In this paper, a class of
4
t
h
-order neutral delay differential equations with continuously distributed delay is studied. We establish a new oscillation criterion using the Riccati transformation. ...An example illustrating the results is also given.
In this paper, we study Fermat-type functional equations fn+gn+hn=1 in the complex plane. Alternative proofs of the known results for entire and meromorphic solutions of such equations are given. ...Moreover, some conditions on degrees of polynomial solutions are given.
A general framework is proposed for pricing both continuously and discretely monitored Asian options under one-dimensional Markov processes. For each type (continuously monitored or discretely ...monitored), we derive the double transform of the Asian option price in terms of the unique bounded solution to a related functional equation. In the special case of continuous-time Markov chain (CTMC), the functional equation reduces to a linear system that can be solved analytically via matrix inversion. Thus the Asian option prices under a one-dimensional Markov process can be obtained by first constructing a CTMC to approximate the targeted Markov process model, and then computing the Asian option prices under the approximate CTMC by numerically inverting the double transforms. Numerical experiments indicate that our pricing method is accurate and fast under popular Markov process models, including the CIR model, the CEV model, Merton’s jump diffusion model, the double-exponential jump diffusion model, the variance gamma model, and the CGMY model.
There are many natural phenomena including a crisis (such as a spate or contest) which could be described in three steps. We investigate the existence of solutions for a three step crisis ...integro-differential equation. We suppose that the second step is a point-wise defined singular fractional differential equation.
In this article, we study the maximal robust invariant set estimation problem for discrete-time perturbed nonlinear systems within the optimal control framework. The maximal robust invariant set of ...interest is a set of all states such that every possible trajectory starting from it never violates a specified state constraint, regardless of actual disturbances. The maximal robust invariant set is shown to be the zero level set of the unique bounded solution to a Bellman type equation, which is a functional equation being widely used in discrete-time optimal control. Consequently, the maximal robust invariant set estimation problem is reduced to a problem of solving a Bellman type equation. This is the main contribution of this article. The uniqueness of bounded solutions enables us to solve the derived Bellman type equation using numerical methods such as the value iteration and policy iteration, which provide an approximation of the maximal robust invariant set. Finally, two examples demonstrate the performance of our Bellman equation based method.
We find on a monoid
M
the complex-valued solutions
f, g : M
→ ℂ such that
f
is central and
g
is continuous of the functional equation
f
(
x
σ
(
y
)
)
+
f
(
τ
(
y
)
x
)
=
2
f
(
x
)
g
(
y
)
,
x
,
y
∈
M
...,
where
σ
:
M
→
M
is an involutive automorphism and
τ
:
M
→
M
is an involutive anti-automorphism. The solutions are described in terms of multiplicative functions, additive functions and characters of 2-dimensional representations of
M
.
We prove that every positive solution to the difference equation
x
n
=
max
A
1
x
n
-
1
α
1
,
A
2
x
n
-
2
α
2
,
…
,
A
k
x
n
-
k
α
k
,
n
∈
N
0
,
where
k
∈
N
,
A
i
>
0
,
α
i
∈
(
0
,
1
)
,
i
=
1
,
…
,
k
..., converges to the following quantity
max
A
1
1
α
1
+
1
,
…
,
A
k
1
α
k
+
1
, confirming a quite recent conjecture of interest. We also prove another result on global convergence which concerns some cases when not all
α
i
,
i
=
1
,
…
,
k
belong to the interval (0,
1).
In this research work, we present a mathematical model for novel coronavirus-19 infectious disease which consists of three different compartments: susceptible, infected, and recovered under convex ...incident rate involving immigration rate. We first derive the formulation of the model. Also, we give some qualitative aspects for the model including existence of equilibriums and its stability results by using various tools of nonlinear analysis. Then, by means of the nonstandard finite difference scheme (NSFD), we simulate the results for the data of Wuhan city against two different sets of values of immigration parameter. By means of simulation, we show how protection, exposure, death, and cure rates affect the susceptible, infected, and recovered population with the passage of time involving immigration. On the basis of simulation, we observe the dynamical behavior due to immigration of susceptible and infected classes or one of these two.
This article deals with the generalization of natural convection flow of
C
u
−
A
l
2
O
3
−
H
2
O
hybrid nanofluid in two infinite vertical parallel plates. To demonstrate the flow phenomena in two ...parallel plates of hybrid nanofluids, the Brinkman type fluid model together with the energy equation is considered. The Caputo–Fabrizio fractional derivative and the Laplace transform technique are used to developed exact analytical solutions for velocity and temperature profiles. The general solutions for velocity and temperature profiles are brought into light through numerical computation and graphical representation. The obtained results show that the velocity and temperature profiles show dual behaviors for
0
<
α
<
1
and
0
<
β
<
1
where
α
and
β
are the fractional parameters. It is noticed that, for a shorter time, the velocity and temperature distributions decrease with increasing values of the fractional parameters, whereas the trend reverses for a longer time. Moreover, it is found that the velocity and temperature profiles oppositely behave for the volume fraction of hybrid nanofluids.
In this paper, we symbolically compute an extended (3+1)-dimensional variable-coefficient modified Kadomtsev-Petviashvili system for the electromagnetic waves in a certain ferromagnetic material, ...ion-acoustic/dust-ion-acoustic/dust-acoustic waves in a sort of plasma, or water waves. Particular instances of the system in magnetooptics, ferromagnetism, plasma mechanics and fluid dynamics are presented, such as one case in magnetooptics, describing the electromagnetic waves in a ferromagnetic thin charge-free isotropic film with the possible application in magnetooptic recording. Making use of symbolic computation, we figure out (1) a set of the variable-coefficient-dependent bilinear forms, (2) two sets of the variable-coefficient-dependent
N
-soliton solutions and (3) two sets of the variable-coefficient-dependent auto-Bäcklund transformations along with some solitons, with
N
denoting a positive integer. Our results, under the involved constraints, rely on the variable coefficients.
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