NUMERICAL SOLUTION OF A TRANSMISSION LINE PROBLEM OF ELECTRODYNAMICS IN A CLASS OF DISCONTINUOUS FUNCTIONS

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1 NUMERICAL SOLUTION OF A TRANSMISSION LINE PROBLEM OF ELECTRODYNAMICS IN A CLASS OF DISCONTINUOUS FUNCTIONS TURHAN KARAGULER AND MAHIR RASULOV A special numerical method for the solution of first-order partial differential equation which represents the transmission line problem in a class of discontinuous functions is described. For this, first, an auiliary problem having some advantages over the main problem is introduced. Since the differentiable property of the solution of the auiliary problem is one order higher than the differentiability of the solution of the main problem, the application of classical methods to the auiliary problem can easily be performed. Some economical algorithms are proposed for obtaining a numerical solution of the auiliary problem, from which the numerical solution of the main problem can be obtained. In order to show the effectiveness of the suggested algorithms, some comparisons between the eact solution and the numerical solution are carried out. Copyright 26 T. Karaguler and M. Rasulov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. The Cauchy problem It is known from the electromagnetic field and the circuit theories that the equations for a current and potential in a transmission line have the following form [1, 3, 4]: L i(,t) C v(,t) + v(,t) + i(,t) + Ri(,t) =, (1.1) + Gv(,t) =. (1.2) Here v(,t) andi(,t) are potential and current at any points and t, R is resistance per unit length, L is inductance per unit length, C is capacitance per unit length, and G is conductance per unit length. These line parameters are taken constant since the medium is assumed as linear and homogeneous. The initial condition for (1.1), (1.2)are i(,)= i (), (1.3) v(,)= v (), (1.4) where i ()andv () are given as continuous or piecewise continuous functions. Hindawi Publishing Corporation Proceedingsof the Conference on Differential & Difference Equations and Applications, pp

2 52 Numerical solution of transmission line problem From the Biot-Savard law of electromagnetic theory, when a wire carrying a current produces a magnetic field around it [1], if this magnetic field changes, the source of it, which is the current, changes too. This results with a wave which travels through the wire. This wave will be eamined in the frame of transmission line problem. Heaviside showed that if G/C = R/L equality esists between the parameter of the transmission line, then (1.1), (1.2) are reduced to the well-known second-order wave equation such that 2 u 2 = a2 2 u 2, (1.5) where a = 1/ LC. Using the general solution of (1.5) whichisobtainedfromd Alembert sformulafor both unknown functions i(,t) andv(,t), we have (see, [2, 4]) v(,t) = e (R/L)t[ φ( at)+ψ( + at) ], (1.6) C i(,t) = L e (R/L)t[ φ( at) ψ( + at) ]. (1.7) Here, the arbitrary functions ϕ(ξ)andψ(ξ) are found from the initial conditions such as v(,)= f ()andi(,)= (C/L)F(). When the transmission line is too long, which is the common case in practice, then the following problem occurs. The functions F()and f () are definite in the (,l)interval, however, the solutions obtained with the formulas of (1.6) and(1.7) requirevaluesof f () andf() functions for arbitrary values. This will enforce the etension of f () and F() functions beyond the interval of (,l). On the other hand, it is known from the literature that the solution of (1.5) has weak discontinuity on the characteristics. This means that the solution is on the characteristics, and continuously differentiable, but its first-order derivatives are piecewise continously differentiable. This property prevents applying well-known numerical methods in the literature to the equation of type (1.5) such as the system of (1.1) and(1.2). Furthermore, if the initial functions posses the singular points, the numerical methods mentioned above fail even worst. The weak solutions of the problem of (1.1) (1.4) are defined as follows. Definition 1.1. The functions i(,t), v(,t) satisfying the initial conditions (1.3), (1.4)are called the weak solutions of the problem (1.1) (1.4) if for any test functions f (,t) which are equal to zero at the value t = T and at the boundary of the plane t+, the integral relations ( Li(,t) ft + v(,t) f Ri(,t) ) + ddt + L i () f (,)d =, D T ( Cv(,t) ft + i(,t) f Gv(,t) ) + (1.8) ddt + C v () f (,)d = D T hold. As seen from (1.8), i(,t)and v(,t) are notnecessarily tobe countinous.

