DNM Mathematics AMA - AESS

74 Pages·2010·0.92 MB·English

MSc
AMA/AESS
Michel
Fournié
I-Introduction
DNM Mathematics
II-Time
independent
PDE AMA - AESS
III-Time
dependent
PDE
IV-Conclusion Michel Fournié
References
[email protected]
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Presentation of the course
MSc
AMA/AESS Duration 16.25h during 6 days !!! (exam 05/10)
Michel
Fournié Contents Mathematical Modelisation
I-Introduction
Plan of the course
II-Time
independent 1- Introduction
PDE
Partial Differential Equation, Classification
III-Time
dependent 2- Time independent PDE
PDE
Laplace’s equation (exercice)
IV-Conclusion
3- Time dependent PDE : Evolution equations
References
Heat equation
Waves equation
Maxwell equation (exercice)
4- Conclusion
Numerical simulations
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MSc
AMA/AESS
Michel
Fournié
I-Introduction
P.D.E.,Notations
Classification
Standardmodels
I - Introduction
II-Time
independent
Partial Differential Equations
PDE
III-Time
dependent
PDE
IV-Conclusion
References
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Mathematical modelisation
MSc
AMA/AESS
Michel
Fournié • The models are Partial Differential Equations (P.D.E.) i.e.
I-Introduction differential equations with several variables (space and time
P.D.E.,Notations
for example)
Classification
Standardmodels
II-Time
independent • A limits problem is a problem formed by one or several
PDE
P.D.E. with boundary conditions given on the totality of the
III-Time
dependent boundary
PDE
IV-Conclusion
• A Cauchy’s problem for an evolution problem (depending
References
on time) with an initial data in time
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Notations
MSc
AMA/AESS
Michel
Fournié Vector, Matrix
I-Introduction For some vectors u and v in IRN we denote
P.D.E.,Notations
Classification • u.v = (u,v) = (cid:80)N uv the scalar product
Standardmodels N i=1 i i
II-Time
iPnDdeEpendent • (cid:107)u(cid:107)N = ((u,u)N)12 = ((cid:80)Ni=1ui2)12 the euclidian norm
IdIIe-pTeinmdeent • u(cid:78)v ∈ IRN,N where (u(cid:78)v)ij = uivj, 1 ≤ i,j ≤ N
PDE
IV-Conclusion For some matrices σ and τ in IRN,N
References
• σ : τ = (cid:80)N σ τ
i,j=1 ij ij
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Differential operators
MSc
AMA/AESS
Michel For a function u from IRN into IR
Fournié
Gradient
I-Introduction
P.D.E.,Notations (cid:18) ∂u ∂u (cid:19)t
Classification grad(u) ∈ IRN = ∇u = (∂u) = ,··· ,
Standardmodels i 1≤i≤N ∂x ∂x
1 N
II-Time
independent Laplacian
PDE
III-Time (cid:88)N ∂2u
dependent ∇2u = ∆u =
PDE ∂x2
i=1 i
IV-Conclusion
References Remark :
These operators must be considered for space variables
(not time)
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Differential operators
MSc
AMA/AESS
For a function u from IRN into IRN
Michel
Fournié
Gradient
I-Introduction
grad(u) ∈ IRN,N = ∇u = (∂u)
P.D.E.,Notations j i 1≤i≤N
Classification  
Standardmodels ∂1u1 ··· ∂Nu1
IinI-dTeipmeendent =  ... ... 
PDE
∂ u ··· ∂ u
III-Time 1 N N N
dependent
PDE Divergence
IV-Conclusion div(u) ∈ IR = ∇.u = (cid:80)N ∂ui with u = (u ,··· ,u )t
i=1 ∂x 1 N
References i
Laplacian
∆u ∈ IRN = ((cid:80)N ∂2ui) = div(grad(u)) = ∆u
j=1 ∂x2 1≤i≤N
j
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Example 1 : The Laplace’s problem
MSc
AMA/AESS
Michel • We consider an elastic membrane which occupies at rest
Fournié
the domain Ω included in the plan (O,x ,x )
1 2
I-Introduction • The membrane is fixed on ∂Ω
P.D.E.,Notations
Classification • We apply a surfacic force f on the membrane
Standardmodels
• We denote u the vertical displacement in each point then
II-Time
independent u is solution of
PDE
III-Time (cid:26)
dependent −∆u = f in Ω,
PDE
u = 0 on ∂Ω
IV-Conclusion
References
We recall that ∆u = ∂2u + ∂2u
∂x2 ∂x2
1 2
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Example 2 - The heat equation
MSc
AMA/AESS
• We consider a material supposed homogeneous and
Michel
Fournié isotropic
• c and k are physical constants (specific heat and thermal
I-Introduction
P.D.E.,Notations conductivity) depending on materials
Classification
Standardmodels • The temperature in the material is governed by
II-Time
independent
PDE
III-Time  c∂T(t,x) −k∆T(t,x) = f(t,x), for (x,t) ∈ Ω×IR+
dependent  ∂t ∗
PDE T(t,x) = 0, for (x,t) ∈ ∂Ω×IR+ boundary condition
∗
IV-Conclusion  T(t = 0,x) = T (x), for (x,t) ∈ Ω initial condition
0
References
Remark : The heat equation is linear in the sens that its
solution depends linearly on the data (f,T )
0
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Example 3 : The Waves equation
MSc
AMA/AESS
Michel
Fournié
• In the electromagnetic, the waves are governed by
I-Introduction
P.D.E.,Notations
Classification  ∂2u −∆u = f on Ω×IR+
Standardmodels  ∂t2 ∗
II-Time  u = 0 on ∂Ω×IR+
∗
independent
PDE  u(t = 0) = u0 on Ω
III-Time  ∂u(t = 0) = u on Ω
dependent ∂t 1
PDE
IV-Conclusion
References Remark : Take your courses to find other models !!!
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