An Operator-Difference Method for Telegraph Equations Arising in Transmission Lines

A second-order linear hyperbolic equation with time-derivative term subject to appropriate initial and Dirichlet boundary conditions is considered. Second-order unconditionally absolutely stable difference scheme in Ashyralyev et al. 2011 generated by integer powers of space operator is modified for the equation. This difference scheme is unconditionally absolutely stable. Stability estimates for the solution of the difference scheme are presented. Various numerical examples are tested for showing the usefulness of the difference scheme. Numerical solutions of the examples are provided using modified unconditionally absolutely stable second-order operator-difference scheme. Finally, the obtained results are discussed by comparing with other existing numerical solutions. The modified difference scheme is applied to analyze a real engineering problem related with a lossy power transmission line.


Introduction
Second-order linear hyperbolic partial differential equations with both constant and variable coefficients arise in many branches of science and engineering, for example, electromagnetic, electrodynamics, thermodynamics, hydrodynamics, elasticity, fluid dynamics, wave propagation, and materials science, see 1-13 .For example, they are used frequently for modelling power transmission lines 7-10, 13 .In numerical methods for solving these equations, the problem of stability has received a great deal of importance and attention see 14-20 .Specially, a suitable model for analyzing the stability is provided by a proper unconditionally absolutely stable difference scheme with an unbounded operator.
In the literature, the work on new finite difference schemes has drawn important attention for numerical solutions of linear hyperbolic partial differential equations, see 21-26 and the references given therein .However, these difference schemes are conditionally stable since the stability estimates of them are based on some restrictions on choice of the grid step sizes τ and h with respect to the time and space variables, respectively.Many scientists have investigated unconditionally stable difference schemes for linear hyperbolic differential equations.Such a difference scheme for approximately solving linear hyperbolic differential equations was studied for the first time in 14 .The first-order difference scheme was constructed for approximately solving the abstract Cauchy problem where A t is an unbounded selfadjoint positive linear operator with domain D A t in an arbitrary Hilbert space H.It is known see 27, 28 that various initial-boundary value problems for hyperbolic equations can be reduced to the initial value problem 1.1 .Note that 1.1 is the well-known wave equation in the special case when A t is equal to Laplace operator Δ.The stability estimates for the solution of this difference scheme and for the firstand second-order difference derivatives were established in Hilbert space.Then, in the past decade, a huge variety of works on finite difference method for numerical solutions of linear hyperbolic partial differential equations were studied.For the problem 1.1 when A t A, Ashyralyev and Sobolevskii 15 developed the firstand two different types of second-order difference schemes.The stability estimates for the solutions of these difference schemes and for the first-and second-order difference derivatives have been established.For the same problem, they developed also the high-order two-step difference methods generated by an exact difference scheme, and by the Taylor expansion on three points in 16 ; here, the stability estimates for approximate solutions by these difference methods are also discussed.In 17, 18 , two different types of secondorder difference methods for the problem 1.1 were studied, and the stability estimates for the solutions of these difference schemes were established.But, the difference methods developed in these references are generated by square roots of A t .Further, an operator A 1/2 t of which construction is not easy is required to realize these difference methods practically.Therefore, in spite of derived theoretical results, it is not very practical to apply these methods for solving an initial-value problem numerically.Another two different types of second-order difference methods generated by integer powers of A t for the problem 1.1 were developed and the stability estimates for the solutions of these difference methods were established in 19, 20 .We should mention that all difference methods in references 15-20 are unconditionally absolutely stable and applicable for multidimensional linear hyperbolic equations with space variable coefficient.Specially, the difference methods in 17-20 are also applicable for multidimensional linear hyperbolic equations with time and space variable coefficients.
For approximately solving the problem when a 0 and b 1, a three-level implicit difference scheme of O τ 2 h 2 was discussed in 29 wheres α and β are real numbers.For α > 0, β 0, and α > β > 0, 1.The only alternative is the application of stable numerical methods for solving the initial-boundary value problem 1.2 since the analytical solution cannot be determined for arbitrary f t, x .Being easy to implement and universally applicable, the finite difference method outstands among many of the available numerical methods.For solving a linear hyperbolic partial differential equation with time or space variable coefficient analytically, there is no specific method in the literature.
In the present paper, the second order in both time and space direction unconditionally absolutely stable operator-difference scheme in 20 generated by integer powers of A t is modified for approximately solving the equation Here, A t is an unbounded selfadjoint positive linear operator with domain D A t in an arbitrary Hilbert space H.The initial conditions of the above equation consists of Numerical solutions of the equations 1.2 , 1.3 , and 1.4 are computed with the modified difference scheme.The results are provided and compared with the difference schemes in 29-34 .The differences among all these difference schemes are illustrated with numerical examples by considering various initial-boundary value problems.