3 T. Karaguler and M. Rasulov Auiliary problem. As it is known, the derivatives of the solutions of (1.1), (1.2) with respect to and t are discontinuous on the characteristics of the equations. This requires that the applied method to find the numerical solution must have high accuracy. Because of this, in this paper, a new numerical method to obtain the weak solution of the problem (1.1) (1.4) in a class of discontinuousfunctionsis suggested. For this aim, along with the references [4 7], a special auiliary problem, as below, L I(,t) + V(,t)+RI(,t) =, (1.9) C V(,t) + I(,t)+GV(,t) =, (1.1) I(,)= I (), (1.11) V(,)= V () (1.12) is introduced. Here I () andv () areanycontinuouslydifferentiable functions which satisfy the equations of di ()/d = i ()anddv ()/d = v (). The auiliary problem has the following advantages. (1) When to employ the i(,t) and v(,t), no need to use their derivativeswith respect to and t. (2) The differentiability property of I(,t), V(,t) functions is one degree higher than differentiability property of i(,t)and v(,t) functions. One of the most significant advantages of the auiliary problem is that the well-known methods are applicable to it. Even on the basis of auiliary problem, the higher-order finite-differences scheme are allowed to develop. Theorem 1.2. If I(,t) and V(,t) are the solutions of the problem (1.9) (1.12), then the functions i(,t) and v(,t) defined by i(,t) = I(,t), (1.13) v(,t) = V(,t) (1.14) epressions are the weak solutions of the problem (1.1) (1.4) only in a sense of (1.8). As it is obvious from (1.9), (1.1) that the equations are freed from the time and space derivative terms of i(,t) and v(,t). Therefore the functions i(,t) and v(,t) can be discontinuous too. This would make it possible to develop accurate and economical algorithms for obtaining the solution which represent the physical properties of the problem. 2. Initial-boundary value problem for transmission line equation As usual, we denote R + ={(,t), >, t>}. In this section, by adding the following boundary conditions, we will investigate the considered initial value problem: i(,t) = i 1 (t), (2.1) v(,t) = v 1 (t). (2.2)

4 54 Numerical solution of transmission line problem Here, i 1 and v 1 are known functions. In general, we assume that the functions i k and v k (k = 1,2) can be discontinuous too. The weak solution of the problem is specified in the following definition as follows. Definition 2.1. The functions i(,t)andv(,t) satisfy the (1.3)-(1.4) initial, and the (2.1) and (2.2) boundary conditions are called the weak solutions of the initial-boundary value problem if for any test function f (,t) satisfying f (,T) = and is equal to zero on the boundary of the t + half-space, the integral relations hold. R t ( Li(,t) ft (,t)+v(,t) f (,t) Ri(,t) ) ddt + L i () f (,)d + T v 1 (t) f (,t)dt =, R t ( Cv(,t) ft (,t)+i(,t) f (,t) Gv(,t) ) ddt + C v () f (,)d + T i 1 (t) f (,t)dt = (2.3) According to [5, 6], the auiliary problem for the initial-boundary value problem which is shown above can be written as L i(ξ,t)dξ + v(,t)+ R i(ξ,t)dξ = v(,t), (2.4) C v(ξ,t)dξ + i(,t)+ G v(ξ,t)dξ = i(,t). (2.5) The initial conditions for the auiliary equations (2.4), (2.5) will be the same as the boundary conditions of the main problem (1.3)-(1.4). 3. Finite-differences schema and numerical eperiments In order to obtain the numerical solution of the problem (1.1) (1.4), at first, we cover the domain of definition of the definition of the solutions by the following grid as ω h,τ = {( i,t k ) i = ih; t k = kτ, i =,±1,±2,...; k =,1,2,...; h>, τ> }. (3.1) Here, h and τ are the steps of the grid ω h,τ with respect to and t variables, respectively. Then the auiliary problem (1.9) (1.12) is approimated at any grid point (i,k), I i,k+1 = I i,k τ L ( vi,k + RI i,k ), (3.2) V i,k+1 = V i,k τ C ( ii,k + GV i,k ), (3.3) I i, = I i, (3.4) V i, = V i. (3.5)

