By S L Sobolev

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Extra resources for Applications of functional analysis in mathematical physics

Sample text

See Fig. 2. 5) have to be adjusted if mass sources or sinks are involved. Let the local creation or annihilation of mass be described by a mass flux density S(x, t). So, the system has locally a mass source or sink, and S specifies the amount of mass added or removed per unit of time and per unit of length. If S > 0, mass is added to the system; if S < 0, the system has locally a mass sink. In the presence of sources/sinks S(x, t), the global form of the conservation equation becomes b {∂t ρ + ∂x Q − S} dx = 0 a and the local form ∂t ρ + ∂x Q = S.

Thus, we argue as follows. The appearance of L in α can easily be scaled away by taking τ = L/c. Then α = 1, β = 1 6 H L 2 , γ = 3 a . 21) h. Show that if L is given the dimension of length, τ will have the dimension of time, and show that the variables x, t, and u and the parameters α, β, γ are dimensionless. i. Observe that now L appears only in the coefficient β. Keeping all other parameters fixed, look at the limit for long waves, and explain that the third order spatial derivative in the KdV equation describes effects that are due to the length of the waves under consideration; the longer the waves, the less this term contributes.

I. Observe that now L appears only in the coefficient β. Keeping all other parameters fixed, look at the limit for long waves, and explain that the third order spatial derivative in the KdV equation describes effects that are due to the length of the waves under consideration; the longer the waves, the less this term contributes. Find the equation obtained in the limit for infinite long waves of finite amplitude. Long waves with small amplitudes j. Consider the limit of infinitesimally small, infinitely long, waves.