Operator Difference Schemes-Stability Estimates
Using the finite difference formulas

2.3
Further, we have

2.4
Neglecting small terms O τ 2 , we obtain the following second-order difference scheme: for approximately solving 1.5 .
Theorem 2.1.For the solution of the difference scheme 2.5 , the stability estimate holds, where u 0 ∈ D A 1/2 0 and C 1 does not depend on u 0 , u 0 , f s 0 ≤ s ≤ N − 1 and τ.
Theorem 2.2.For the solution of the difference scheme 2.5 , the stability estimate holds, where u 0 Proof.Theorems of the existence and uniqueness of the classical solutions of the initial value problem for the hyperbolic equation in a Hilbert space H with the selfadjoint positive operator A t with a region of definition that does not depend on t under some additional condition of differentiability on the nonlinear maps were established in 37 .
The proofs of the above Theorems are based on the symmetry properties of the operators A x h t defined by formula 2.5 and the following theorem on the coercivity inequality for the solution of the elliptic difference problem in L 2h .

Theorem 2.3. For the solutions of the elliptic difference problem
the following coercivity inequality holds (Sobolevskii) [38]:

Numerical Examples and Discussion
We have not been able to obtain a sharp estimate for the constants figuring in the stability inequalities.So, in this section, the numerical results are presented by considering various initial-boundary value problems.
with the exact solution u t, x e −2t sinh x is considered.According to the problem 3.3 , the differential operator can be defined follows: Concerning the above differential operator, discrete operators can be defined as

3.5
For the approximate solution of the problem 3.3 and the ones appearing in the sequel, using the second order of approximation of second and fourth derivatives in the above discrete operators and applying the second-order difference scheme 2.5 , the system of linear equations is obtained.Then, writing the system in the matrix form, a second-order difference equation with respect to k with matrix coefficients is arrived.To solve this resulting difference equation, iterative method is applied see the third Chapter of 40 .
The exact and numerical solution obtained for α 40 and β 4 by using the secondorder difference scheme 2.5 with h 1/5 and τ 1/3 λ 1.6 is shown in Figures 1 and  2, respectively, as an example.The exact and numerical solutions of the next examples are shown in odd and even figure number, respectively.The difference between Figures 1 and 2 is not clearly obvious though the largest stepsize values which correspond to minimum grid numbers, N 3, M 5, are used to graph them.The errors should be computed for the accurate comparison of the numerical and exact solutions as well as for the comparison of the two different difference schemes.RMS errors at t 1, t 2 for different values of α and β α > β ≥ 0 are tabulated in Tables 1 and 2, and the results are compared with the solutions obtained by 29 .
Though both difference schemes are of O τ 2 h 2 , considering the errors in the numerical results tabulated in Tables 1 and 2 for the different values of the parameters α, β and the step sizes h, τ hλ, it can be seen easily in Table 1 that the present scheme is approximately 2 times better in the sense of accuracy than the scheme in 29 for α 10, β 0. For the values of parameters indicated in Table 2, the present scheme is a little bit better than the scheme in 29 for h 1/32, but it yields approximately 1.5 times smaller error for h 1/64.
Although both of the schemes compared in this example are of O τ 2 h 2 , the scheme in 29 requires adequate choice of its free parameters.Otherwise, the results come out to be worse than the results of this paper.
Example 2. When the following equation 31, 32, 34 : with initial conditions u 0, x sin x, u t 0, x − sin x, 0 ≤ x ≤ π 3.8  and boundary conditions u t, 0 u t, 1 0, t ≥ 0 3.9 is considered, then the analytical solution of this problem is u t, x e −t sin x.We apply the same procedure 3.3 for the approximate solution of the above problem.
The exact solution and the numerical solution obtained for α 2 and β √ 2 with the largest step sizes h π/5 and τ hλ 1/3 are shown in Figures 3 and 4.
The difference between the exact solution and numerical solutions for the given example is not obvious though the largest step-size values are used to graph them.For smaller values of h and τ, there is apparently no difference between the graphs of the exact and numerical solutions without depending on the values of the parameters α and β.So again, the errors should be computed for the accurate comparison.The absolute errors involved by the scheme 2.5 and by the scheme 16 in 31 at t 1, t 2 are listed in Tables 3, 4 and 5

