Computational Physics - Department of Physics

9.2 Shooting methods 289 Table 9.2Integrated and exact solution of the differential equation−y′′+x^2 y= 2 εywith boundary condit ...

290 9 Two point boundary value problems Ĥψ(r) = (̂T+V̂)ψ(r) =Eψ(r). (9.4) In detail this gives ( − ̄ h^2 2 m ∇^2 +V(r) ) ψ(r) = ...

9.3 Numerical procedure, shooting and matching 291 The eigenfunctions in Eq. (9.5) are subject to conditions which limit the pos ...

292 9 Two point boundary value problems d dρu <(ρ) u<(ρ) = d dρu >(ρ) u>(ρ) at ρ=ρm. (9.19) We can modify this expre ...

9.3 Numerical procedure, shooting and matching 293 we have a solution. The matching code is given below. To choose the matching ...

294 9 Two point boundary value problems point as the midpoint of the integration interval and compute safely the solution. This ...

9.4 Green’s function approach 295 If we then define the Green’s function as G(x,y) = { y( 1 −x)if 0≤y≤x x( 1 −y)if x≤y≤ 1 we can ...

296 9 Two point boundary value problems u(x) =u>(x) ∫x a u<(x′)f(x′)dx′+u<(x) ∫b x u>(x′)f(x′)dr′ (9.25) The algorit ...

9.5 Exercises 297 // Compute Wronskian at matching mid-point midpoint = (max_step)/2; // first part of Wronskian wronskian = (ui ...

298 9 Two point boundary value problems Compare these results with those obtained by solving the above differential equation as ...

9.5 Exercises 299 d^2 u(r) dr^2 + m h ̄^2 (V 0 +E)u(r) = 0 r<a, (9.29) and d^2 u(r) dr^2 +m h ̄^2 E u(r) = 0 r>a. (9.30) ...

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Chapter 10 Partial Differential Equations AbstractPartial differential equations play an important role in our modelling of phys ...

302 10 Partial Differential Equations or if we have a finite electric charge represented by a charge densityρ(x)we have the fami ...

10.2 Diffusion equation 303 and if we set B=C= 0 , we recover the 1 + 1 -dimensional diffusion equation which is an example of a ...

304 10 Partial Differential Equations the diffusion equation, or heat equation. If we specialize to the heat equation, we assume ...

10.2 Diffusion equation 305 withL= 1 the length of thex-region of interest. The boundary conditions are u( 0 ,t) =a(t) t≥ 0 , an ...

306 10 Partial Differential Equations a(t) t g(x) b(t) x ui− 1 ,j ui,j ui,j+ 1 ui+ 1 ,j ✲ ✻ Fig. 10.1Discretization of the integ ...

10.2 Diffusion equation 307 whereV 0 is the initial vector at timet= 0 defined by the initial valueg(x). In the numerical implem ...

308 10 Partial Differential Equations bi j= 2 δi j−δi+ 1 j−δi− 1 j, meaning that we have the following set of eigenequations for ...