Classical and generalized of a mixed problem– solutions for a non-homogeneous wave equation

2019 ◽  
Vol 484 (1) ◽  
pp. 18-20
Author(s):  
A. P. Khromov ◽  
V. V. Kornev
Keyword(s):  
Wave Equation ◽  
Fourier Series ◽  
Mixed Problem ◽  
Homogeneous Equation ◽  
Generalized Solution ◽  
Problem Solution ◽  
Homogeneous Wave ◽  
Homogeneous Wave Equation ◽  
Explicit Expressions

This study follows A.N. Krylov’s recommendations on accelerating the convergence of the Fourier series, to obtain explicit expressions of the classical mixed problem–solution for a non-homogeneous equation and explicit expressions of the generalized solution in the case of arbitrary summable functions q(x), ϕ(x), y(x), f(x, t).

scholarly journals Mixed Problem for a Homogeneous Wave Equation with a Nonzero Initial Velocity and a Summable Potential

2020 ◽  
Vol 20 (4) ◽  
pp. 444-456
Author(s):  
Vitalii P. Kurdyumov ◽  
◽  
Avgust P. Khromov ◽  
Victoria A. Khalova ◽  
◽  
...  
Keyword(s):  
Wave Equation ◽  
Mixed Problem ◽  
Initial Velocity ◽  
Formal Solution ◽  
Fourier Method ◽  
Generalized Solution ◽  
Initial Position ◽  
Homogeneous Wave ◽  
Homogeneous Wave Equation ◽  
Summable Potential

For a mixed problem defined by a wave equation with a summable potential equal-order boundary conditions with a derivative and a zero initial position, the properties of the formal solution by the Fourier method are investigated depending on the smoothness of the initial velocity u′t(x, 0) = ψ(x). The research is based on the idea of A. N. Krylov on accelerating the convergence of Fourier series and on the method of contour integrating the resolvent of the operator of the corresponding spectral problem. The classical solution is obtained for ψ(x) ∈ W1p (1 < p ≤ 2), and it is also shown that if ψ(x) ∈ Lp[0, 1] (1 ≤ p ≤ 2), the formal solution is a generalized solution of the mixed problem.

scholarly journals A minimization approach to the wave equation on time-dependent domains

2020 ◽  
Vol 13 (4) ◽  
pp. 425-436 ◽  
Author(s):  
Gianni Dal Maso ◽  
Lucia De Luca
Keyword(s):  
Wave Equation ◽  
Weak Solutions ◽  
Space Time ◽  
Time Dependent ◽  
Existence Of Weak Solutions ◽  
Homogeneous Wave ◽  
Minimization Approach ◽  
Homogeneous Wave Equation

AbstractWe prove the existence of weak solutions to the homogeneous wave equation on a suitable class of time-dependent domains. Using the approach suggested by De Giorgi and developed by Serra and Tilli, such solutions are approximated by minimizers of suitable functionals in space-time.

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