for different values of h and τ.
All the results obtained by the scheme 2.5 are almost the same with those obtained by using the scheme in 31 except the results listed in the first rows of Tables 4 and 5.Although the order of accuracy O τ 5 h 2 of the scheme 16 in 31 is much better than O τ 2 h 2 of the present paper, this is not apparent in the results of Tables 3, 4 and 5.In fact, its effect shows up when larger enough τ values e.g., τ 0.1 are used 31, 32 .
The absolute errors given by the scheme 2.5 and by the scheme in 32 in numerical solutions of the same example are listed in Tables 6 and 7 for h π/30 and h π/300, respectively, with the same step size τ 0.1 for time variable.For all values of h, the errors at t 1.0 of the scheme in 32 are much smaller than those of the present scheme 2.5 .This is naturally expected since the order of accuracy of the scheme in 32 is O τ 2 h 4 which is better than that of 2.5 , namely, O τ 2 h 2 .This difference in the errors disappears at t 2.0 on behalf of the scheme 2.5 .In other words, as t increases to 2.0, the errors of the scheme 2.5 are decreasing quietly for all values of h, but the errors of the scheme in 32 remain almost at the same level when compared with those at t 1.0.Now, consider the main equation in 1.2 in two cases α 4, β 2 and α 6, β 2 with the same exact solution and initial-boundary conditions of the problem 3.7 .The errors are tabulated in Table 8 for h 1/50 and τ 1/10000 and compared with those found by using the radial basic function method in 34 .
It is obvious that the scheme 2.5 is almost 5 and 10 times better in the sense of error size than the scheme in 34 for α 4, β 2 and α 6, β 2 respectively.However, the computation time of the solution in 34 is quite smaller than that of the present method.

3.11
The analytical solution given in 32, 33 is u t, x sinh x cos t.The exact and numerical solutions obtained for α 10 and β 2 with h 1/5 and τ 1/3 are shown in Figures 5 and  6.The difference between the exact and numerical solutions for the given example is not obviously apparent though the largest step-size values are used again to graph them.Numerical solutions for different values of α and β when h 1/8.2 k , for k 0, 1, 2, 3, and τ 3.2h, obtained by using the scheme 2.5 are compared with those obtained by using the schemes in 32, 33 .The RMS errors in solutions are tabulated in Tables 9, 10 and 11 at t 2.0.
The errors in Tables 9, 10 and 11 involved in the results obtained by using the schemes of 32, 33 are naturally much smaller than those in Table 11.The reason of this huge difference is based on that the order of accuracy of the schemes in 32, 33 is O τ 2 h 4 , whilst it is O τ 2 h 2 in the present paper.
Example 4. Finally, the following equation 30 : x t e −2t sinh x 3.12 with time and space variable coefficients that are defined in the region Ω 0 < x < 1 × t > 0 is solved numerically.The initial and boundary conditions of this problem are the same as the initial and boundary conditions of the first example because they have the same exact solution.RMS errors are tabulated at the points t 1.0 and t 2.0 for a fixed values λ 1.6, and the results are listed in Table 12.
Apparently, for the given equation with both time and space variable coefficients, the results of the scheme 2.5 are much more satisfactory than the results of the scheme in 30 for all values of h shown in Table 12 although the order of accuracy for both schemes is O τ 2 h 2 .On the other hand, when applied to solve an equation with constant coefficients, both schemes result with the same order of errors.

