equations are popular techniques for analysis of electricalcircuits. Nodal analysis begins by assigning different voltages to the different nodesin the circuit, as shown in figure 1, and writing the Kirchoff Current Law (KCL)equation for each one of the nodes, which essentially states that the sum of allcurrents entering the node must equal the sum of all currents leaving the node. Onthe other hand, mesh analysis begins by assigning currents to the different loops inthe circuit, as shown in figure 2, and writing the Kirchoff Voltage Law (KVL) foreach closed loop or mesh. The KVL dictates that the sum of all voltage drops in aclosed loop be zero. In either technique, a second-order discrete difference equationis obtained, where the index j of the equation represents, respectively, the node andloop numbers as shown below. The classification ‘‘boundary value’’ becomesapparent when we impose conditions on the voltages (currents) at two ends

DOI: 10.1080/00207390701240877

*Corresponding author. Email: rabusaris@sharjah.ac.ae

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(loops) of the circuit, e.g. Vð0Þ ¼ V0ðIð0Þ ¼ I0Þ and VðN Þ ¼ VNðI ðN Þ ¼ INÞ where Nis one of the circuit nodes (loops). The voltages V(0) and V(N ) (currents I(0) andIðN ÞÞ are, respectively, called initial and final (or boundary) conditions. A commonscenario might be VðN Þ ¼ 0ðIðNÞ ¼ 0Þ which signifies a short circuit (open switch, S)condition imposed at node (loop) N, thus bringing the voltage (current) down tozero. This could represent a short-circuit fault in the first case, or a load beingdisconnected in the second case. Hence, the response of the difference equation isrestricted at the two ends, and there is a ‘‘gap’’ between the initial and final indices.Within the gap, the response is not restricted.

The equations representing the two circuits above can be easily derived asfollows. Applying KCL to the circuit in figure 1 yields:

Vðj Þ � Vð jþ 1Þ

R2j¼

Vð jþ 1Þ

R2jþ1þVð jþ 1Þ � Vð jþ 2Þ

R2jþ2, j ¼ 0, 1, . . . ,N� 2

Hence, the following DBVP is obtained:

Vð jþ 2Þ þ a1ð j ÞVð jþ 1Þ þ a0ð j ÞVð j Þ ¼ 0 ð1Þ

Vð0Þ ¼ V0 and VðN Þ ¼ 0 ð2Þ

where

a1ðj Þ ¼ � 1þR2jþ2

R2jþR2jþ2

R2jþ1

� �and a0ð j Þ ¼

R2jþ2

R2j: ð3Þ

Similarly, applying KVL to the circuit in figure 2, we get:

Rjþ1Ið jþ 2Þ � Rj þ Rjþ1

� Ið jþ 1Þ þ RjIð j Þ ¼ 0, j ¼ 0, 1, . . . ,N� 2 ð4Þ

Figure 1. Nodal setup.

Figure 2. Mesh setup.

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and so we have the following DBVP:

Ið jþ 2Þ þ a1ð j ÞIð jþ 1Þ þ a0ð j ÞIð j Þ ¼ 0, ð5Þ

Ið0Þ ¼ I0 and IðN Þ ¼ 0 ð6Þ

where

a1ðjÞ ¼ � 1þRj

Rjþ1

� �and a0ðj Þ ¼

Rj

Rjþ1ð7Þ

Notice that the linear difference equations (1) and (5) stated above arenon-autonomous owing to the fact that their coefficients depend on the index j.

In this paper, we are concerned with the fundamental question of existence anduniqueness which is a corner stone for well-posedness of a problem. For definiteness,we shall consider a non-autonomous linear difference equation of order k52 givenin the recursive form:

Xki¼0

aiðtÞyðtþ iÞ ¼ gðtÞ, t ¼ 0, 1, 2, . . . ð8Þ

and subject to separable boundary conditions:

yðni þ jÞ ¼ �ij, i ¼ 1, . . . , r and j ¼ 0, . . . , ki � 1 ð9Þ

or to boundary conditions of the type

�jyðniÞ ¼ �ij, 14i4r, 04j4ki � 1 ð10Þ

where

0 ¼ n14n1 þ k1 � 1 < n24n2 þ k2 � 1 < � � � < nr�14nr�1 þ kr�1 þ 1 < nr

and ki are positive integers such that 14r4k and k1 þ � � � þ kr ¼ k. This type ofboundary conditions is commonly known in the literature as Hermite type. Our goalis to develop a necessary and sufficient condition for the existence of a uniquesolution of DBVP (8), (9) and DBVP (8)–(10).

