Theory And Applications Of Numerical Homotopy Method

This paper mainly investigates how to solve nonlinear equations, find the periodic solutions of high order Duffing equation and solve the Picard Boundary Value by using the homotopy method.In chapter 1, we introduce the development of homotopy algorithm, and then introduced some important ideas of homotopy algorithms and how to determine the quality of a homotopy method.In chapter 2,we consider the following nonlinear equationswhich f_i(i-1,2,…, n) defined as real function in open domain of n-dimensional Euclid space.We described in detail how to use homotopy method to find the zero value of this system. We first provide a criterion for obtaining tangent vector in the direction of increasing arch length. then, we track the homotopy curve using the estimates-correction method, and give some detailed discussions for the change of step length to improve the computational efficient.In chapter 3, we consider the high-duffing equation:x⁽²ⁿ⁾+g(x)=e(t), x∈R, (2) which g:R→R in a C¹ function,e:R→R is a continuous function,e(t) is a 2Ï€periodic function,e(t) = e(t + 2Ï€), t∈R,we give the conditions of the the existence of solutions:Theorem 5.1. If there is a A>0,such thatN²ⁿ<α≤(-1)_n+1 g(x)/x≤β<(n+1)²ⁿ, |x|≥AWhere N is a non-negative integer,α>0,β>0,Then (2) at least one periodic solution.We give a strict proof about this issue ,and we construct a homotopy to change this problem to find ap=(p₀,…,p_2n)^T,p_i=x⁽ⁱ⁾(0)such that this equation is a periodic equation.For finding the periodic solutions numerically, we first convert this equation into n first-order differential equations. Then we make numerical iteration for the given initial point and obtain a nonlinear map. Next, we use homotopy method to find the zero of the nonlinear map.In chapter 4, we consider the Picard Boundary Value equation:On this issue,we introduced a previous results about the existence of solutionsTheorem 5.2. Assume that there exist a twice differentiable function(?):[a,b]→R⁺\{0}and a continuous functionF:[a,b]×R⁺×R→R for any(t,x,y),such that t∈[a, b], |x|=(?)(t),〈x, y〉=|x|(?)'(t),we have〈x, f(t,x,y)〉≤(?)(t)F(t, (?)(t), (?)'(t))+|y|²-(?)'(t)²(?)"+F(t,(?)(t),(?)'(t))≤0.Assume moreover that there exist numbersα∈[0,1),β≥0,for any (t,x,y),when t∈[0, T], |x|≤(?)(t),y∈Rⁿ,then〈(x,f(t,x,y)〉≤α|y|²+β,|〈y,f(t,x,y)〉|>≤h(|y|)|y|where,h:R⁺→R⁺\{0}progressive, continuous and such thatintegral from n=0 to +∞(s²ds/h(s))=+∞then functions(3)has a solution x^* ,such that |x^*(t)|≤(?)(t),t∈[a,b].We convert this problem into a problem of nonlinear equations using thedifference method. especially. we get some points t_i=ih,h=1/(m+1), i=0,1,…, m+1.On the point tⁱ,we can change the equations into(x_i+1-2x_i+x_i-1)/h²-f(t_i,x_i,(x_i+1-x_i-1)/2h)=0,i=0,1,…,mthe boundary conditions is x₀=0,x_m+1=0.. Then by means of homotopy method, following the path of homotopy of numerical value until the hyperplane ofλ=1.Concretely, we begin at y⁰=(x⁰,λ₀), compute the string of points y¹,y²,…, on the curve, such that, each point yⁱ⁺¹ is obtained throw guessing yⁱ=(xⁱ,λ_i),to get z₀, and correcting it by Newton's method.then we make some numerical experiments for some specific examples.In the final chapter,We get a conclusion of the full text.