The Use of Homotopy Analysis Method to Solve the Time-Dependent Nonlinear Eikonal Partial Differential Equation Mehdi Dehghan and Rezvan Salehi Department of Applied Mathematics, Faculty of Mathematics and Computer Science, Amirkabir University of Technology, No. 424, Hafez Ave., 15914, Tehran, Iran Reprint requests to M. D.; E-mail: [email protected], [email protected] or R. S.; E-mail: [email protected] Z. Naturforsch. 66a, 259 – 271 (2011); received April 8, 2010 / revised July 21, 2010 In this research work a time-dependent partial differential equation which has several important applications in science and engineering is investigated and a method is proposed to find its solution. In the current paper, the homotopy analysis method (HAM) is developed to solve the eikonal equation. The homotopy analysis method is one of the most effective methods to obtain series solution. HAM contains the auxiliary parameterh ¯, which provides us with a simple way to adjust and control the convergence region of a series solution. Furthermore, this method does not require any discretization, linearization or small perturbation and therefore reduces the numerical computation a lot. Some test problems are given to demonstrate the validity and applicability of the presented technique. Key words: Homotopy Analysis Method; Eikonal Equation; Semi-Analytic Approaches; Time-Dependent Partial Differential Equations; Applications; Adomian Decomposition Method; Hamilton-Jacobi Equation. AMS subject classifications: 74G10, 35C10, 70H20 1. Introduction travelling in a medium with variable speed of propa- gation. The Hamilton-Jacobi time-dependent partial differ- Two different type of methods can be found to solve ential equation, the eikonal equation. One approach is to treat the problem as a static (time-independent) boundary value ψt + H(x, ψ(x,t)) = 0, (1) problem and design an efficient numerical algorithm arises in many applications ranging from classical me- to solve the system of nonlinear equations after dis- chanics to contemporary problems of optimal control. cretization. For example, the fast marching [9, 10] and These include geometrical optics, crystal growth, etch- fast sweeping methods [11, 12] are of these types. The ing, computer vision, obstacle navigation, path plan- fast marching method employs a heap to sort points on ning, photolithography, and seismology. the moving wavefront. This is based on the property of A very important member of the family of the static the solution that guarantees the characteristic steepest Hamilton-Jacobi equations is the eikonal equation. The descent on u. The solution at each point depends on stationary eikonal equation is points with smaller values, and updating the minimum | u(x)| = η(x), x ∈ Rn, (2) value on the wavefront, using the heap-sort maintains this condition. The complexity of this algorithm is of with a boundary condition u(x)=φ(x), x ∈ Γ ⊂ Rn. order O(N logN) for N grid points, where the logN The eikonal equation has many applications in opti- factor comes from the heap-sort algorithm. On the mal control [1, 2], computer vision [3 – 5], geometric other hand, one can update solutions along a specific optics [6], path planning [7, 8], etc. The equation is direction without explicit checks for causality prop- closely related to conservation laws, and information erty. This is the main idea behind the fast sweeping travels with characteristics or rays from the boundary. method which solves the problem on an n-dimensional If η = 1andφ = 0 then the solution u(x) is the dis- grid using at least 2n directional sweeps, one per quad- tance between the point x and the boundary. If η de- rant, within a Gauss-Seidel update scheme. The fast pends on x, u(x) is the phase of high frequency wave sweeping is optimal in the sense that a finite number of 0932–0784 / 11 / 0500–0259 $ 06.00 c 2011 Verlag der Zeitschrift f¨ur Naturforschung, T¨ubingen · http://znaturforsch.com 260 M. Dehghan and R. Salehi · HAM for Time-Dependent Nonlinear Eikonal Equation iterations is needed [12], so that the complexity of the a time-dependent Hamilton-Jacobi equation, algorithm is O(N) for a total number of N grid points, although the constant in the complexity depends on the ϕt + H(x,y,t,ϕx,ϕy)=0, (4) equation. Ω ⊆ Rn The high-order finite difference type fast sweep- on domain and subject to initial conditions ϕ( , = )=ϕ ( ) ∈ Γ ⊂ ∂Ω ing method developed in [11] provides a quite gen- x t 0 0 x for x . Osher and Sethian eral framework, and it is easy to incorporate any or- developed a Hamilton-Jacobi scheme with second- der of