Exact Solutions of Coupled Parallel Resonant Circuits Equations by Decomposition Method

In this paper, we provide the solution of a modeled parallel resonant circuit using the decomposition method. The parallel resonant circuit also known as resistor (R) in ohms, inductor (L) in Henry and capacitor (C) in farads (RLC) circuit is used in turning radio or audio receivers. The mathematical models are in a system of ordinary differential equations (ODE), which we solve using the Adomian Decomposition Method (ADM). We discovered that this method gave exact solutions as can be obtained using any traditionally known analytic method. These findings are illustrated in three test problems. Keywords— Adomian Decomposition Method, Resonant Circuits, System of Differential equations.


I. INTRODUCTION
The RLC circuit is one in which the fundamental elements resistor, inductor and capacitors are connected linearly and passively in series or parallel in nature across a voltage supply.A model for parallel RLC circuit using Kirchhoff's voltage and current law is stated in literature with the system of equations.See [7], [8], [9] and [10].
, E is the battery that provide the voltage, R, L and C are as defined previously.Equation (1) has also been used to model a hydraulic system.[8] stated that wires are pipes filled with fluid, the switches correspond to valves and the battery is analogous to a pump which maintains a pressure E. When the pressure is increased it forces the molecules of the fluid to move in a pipe.This paper is divided into sections; in the next section we give the mathematical concept of ADM on a system of differential equations.Subsequently, we give illustration in form of examples using the decomposition method on equation (1) and we conclude.

II. MATHEMATICAL CONCEPT OF ADM ON SYSTEM OF DIFFERENTIAL EQUATIONS
The compact form a system of differential equations can be given as  [6] and further simplified by applying in various form in [2], [3], [4], [5], equation ( 2) is given as where L is a linear operator with an inverse By ADM, i  an the integrand in equation ( 4) are give as where j , i A are the Adomian polynomials see [5] and [6].
Putting equations ( 5) and ( 6) in equation ( 4), we obtain From equation (7), the source term and other terms are defined.A comprehensive analysis has been given by [1] and [11] to prove the convergence of equation ( 5).In the following section, we give illustrations to demonstrate the efficiency of ADM in obtaining exact solutions to system of differential equations.And, to validate the theoretical aspect just discussed in this section.

III. ILLUSTRATIVE EXAMPLE
In this section, we adapt the problems given in [8] and apply the ADM.First we apply the concept of ADM on the governing equation (1).We obtain, on applying the inverse operator, that

Example 1
In relation to the coupled equation ( 1), it was stated in [ We give the Taylors series form of equation ( 9 Applying the concept of ADM in this example we have   12).Also, Fig. 2a, 2b, 2c and 2d shows the plots of the exact solutions and that of ADM results.The variation in the plots is as a result of using very few terms in the series solution of ADM and infinite terms of the exact solution equation (11).

2
the second line of equation(10).The result of the analytical solutions and that of ADM are clearly shown in Fig.1a, 1b, 1c and 1d.The unevenness in the plot is because few terms of the series solution of ADM were applied compare to the whole infinite terms of the exact solution equation(9).Example Similarly, in relation to the coupled equation (1), it was stated in[8] that for , β in equation (13).Similarly, Fig. 3a, 3b, 3c and 3d depict the plots of the analytical solutions equation (13) and ADM solutions.For ADM only few International journal of Chemistry, Mathematics and Physics (IJCMP) [Vol-2, Issue-2, Mar-Apr, 2018] https://dx.doi.org/10.22161/ijcmp.2.2.2 ISSN: 2456-866X