DBVPs received a great deal of attention recently (see [1, pp. 629–744] and[2, pp. 229–278, 327–341], and the references cited therein). In general, the resultsobtained thus far provide sufficient conditions for the existence and/or uniquenessof solution(s) of a DBVP. Recently, the authors in [3, 4] have developed acomputationally tractable and easy-to-apply necessary and sufficient condition forthe existence of a unique solution of DBVP (8), (9) in which the governing differenceequation is autonomous. However, the proofs presented therein are computationallyinvolved and employ a generalization of what is called the exponential Vandermondedeterminant in which the entries are powers of the roots of the associated character-istic equation. This approach, unfortunately, is not valid for the non-autonomouscase, as it is meaningless to speak of characteristic equations for non-autonomousdifference equations.

This paper is organized as follows: in section 2 we establish the main result.In section 3, we give illustrative numerical examples. Finally, in section 4, we drawconclusions.

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2. Existence and uniqueness results

First, it is known (see [2, p. 14]) that

yðtþ jÞ ¼Xj‘¼0

j

‘

� ��‘yðtÞ

where �‘ is the ‘th-order forward difference operator. Therefore, for each 14i4r,equation (10) can be rewritten as

yðni þ jÞ ¼Xj‘¼0

j

‘

� ��‘yðniÞ ¼

Xj‘¼0

j

‘

� ��i‘ ¼ �ij, j ¼ 0, . . . , ki � 1

This means that the difference between Hermite boundary conditions (10) andseparable boundary conditions (9) is rather symbolic, and the two types are,essentially, equivalent.

Now, we are ready to establish our existence and uniqueness theorem.Theorem 2.1: Let s‘ ¼ n‘þ1 � n‘ � k‘, ‘ ¼ 1, . . . , r� 1, and let s ¼

Pr�1‘¼1 s‘ ¼ nrþ

kr � k. Then the existence of a unique solution of DBVP (8), (9) is equivalent to thenonsingularity of the block matrix

A ¼ A1 � � � Ar�1

where Aj is the s sj matrix given by

Aj ¼

anjþkj ð0Þ anjþkjþ1ð0Þ � � � anjþ1�1ð0Þ

anjþkj�1ð1Þ anjþkj ð1Þ � � � anjþ1�2ð1Þ

..

. ... ..

.

anjþkj�sþ1ðs� 1Þ anjþkj�sþ2ðs� 1Þ � � � anjþ1�sðs� 1Þ

0BBBBB@

1CCCCCA

j ¼ 1, . . . , r� 1.

Proof: First, to simplify notation we define ajðtÞ ¼ 0 if j<0 or j> k. Using thisindex extension and equation (8), we have

gðtÞ ¼Xkþtj¼t

aj�tðtÞyð jÞ, t ¼ 0, 1, 2, . . .

¼Xk1�1j¼t

aj�tðtÞyðj Þ

þXr�1‘¼1

Xn‘þ1�1j¼n‘þk‘

aj�tðtÞyðj Þ þXn‘þ1þk‘þ1�1

j¼n‘þ1

aj�tðtÞyðj Þ

" #

þXkþt

j¼nrþkr

aj�tðtÞyðj Þ

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It is worth mentioning that some of these sums are zeros due to either beingempty sums (upper index is less than lower index) or due to the index extensionintroduced above. Furthermore, the purpose of the sum-splitting presented above isto facilitate the use of boundary conditions given in equation (9) as will beseen below.

Now, a unique solution of DBVP (8), (9) exists if and only if the missing y-valuesat intermediate time instances, i.e. the ‘gaps’, are uniquely determined from the givenboundary conditions. To this end, rearranging the above equations fort ¼ 0, 1, . . . , nr þ kr � k� 1, we obtain the following system of linear equations:

Xr�1‘¼1

Xn‘þ1�1j¼n‘þk‘

aj�tðtÞyðj Þ ¼ gðtÞ �Xk1�1j¼t

aj�tðtÞ�1j �Xr�1‘¼1

Xn‘þ1þk‘þ1�1

j¼n‘þ1

aj�tðtÞ�ð‘þ1Þj

Observe that the coefficient matrix of this system of linear equations is given by

ak1 ð0Þ � � � an2�1ð0Þ j � � � j anr�1þkr�1 ð0Þ � � � anr�1ð0Þak1�1ð1Þ � � � an2�2ð1Þ j � � � j anr�2þkr�1�1ð1Þ � � � anr�2ð1Þ

..