5 T. Karaguler and M. Rasulov 55 Here, I i,k and V i,k represent the approimate values of the functions I(,t)andV(,t) at the point (i,k)ofthew h,τ grid. Theorem 3.1. If the mesh functions I i,k and V i,k are the numerical solutions of the auiliary problem (3.2) (3.5), then i i,k = I, v i,k = V (3.6) are the numerical solutions of the main problem. Equations (3.2), (3.3) suggest that the algorithm is very simple and economical. Furthermore, on the basis of auiliary equations (1.9), (1.1), higher-order finite-difference schemes with respect to t can be developed. In order to approimate the auiliary equations (2.4), (2.5) by means of finite-difference method, the quadrature formula is applied to the φ(z)dz as φ(z)dz hσ i j=1φ(z j ). (3.7) Considering (3.7), (2.4), and (2.5) can be approimated by means of finite difference as I i,k+1 = h t L V i,k+1 = h t C ( ) i 1 V1 tk RIi,k V i,k Ij,k+1 I j,k + Ii,k, j= ( ) i 1 I1 tk Ii,k GV i,k Vj,k+1 V j,k + Vi,k. j= (3.8) In order to demonstrate the effectiveness of the suggested algorithm, firstly, we investigate (1.5) with the initial conditions (1.3), (1.4). As initial functions, we have a piecewise differentiable functions given as, < 1, u 1 1, 1 3, f () = 3 1 u 1 2, 3 2, 3 2, > 2,, < 1, F() = c, 1 2,, > 2. (3.9) The numerical eperiments have been carried out for the data 1 = 2., 2 = 2., 3 =., u 1 = 1., c = 2..

6 56 Numerical solution of transmission line problem u(, t).2 u(, t) (a) (b) Figure 3.1. The graphs of the eact solution at T = 4 for (a) the case f () andf() = ; (b) the case f () = andf(). In order to implement the auiliary problem, at first the functions I, V are obtained as follows:, < 1, 3 1 A, 1 3, I () = 6 B 3 u E, 3 2, 6 2 u , >2,, < 1, 2 V 1 () = 1 c, 1 2, 2 c c ( ), >2. (3.1) Here A = u 1 /( 3 1 ), B = u 1 /( 3 2 ), E = (B/6) ( 2 1 ) 2 2. In Figure 3.1, the solution of the Cauchy problem of (1.5)definedbytheD Alembert s formula is illustrated. Figure 3.2 shows the solution of the auiliary problem having the same initial data as the main problem. As these graphs, Figures 3.1 and 3.2 illustrate that the solutions of the auiliary and main problems match very well. This clearly proves the usefulness of the auiliary problem. The graphics of solutions obtained by the auiliary problem are given in Figure 3.3. Comparing the results shown in Figures 3.1(a) and 3.3(b), we observe that the numerical solution obtained from the auiliary problem is very much similar to the solutions obtained from the classical methods. The graphs of solutions of the problem (1.5) with the data f () = 2cos, F() = 1cos, R =.2, L = , C = 1 5, G =.2aredemonstratedinFigure 3.4.

7 T. Karaguler and M. Rasulov 57 u(, t) (a) u(, t) (b) Figure 3.2. (a) The time evolution of the solution of the auiliary problem; (b) the graph of u(,t) = 2 v(,t)/ 2, f (), F() = att = 4. Vi,k i (a) Ui,k i (b) Figure 3.3. The time evolution of the numerical solution obtained by using the auiliary problem f (), F() =. (a) Solution of the auiliary problem; (b) the graph of U i,k = V. v(, t) Data 1 Data 2 (a) i(, t) Data 1 Data 2 (b) Figure 3.4. The graphs of v(, t) and i(, t) solutions.

8 58 Numerical solution of transmission line problem 4. Conclusion Introduction of the auiliary problem has several advantages over the main problem. Firstly, the well-known numerical methods can easily be applied to the auiliary problem. Further, the auiliary problem lets higher-order finite-differences schemes be used. This leads to developing simple algorithms. References [1] D. K. Cheng, Field and Wave Electromagnetics, Addison-Wesley, New York, 21. [2] S. K. Godunov, Equations of Mathematical Physics, Nauka, Moscow, [3] W. H. HaytandJ. A. Buck, Engineering Electromagnetics, 6th ed., McGraw-Hill, New York, 21. [4] N. S. Koshlakov, Partial Differential Equations of Mathematical Physics, Bishaya Skola, Moscow, 197. [5] M. Rasulov, On a method of solving the Cauchy problem for a first order nonlinear equation of hyperbolic type with a smooth initial condition, Soviet Mathematics Doklady 43 (1991), no. 1, [6], Finite Difference Scheme for Solving of Some Nonlinear Problems of Mathematical Physics in a Class of Discontinuous Functions, Baku, [7] M. Rasulov and T. A. Ragimova, A numerical method of the solution of the nonlinear equation of a hyperbolic type of the first order differentialequations,minsk 28 (1992), no. 7, Turhan Karaguler: Department of Mathematics and Computing, Faculty of Science and Letters, Beykent University,Beykent 345, Büyükçekmece-Istanbul, Turkey address: turank@beykent.edu.tr Mahir Rasulov: Department of Mathematics and Computing, Faculty of Science and Letters, Beykent University,Beykent 345, Büyükçekmece-Istanbul, Turkey address: mresulov@beykent.edu.tr