accuracy and any type of numerical Hamil- order viscosity for a curve propagation with curvature- tonian into the framework. Much faster convergence dependent speed. In that work, they considered a small = ( ) speed than that by the time-marching approach can section of curve t u X ,asu is satisfied in the eikonal be achieved. Several other numerical schemes are ex- equation, and produced an evolution equation of the tended to solve Hamilton-Jacobi equations for example form the essentially non-oscillatory (ENO) scheme [13], the / Ψ +F(K)(1+| Ψ|2)1 2 = G(X, Ψ,ε), X ∈ Rd, (5) weighted ENO (WENO) scheme [14], the discontinu- t ous Galerkin method [15], etc. where xd = Ψ(x1,...,xd−1,t) and K is the curvature. The other class of numerical methods for static At the same time, if we view the curve t = u(X) as a Hamilton-Jacobi (HJ) equations is based on the refor- level set of the function Φ(X,t)=C,weareledtothe mulation of the equations into suitable time-dependent Hamilton-Jacobi equation problems. One technique to obtain a time-dependent d Hamilton-Jacobi equation is using the so called parax- Φt + c(X)| Φ| = 0, X ∈ R . (6) ial formulation [13, 16 – 18]. Another approach is the level set method. A large number of applications re- 2. The Main Problem quire the development of optimal algorithms for track- ing moving interfaces (that is, advancing fronts). Ad- In this paper, we consider the time-dependent vances in numerical analysis have led to computation- eikonal equations (7) and (8) that are formulated by the ally efficient tools for tracking interface motion by us- aid of the level set method. According to the level set ing level set methods. The level set method first intro- framework, if we view the curve t = u(x) as a level set duced by Osher and Sethian [19] in 1988 is a simple of the function φ(x,t)=c, we are led to the Hamilton- and adaptable method for computing and analyzing the Jacobi equation motion of an interface in two or three dimensions and following the evolution of interfaces [20]. The main φt (x,t)+c(x)| φ(x,t)| = 0, in Ω × [0,T ]. (7) idea of the level set method is to embed the propa- gating interface as the zero level set of a continuous At the same time, if considering a small section of ϕ real valued function, called a level set function. Let curve t = u(X),asu is satisfied in (2), then we can ϕ Γ denote this function then embeds the interface produce an evolution equation of the form as its zero level set Γ (t = 0)={x ∈ Rn | ϕ(x)=0}. 2 1/2 Furthermore, by adding a time variable, the level set ψt + f (X)(1+| ψ| ) = 0, in Ω ×[0,T], (8) function can be used to capture a given dynamics of the interface using a time dependent partial differential where xd = ψ(x1,···,xd−1,t), d = 2, 3, and subject to equation (PDE) in ϕ. The location of the interface at the initial conditions time t in this case is the zero level set of ϕ at that time: n Γ (t)={x ∈ R | ϕ(x,t)=0}. φ(x,t0)=g(x), ψ(X,t0)=h(X), x ∈ Ω, (9) Osher [21] provided a link to the time dependent ( )= 1 ( ) ( ) Hamilton-Jacobi equations by proving that the t-level where c x η(x) , g x and h x are Lipschitz contin- set of ϕ(x,y) is the zero level set of the viscos- uous functions. ity solution of the evolution equation at time t.In In this investigation, we focus on an eikonal equa- that paper, Osher derived from the general first-order tion which is transformed in the form of a Hamilton- equation Jacobi equation by the level set formulation. These equations are solved numerically by several authors. F(x,y,u,ux,uy)=0(3)Interested reader can see [16, 22] and the references M. Dehghan and R. Salehi · HAM for Time-Dependent Nonlinear Eikonal Equation 261 therein. But the current paper proposes a different ap- solving the Laplace equation with Dirichlet and Neu- proach. The main idea behind this work is to use a mann boundary conditions. This technique is also ap- semi-analytic (or quasi-numerical) technique [23]. The plied in [34] to compute an explicit series solution of solution is given by means of the homotopy analysis travelling waves with a front of the Fisher equation. method (HAM). Several examples are given to show In [35] the heat transfer analysis is investigated for the efficiency of this method for solving the studied magnetohydrodynamic (MHD) flow in a porous chan- model. nel, and the homotopy analysis method is employed The rest of this paper is arranged as follows: In Sec- to obtain the expressions for velocity and tempera- tion 3, we present the mathematical framework of the ture fields. Authors of [36] presented an efficient nu- homotopy analysis method.