. ...

j j ... ..

.

ak1�sþ1ðs� 1Þ � � � an2�sðs� 1Þ j � � � j anr�2þkr�1�sþ1ðs� 1Þ � � � anr�sðs� 1Þ

0BBB@

1CCCA

Hence, the result follows. œ

3. Applications

To illustrate the applicability of Theorem 2.1, we present the following examples.

Example 3.1: Consider the DBVP

yðtþ 5Þ þ a4ðtÞyðtþ 4Þ þ a3ðtÞyðtþ 3Þ þ a2ðtÞyðtþ 2Þ

þa1ðtÞyðtþ 1Þ þ a0ðtÞyðtÞ ¼ gðtÞ, t ¼ 0, 1, 2, . . .

yð0Þ ¼ �10, yð2Þ ¼ �20, yð4Þ ¼ �30, yð6Þ ¼ �40, yð8Þ ¼ �50:

8><>:

In this case, we have k¼ 5, n1¼ 0, n2¼ 2, n3¼ 4, n4¼ 6, n5¼ 8, andk1 ¼ � � � ¼ k5 ¼ 1. Thus, s1 ¼ s2 ¼ s3 ¼ s4 ¼ 1, and there are four blocks each ofsize 4 1. Therefore,

AT1 ¼ a1ð0Þ a0ð1Þ a�1ð2Þ a�2ð3Þ

¼ a1ð0Þ a0ð1Þ 0 0

AT2 ¼ a3ð0Þ a2ð1Þ a1ð2Þ a0ð3Þ

AT3 ¼ a5ð0Þ a4ð1Þ a3ð2Þ a2ð3Þ

¼ 1 a4ð1Þ a3ð2Þ a2ð3Þ

AT4 ¼ a7ð0Þ a6ð1Þ a5ð2Þ a4ð3Þ

¼ 0 0 1 a4ð3Þ

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and

A ¼ A1 A2 A3 A4

¼

a1ð0Þ j a3ð0Þ j 1 j 0

a0ð1Þ j a2ð1Þ j a4ð1Þ j 0

0 j a1ð2Þ j a3ð2Þ j 1

0 j a0ð3Þ j a2ð3Þ j a4ð3Þ

0BBB@

1CCCA

Hence, the non-autonomous DBVP above has a unique solution if and only if theabove-constructed A is nonsingular. In particular, if a4ðtÞ ¼ a3ðtÞ ¼ a2ðtÞ ¼ a1ðtÞ ¼ 0and a0ðtÞ ¼ cosð�tÞ, then

jAj ¼

0 0 1 0

�1 0 0 0

0 0 0 1

0 �1 0 0

���������

���������¼ �1 6¼ 0

and so a unique solution is guaranteed to exist.

Example 3.2: We turn back to our motivating example in Section 1. In view ofTheorem 2.1, DBVPs (1), (2) and (5), (6) have unique solutions if and only if thefollowing tridiagonal matrix

A ¼

a1ð0Þ 1

a0ð1Þ a1ð1Þ. .

.

. .. . .

.1

a0ðN� 2Þ a1ðN� 2Þ

0BBBBB@

1CCCCCA

is nonsingular.Equation (3) implies that the matrix A is strictly diagonally dominant and so it is

nonsingular [5, Theorem 6.20, pp. 404, 405]. Therefore, DBVP (1), (2) always has aunique solution. On the other hand, as can be seen from equation (7), thecorresponding matrix A is diagonally dominant but not a strict one. However,nonsingularity is still valid. The justification is as follows.

