14:00h – 14:40h

Solving Differential Equation-Based Optimization Problems via the Cross-Entropy Method

14:00h – 14:40h

The nonlinear fractional relativistic Schrodinger-Choquard equation

14:00h – 14:40h

Recent results on small-time controllability of nonlinear parabolic equations with bilinear controls.

14:50h – 15:30h

New formalism in finite difference:the complete difference method for Helmholtz equation

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A new formalism in the finite difference framework is developed, which consists of three
steps: choosing the dimension of the local approximation subspace, constructing a vector
basis for this subspace, and determining the coefficients of the linear combination [1, 3]. This new approach called the Complete Method is capable to generate any finite difference scheme that belongs to this subspace, and can be applied to any PDE. The Helmholtz equation is the PDE that describes the time-harmonic of the wave equation. It is well known that finite difference and finite element methods exhibit the ‘error pollution effect’ for medium and high wavenumber. When applied to the Helmholtz equation on uniform meshes, the Complete Finite Difference Method is both consistent and capable of minimizing the dispersion relation for all stencils in all dimensions [1, 3]. In this case, the vectors used to form the basis of the complete method are the classical centered scheme and new schemes developed by modifying only the k 2 u term of the PDE [1, 2, 3, 4]. The coefficients of the linear combination are chosen in such a way as to minimize the dispersion relation. In the 1D case and 3-point stencil, pollution error is eliminated. In the 2D case and 5-point stencil, the Complete Centered Finite Difference Method presents a dispersion relation equivalent to Galerkin/Least-Squares Finite Element Method. In the 2D case and 9-point stencil, two versions were developed using two different bases for the local approximation subspace. Both versions are equivalent and exhibit a dispersion relation similar to Quasi Stabilized Finite
Element Method. Numerical results confirm the good performance of the new formalism.
References
[1] ALVAREZ GB, NUNES HF & MENEZES WA. 2024. Complete centered finite difference
method for Helmholtz equation. An Acad Bras Cienc 96: e20240522. DOI 10.1590/0001-
3765202420240522.
[2] ALVAREZ GB & NUNES HF. 2024. Novos Esquemas de Diferenças Finitas para a
Equação de Helmholtz. REMAT: Revista Eletrônica da Matemática 10: e4001. DOI
10.35819/remat2024v10iespecialid7019
[3] NUNES HF. 2024. Método Completo de Diferenças Finitas Centradas para a Equação de
Helmholtz. Volta Redonda: Master’s Thesis, Universidade Federal Fluminense, 133 p.
[4] ALVAREZ GB & NUNES HF. 2023. Novos esquemas de diferenças finitas para a equação
de Helmholtz. Encontro Regional de Matemática Aplicada e Computacional (ERMAC-RJ) &
Simpósio Primeira Década PPG-MCCT, 2023.

14:00h – 14:40h

Shock formation and maximal hyperbolic development in multi-D gas dynamics.

14:50h – 15:30h

The k-Yamabe flow and its solitons.

The Yamabe problem is a classical question in conformal geometry that has promoted a fruitful interaction between geometry and analysis. In this talk I will briefly introduce this problem and a fully-nonlinear extension of it, known as the k-Yamabe problem. I will finish the talk  by discussing recent results obtained with Maria Fernanda Espinal related to the classification of soliton solutions to this equation.

14:50h – 15:30h

PARABOLIC SYSTEM WITH BOUNDARY CONDITIONS:WELL-POSEDNESS AND NULL CONTROLLABILITY

In this talk, we present some results on wellposedness and (internal
and boundary) null controllability of the parabolic equation with dynamic
boundary conditions and drift terms

∂ty − d∆y + B(x).∇y + c(x).y = 1ωu + f in ΩT ,
∂tyΓ − δ∆ΓyΓ + d∂νy + b(x).∇ΓyΓ + l(x)yΓ = 1Γ0v +g on ΓT 

y|Γ(t, x) = yΓ(t, x) on ΓT ,

y(0, ·) = y0 in Ω,
y|Γ(0, ·) = y0,Γ on Γ,
where Ω is a bounded domain of RN , with smooth boundary Γ = ∂Ω of
class C2
, ν(x) is the outer unit normal field to Ω in the point M(x) of Γ,
∂νy := (ν.∇y)|Γ, d, δ are positive real numbers, c ∈ L∞(Ω), l ∈ L∞(Γ),
B ∈ L∞(Ω)N , b ∈ L∞(Γ)N , f ∈ L2((0, T) × Ω) and g ∈ L2((0, T) × Γ). The
functions u and v are internal and boundary controls, acting on small regions ω and Γ0, respectively.
We study first the wellposedness of the above system and its associated adjoint system. To obtain boundary and null controllability, we prove adequate observability inequalities, which are obtained by establishing new internal and boundary Carleman estimates for the backward adjoint problems.