Numerical Algorithm of Solving the Forced sin-Gordon Equation

260 PROCEEDINGS OF THE IMCSIT. VOLUME 3, 2008This transformation allows simplifying significantly numericalintegration and speed up the algorithm of calculationof functions u 1,2.(χ, t). The area of integration can becovered by a rectangular mesh and then, the standardSimpson method can be used [9].Fig. 4. The Characteristic triangle with the family of curvesconst γ = .Fig. 6. The rectangular integration area R(v, u).The first integral (13) in the new variables (v, u) hasthe form(21)Fig. 5. The solution surface of (15) representing the Riemannfunction. Each curve is a solution of (19), (20) for particular fixedvalue of γ .The boundary value problem (19), (20) can be solvednumerically using the relaxation method [9]. Since the solutionof this BVP approaches zero when ±∞ thenfor numerical calculations this problem can be solved forfinite values of χ which are found manually from thelimcondition .In the numerical calculations of the double integrals(13) and (14) one of the most difficult tasks is integrationover the triangle A . First, the integral needs to becalculated for various values of ( χ0, τ0). The area of therectangle A becomes larger when the value of τ0 increases.The second difficulty is that changing the value ofthe parameter a leads to changing the rectangle A shape.Therefore, it is very difficult to develop a universalalgorithm for various values of ( χ0, τ0) and the parametera . The integration area A (Fig. 3) can be mapped into awhere the double integral is calculated using the Simpsonmethod and the value of ϕ ( v, u)at each point of the integralsum is a solution of the boundary value problem(19), (20).The numerical algorithm was implemented in a programwritten in C++. For calculating values of the Besselfunctions, double integrals and solving ordinary differentialequations the code given in [9] was used.As an example, the function Φ ( t) = sin(10 t)was considered.Two graphs in a plane ( x, u1) representing resultsof calculations of the integral (21) forareshown in Fig. 7 and Fig. 8.rectangle(Fig. 6) with coordinates(v, u) by means of transformationFig. 7. V = 0.95, t = πwhere t 0= τ 0+ V χ 0/ a .