Page 665 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 665
PARTIAL DIFFERENTIAL EQUATIONS AND APPLICATIONS
y
ux(−0, y) = µ1(y)ux(+0, y) + µ2(y)uy(0, y) + µ3(y) r2(t)u(0, t)dt+
0
+ µ4(y)u(0, y) + µ5(y), 0 < y < h (4)
where θ1(x) = θ1 x ; − x , θ2(y) = θ2 − y ; y , ai(x), bi(y), (i = 1, 2, 3), λj(x), µj(y) (j =
2 2 2 2
4 4
1, 2, 3, 4, 5), ϕ1(y) are given functions, besides k=1 λ2k(x) = 0, k=1 µ2k(x) = 0,
3 a2k(x) = 0 and 3 bk2 (x) = 0.
k=1 k=1
On the certain conditions to given function, we can prove uniqueness of solution of the
Problem I applying the method of integral energy. Existence of solution, reduced to the Volterra
and Fredholm type non linear integral equation respected to uy(0, y) = τ2(y) and u(x, 0) =
τ1(x) accordingly.
Keywords: Loaded equation, Caputo operator, non-local condition, integral gluing condition,
non-linear integral equations.
Two-Dimensional Time-Fractional Telegraph Equation of Distributed
order in Polar Coordinate
Alireza Ansari, alireza_1038@yahoo.com
Department of Applied Mathematics, Faculty of Mathematical Sciences,
Shahrekord University, P.O. Box 115, Shahrekord, Islamic Republic of Iran
Coauthor: Fatemeh Alibeigi
In this paper, we propose an analytical approach for solving the two-dimensional time-fractional
telegraph equation of distributed order in the sense of the Caputo fractional derivative
1 2 1 1 , t > 0, r > 0,
ν2 urr(r, t) + r ur(r, t)
c(α)CDtαu(r, t)α + b b(α)CDtαu(r, t)dα = a2
1 0
with the initial conditions u(r, 0) = f (r), ur(r, 0) = 0. The Hankel and the Laplace transforms
are used to get an integral representation for the solution, particulary in terms of the Prabhakar
function (the Mittag-Leffler function with three parameters) for the single order. Other special
cases of the weight functions c(α), b(α) are also investigated.
Relativistic Hydrodynamics: Geometric Analysis Meets Observational
Astrophysics
Shabnam Beheshti, s.beheshti@qmul.ac.uk
Queen Mary University of London, United Kingdom
Relativistic hydrodynamics describes the motion of fluids in regimes including flow velocities
close to the speed of light (e.g., relativistic plasmas) and fluids interacting with strong gravi-
tational fields (e.g. neutron star mergers, black hole accretion disks). Mathematical research
in this area serves as an essential tool in high-energy nuclear physics, cosmology, and astro-
physics, offering opportunities for strong interplay between mathematical analysis, numerical
simulation, theoretical and experimental physics. In this talk, I shall survey recent progress
on well-posedness theorems for relativistic viscous hydrodynamics and discuss related open
problems in both mathematics and astrophysics.
663
y
ux(−0, y) = µ1(y)ux(+0, y) + µ2(y)uy(0, y) + µ3(y) r2(t)u(0, t)dt+
0
+ µ4(y)u(0, y) + µ5(y), 0 < y < h (4)
where θ1(x) = θ1 x ; − x , θ2(y) = θ2 − y ; y , ai(x), bi(y), (i = 1, 2, 3), λj(x), µj(y) (j =
2 2 2 2
4 4
1, 2, 3, 4, 5), ϕ1(y) are given functions, besides k=1 λ2k(x) = 0, k=1 µ2k(x) = 0,
3 a2k(x) = 0 and 3 bk2 (x) = 0.
k=1 k=1
On the certain conditions to given function, we can prove uniqueness of solution of the
Problem I applying the method of integral energy. Existence of solution, reduced to the Volterra
and Fredholm type non linear integral equation respected to uy(0, y) = τ2(y) and u(x, 0) =
τ1(x) accordingly.
Keywords: Loaded equation, Caputo operator, non-local condition, integral gluing condition,
non-linear integral equations.
Two-Dimensional Time-Fractional Telegraph Equation of Distributed
order in Polar Coordinate
Alireza Ansari, alireza_1038@yahoo.com
Department of Applied Mathematics, Faculty of Mathematical Sciences,
Shahrekord University, P.O. Box 115, Shahrekord, Islamic Republic of Iran
Coauthor: Fatemeh Alibeigi
In this paper, we propose an analytical approach for solving the two-dimensional time-fractional
telegraph equation of distributed order in the sense of the Caputo fractional derivative
1 2 1 1 , t > 0, r > 0,
ν2 urr(r, t) + r ur(r, t)
c(α)CDtαu(r, t)α + b b(α)CDtαu(r, t)dα = a2
1 0
with the initial conditions u(r, 0) = f (r), ur(r, 0) = 0. The Hankel and the Laplace transforms
are used to get an integral representation for the solution, particulary in terms of the Prabhakar
function (the Mittag-Leffler function with three parameters) for the single order. Other special
cases of the weight functions c(α), b(α) are also investigated.
Relativistic Hydrodynamics: Geometric Analysis Meets Observational
Astrophysics
Shabnam Beheshti, s.beheshti@qmul.ac.uk
Queen Mary University of London, United Kingdom
Relativistic hydrodynamics describes the motion of fluids in regimes including flow velocities
close to the speed of light (e.g., relativistic plasmas) and fluids interacting with strong gravi-
tational fields (e.g. neutron star mergers, black hole accretion disks). Mathematical research
in this area serves as an essential tool in high-energy nuclear physics, cosmology, and astro-
physics, offering opportunities for strong interplay between mathematical analysis, numerical
simulation, theoretical and experimental physics. In this talk, I shall survey recent progress
on well-posedness theorems for relativistic viscous hydrodynamics and discuss related open
problems in both mathematics and astrophysics.
663
