Abstract: Abstract
Trefftz method (TM) is well known and has been described in many articles and monographs , TM applied to nonlinear problems can be found e.g. in [1]. The homotopy perturbation method (HPM), introduced by He, [2], is a very useful tool to solve nonlinear PDE, e.g. [3]. Generally, the HPM leads from nonlinear equation (or system of equations) of to the form
L(u) + N(u) = f(r), r ∈ Ω (1)
with conditions B(u(r),u(r)/n)=0 , r ∈ Γ, to a system of equations. To this end one constructs a homotopy v(r, p) : Ω× [0, 1] → R which satisfies:
H(v, p) = (1 − p) [L(v) − L(u0)] + p [L(v) + N(v) − f(r)] = 0 (2)
where u, v can be vector-functions. Expressing v(r, p) as a power series in p
v = v0 + pv1 + p2v2 + ...
and substituting such expansion to the equation leads to a system of PDEs, resulting from the necessity of zeroing the coefficients of pn , n = 0,1,2,… (equation (2) must be met for any p[0,1]). Thanks to TM a linear part of Eq. (1), L(u) can be choosen this way that finding the initial guess in HPM becomes easy and efficient.
Three direct and inverse problems have been solved. At first an inverse problem for the equation
(3)
is solved. Then, some direct and inverse problems for one-dimensional coupled Burgers’ equations
ut − uxx − 2uux + (uv)x = 0, (4)
vt − vxx − 2vvx + (uv)x = 0, (5)
are considered. Finally, some direct and inverse problems for the two-dimensional coupled Burgers’ equations
ut −∇2u − 2u∇u + (uv)x + (uv)y = 0,
vt −∇2v − 2v∇v + (uv)x + (uv)y = 0,
are discussed.
Presented approach can be efficiently used to solve in an approximate way the direct (initial-boundary) and boundary inverse problems. In simpler cases it quickly leads to exact solutions.
References
1.K. Grysa, A. Maciag and A. Pawińska, Solving nonlinear direct and inverse problems of stationary heat transfer by using Trefftz functions. Int. J. Heat and Mass Transfer 55, 7336–7340 (2012)
2. J.-H He, Homotopy perturbation technique. Comput. Math. Appl. Mech. Eng. 178; 257–262 (1999)
3. A. A. Hemeda, Homotopy Perturbation Method for Solving Systems of Nonlinear Coupled Equations. Appl. Math. Sciences, 6, 4787 – 4800 (2012)
