不定常自由边界问题研究

摘要

本文建立了热传导方程的奇异内边界问题：求 {u(x,t),x(t)} ， ?u/ ?t = a 2 ? 2 u/ ?x 2 -b ?u/ ?x - ru+γ( t) δ( x-x( t)) ,-∞ x 0 u(x,0)=0, - ∞ x ∞ u(x(t),t)= max _{- ∞ x + ∞} u( x,t) =φ( t) ,t≥0 ?u/ ?t( x( t) ,t) =o,t ≥ 0 lim _x→-∞ |u| ∞ , lim _x→+∞ |u| ∞ 使其满足 (I) 其中 φ (t) 为待求函数。并获得奇异内边界的线性函数表达式 x ( t) = x ₀ +bt ，且解函数 u(x,t) 满足 u(x(t),t)= max _{- ∞ x + ∞} u( x,t) 。同时获得了热传导方程的问题A (在区域 - ∞ x x(t) ,t≥0 上的自由边界问题)和问题B(在区域 x(t) ≤ x 上的自由边界问题)的自由边界皆为 x ( t) = x ₀ +bt ，问题A和问题B的自由边界与奇异内边界重合；线性函数表达式 x ( t) = x ₀ +bt 为最佳热源位置边界。完全类似地，我们建立了Black-Scholes方程的奇异内边界问题： 求{u(s,t),s(t)},使其满足 ?u/ ?t +( σ 2/ 2) s 2( ? 2 u/ ? s 2) +( r-q) s( ?u/ ?s) -ru=-γ( t) δ( s-s( t)) , 0 s + ∞,0 t T u( s,T) =φ( s) ,0≤s +∞ u( s( t) ,t) =φ( t) , 0 t T (?u/ ?s)( s( t) ,t) =v( t) , 0 t T lim s→ o + |u| +∞ , lim _s→+∞ |u| +∞ (II) 其中 φ (t), v( t) 为待求函数。 获得： 1 ^{0 当终值函数 φ =0 且边值函数 v=0 时；奇异内边界为 s( t) = s _T e σ 2 ω(T-t) ，且解函数 u( s,t) =w( s,t) 满 w( s( t) ,t) = max _{0≤s w(s,t) ; 2 0当终值函数 φ ≠0 时；获得奇异内边界为 s( t) = s T e σ 2 ω(T-t) ，且解函数 u(s,t)=v(s,t)+w(s,t) 满足 w( s( t) ,t) = max 0≤s w(s,t) ， v( t) =( ?v/ ?s)( s( t) ,t) ,φ( t) =v( s( t) ,t) +w(s( t) ,t) 。同时建立了自由边界问题A(在区域 0≤s≤s( t) , ( 0 , T) 上)和自由边界问题B(在区域 s( t) ≤s ∞, ( 0 , T) 上)。获得问题A和问题B在齐次终值条件下确定的自由边界都为 s( t) = s T e σ 2 ω(T-t) 。终值函数 φ 满足 =[K, +∞ ) 或 =[ 0,K] 得到 sT=K 。从而问题A和问题B具有公共自由边界 s( t) =K e θ(T-t) ，满足条件( 1/ s(t))( ds(t)/ dt) ≡- θ ，常数 θ=q-r+( 1/ 2) σ 2 由Black-Scholes方程中的参数 q，r， σ 2 唯一确定。 In this paper, the singular interior boundary problem of heat conduction equation is established: seeking {u(x,t),x(t)} , satisfy ?u/ ?t = a 2 ? 2 u/ ?x 2 -b ?u/ ?x - ru+γ( t) δ( x-x( t)) ,-∞ x 0 u(x,0)=0, - ∞ x ∞ u(x(t),t)= max - ∞ x + ∞ u( x,t) =φ( t) ,t≥0 ?u/ ?t( x( t) ,t) =o,t ≥ 0 lim x→-∞ |u| ∞ , lim x→+∞ |u| ∞ (I) which φ (t) is the function to be determined. And the linear function expression x ( t) = x 0 +bt of singular inner boundary is obtained, satisfying u(x(t),t)= max - ∞ x + ∞ u( x,t). We establish free boundary problem A and free boundary problem B on homogeneous heat conduction equation. The problem A is free boundary problem in region - ∞ x x(t) ,t≥0 . The problem B is free boundary problem in region x(t) ≤ x . It is obtained that free boundary about problem A and problem B are the linear function x ( t) = x 0 +bt . The free boundary about problem A and problem B coincides with the singular inner boundary. Similarly,we establish the singular interior boundary problem of Black-Scholes equation： seeking {u(s,t),s(t)} , satisfy ?u/ ?t +( σ 2/ 2) s 2( ? 2 u/ ? s 2) +( r-q) s( ?u/ ?s) -ru=-γ( t) δ( s-s( t)) , -∞ x 0 u( s,T) =φ( s) ,0≤s +∞ u( s( t) ,t) =φ( t) , 0 t T (?u/ ?s)( s( t) ,t) =v( t) , 0 t T lim s→ o+ |u| +∞ , lim s→+∞ |u| +∞ (II) which φ (t) and v( t) are functions to be determined. When final function φ =0 , and boundary value function v=0 , the singular inner boundary s( t) = s T e σ 2 ω(T-t) is obtained, and the solution function u( s,t) =w( s,t) satisfies w( s( t) ,t) = max 0≤s w(s,t) and when final function φ ≠0 , the singular inner boundary s( t) = s T e σ 2 ω(T-t) is obtained, and the solution function u(s,t)=v(s,t)+w(s,t) satisfies w( s( t) ,t) = max 0≤s w(s,t) and boundary value function v( t) =( ?v/ ?s)( s( t) ,t) ,φ( t) =v( s( t) ,t) +w(s( t) ,t) . We establish free boundary problem A and free boundary problem B on homogeneous Black-Scholes equation. The problem A is free boundary problem in region 0≤s≤s( t) , ( 0 , T) . The problem B is free boundary problem in region s( t) ≤s ∞, ( 0 , T) . It is determined s( t) = s T e σ 2 ω(T-t) that free boundary about problem A and problem B. The conclusion s T =K by the final value function φ(s) satisfies =[K, +∞ ) or =[ 0,K] . Thus the conclusion s( t) =K e θ(T-t) that is the public free boundary about the problem A and problem B,which satisfies the condition ( 1/ s(t))( ds(t)/ dt) ≡- θ , constant θ=q-r+( 1/ 2) σ 2 determined by the parameters q，r， σ 2 in the Black-Scholes equation.}}