Application
As an example, consider the power transmission line having a length of d 150 km with the resistance, conductance, and capacitance parameters R 0.03 Ω/km,L 1 mH/km, G 0.001 μS/km, and C 0.01 μF/km in 8 .
The line is represented by the following telegraphic equation: When the line parameters are normalized with respect to the impedance L/C, frequency w 1/ d √ LC , and distance d, then the above second-order partial differential equation becomes

4.2
Due to the slight losses of the line the coefficients of V t and V in the above equation are quite small.When the relaxed line is excited at one end sending end by a unit pulse of duration 0.75/ √ 10 ms which corresponds to a normalized time of 0.5 s, the voltage at the receiving end x 1 is computed by solving 4.2 with the boundary and initial conditions The problem is solved by the difference scheme 2.5 for the intervals t ∈ 0, 4 for time and x ∈ 0, 1 for space with number of grids 1000 and 100, respectively.The plot of the computed receiving-end voltage is shown in Figure 7 a including the exciting voltage as well.It is obvious that the input pulse is reached at the end of the line at t 1, and it is doubled in magnitude by the reflection at the open end.The wave travels back to the sending end, it gets inverted due to the short-circuit property of the sending end, and reflects back to the receiving end after inversion.Since the travelling time along the line is 1, the wave reaches the receiving end at t 3 as the second time.Due to the very slight loss in the line parameters, the attenuation is not obvious in Figure 7 a .The same problem is solved by increasing the line losses R and G parameters by 10 7 times, though this is not a realistic case.The attenuation in the receiving-end voltage is now obvious as plotted in Figure 7 b .

Concluding Remarks
In this work, the second-order unconditionally absolutely stable difference scheme in 20 generated by integer powers of A t is modified for solving the abstract Cauchy problem for the second-order linear hyperbolic differential equations containing the unbounded selfadjoint positive linear operators.The modified difference scheme is applied for solving various initial-boundary value problems and compared with other published papers in the literature.It is observed that this scheme is more accurate as compared to the other difference schemes whose order of accuracy is the same.Moreover, the present difference scheme 2.5 is compared with difference schemes whose order of accuracy is higher, and it is observed that the good results can be obtained using the difference scheme in the paper depending on the type of the problem and solution parameters of the problem.The difference scheme is applied to a linear hyperbolic equation with variable coefficients, and it is observed that it is much more satisfactory than the scheme obtained by radial basic function, having the same order of accuracy.The modified difference scheme 2.5 is shown to be well applied to analyze a real engineering problem involving a lossy power transmission line with constant coefficients.Being defined by an operator, it can be applied to multidimensional linear hyperbolic differential equations with time and space variables coefficients.

Figure 7 :
Figure 7: a The receiving-and sending-end voltages of the line in the case of slight loss.b The receivingand sending-end voltages of the line in the case of strong loss.
2 is called as a damped wave equation and as a telegraph equation, respectively, see 22 .In 30 , also a three-level implicit difference scheme, whose order is the same as the previous literature for approximately solving the equation The executions in all examples are carried out by MATLAB 7.01 and obtained by a PC Pentium R 2CPV, 2.00 6 Hz, 2.87 GB of RAM.The errors in the numerical solutions are computed by the root mean square RMS error 39 n and u T, x n represent the exact and numerical solutions at final time, respectively.Example 1.First, the following initial-boundary value problem 29 :

Table 8 :
RMS errors of scheme 2.5 and the scheme in 34 .

Table 9 :
RMS errors of the scheme in 32 .

Table 10 :
RMS errors of the scheme in 33 .

Table 12 :
RMS errors of scheme 2.5 and the scheme in 30 .