Suppose to the contrary that ¼ 0 is an eigenvalue of A and ~x ¼ ðx1, . . . , xN�1Þ isthe corresponding eigenvector. Then A~x ¼ ~0, or more explicitly

� 1þR0

R1

� �x1 þ x2 ¼ 0

Rj

Rjþ1xj � 1þ

Rj

Rjþ1

� �xjþ1 þ xjþ2 ¼ 0, j ¼ 1, . . . ,N� 3

RN�2

RN�1xN�2 � 1þ

RN�2

RN�1

� �xN�1 ¼ 0

ðÞ

8>>>>>>>>>>>>><>>>>>>>>>>>>>:

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Since the resistances R0js are positive, the first equation of () implies thatjx2j > jx1jy. Combining this fact with Triangle Inequality (TI), we conclude:

jx3j5 1þR1

R2

� �jx2j �

R1

R2jx1j > jx1j

This, in turn, leads to the conclusion that jx3j > jx2j. Because otherwise we have,again, by TI

R1

R2jx1j5 1þ

R1

R2

� �jx2j � jx3j5

R1

R2jx3j

which is a contradiction to what we found so far. Following this line of reasoning forj ¼ 1, . . . ,N� 3, we get jx1j < jx2j < � � � < jxN�1j. But, from the last equation of (),we have

1þRN�2

RN�1

� �xN�1 ¼

RN�2

RN�1xN�2

which implies that jxN�1j < jxN�2j which is a contradiction!From a circuit theory standpoint this makes perfect sense, since a singular

matrix A would imply multiple solutions to the circuit, which implies that the meshcurrents are not unique! This, of course, cannot happen in a real circuit wherecurrents are determined uniquely by the values of the resistors and the voltagesimposed on them.

4. Conclusions

We have established a necessary and sufficient condition for the existence of a uniquesolution for linear non-autonomous DBVPs. The boundary conditions examined areof Hermite and separable types. The condition thus established is easy-to-apply andlends itself nicely to computations. It is directly linked to the very structure ofthe difference equations. Calculating a determinant obtained from the equationcoefficients is all that is required. We have demonstrated the validity of the resultvia an example from circuit analysis where the results obtained here supportwell-established electrical circuit theory.

The results obtained in this research degenerate to the authors previouslyestablished results for autonomous DBVPs in [4], with simplified proofs.

As for future research, the authors are currently studying other structures ofboundary conditions.

yExamining the circuit in figure 2 and its corresponding equation (4), we can see from theoutset that this scenario leads to violation of circuit theory. It is straightforward to showthat the ‘mesh’ currents become bigger as we get closer to the current source of the circuit,i.e. as we move to the left. Hence, a lower-index ‘‘mesh’’ current is bigger than a higher-index one, which is contrary to the above implication.

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References

[1] Agarwal, R., 2000, Difference Equations and Inequalities: Theory, Methods, andApplications, 2nd edn (New York: Marcel Dekker).

[2] Kelley, W. and Peterson, A., 2001, Difference Equations: An Introduction withApplications (New York: Harcourt Academic Press).

[3] Abu-Saris, R. and Ahmad, W., 2003, A necessary and sufficient condition for existence ofunique solution of discrete boundary value problem, International Journal of Mathematicsand Mathematical Sciences, 39, 2455–2463.

[4] Abu-Saris, R. and Ahmad, W., 2004, Generalized exponential Vandermonde determinantand Hermite multipoint discrete boundary value problem, SIAM Journal of MatrixAnalysis and Applications, 25, 921–929.

[5] Burden, R and Faires, J., 1997, Numerical Analysis, 6th edn (New York: Brooks/Cole).[6] Abu-Saris, R., 2006, On solving non-autonomous linear difference equations with

applications, International Journal of Mathematical Education in Science and Technology,37, 618–623.

[7] Tay, E., Toh, T., Dong, F. and Lee, T., 2004, Convergence of a linear recursive sequence,International Journal of Mathematical Education in Science and Technology, 35, 51–63.

A refined Cauchy–Schwarz inequality

PETER R. MERCER*

Department of Mathematics,Buffalo State College, Buffalo, NY 14222, USA

(Received 5 July 2006)

Much of what follows is inspired by [1], in which refinements of the triangleinequality and its reverse inequality are obtained for nonzero x and y in a normedlinear space. The refinements are given in terms of the angular distance

x

xk k�

y

y�� ��

����������

and they have some interesting consequences.Now upper and lower bounds involving kx==kxk � y==kykk can be just as well

regarded as upper and lower bounds for kx==kxk � y=kykk by way of algebraicmanipulations. And if we are not just in a normed linear space, but a real innerproduct space then a computation reveals that

1�1

2

x

xk k�

y

y�� ��

����������2

¼x, y� �xk k y�� ��

*Email: mercerpr@math.buffalostate.edu

DOI: 10.1080/00207390701359370

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