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Harmonic Wavelet Solutions of the Schrodinger Equation

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Carlo Cattani at University of Tuscia
Carlo Cattani
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Abstract

In this paper nonperiodic harmonic wavelets, with band-limited Fourier transform, are considered. Their connection coefficients are explicitly computed at any order, thanks to a recursive formula. As application the Schrödinger equation is solved at the lowest scale.
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HARMONIC WAVELET SOLUTIONS OF THE
SCHR
¨
ODINGER EQUATION
CARLO CATTANI
DiFarma, University of Salerno,
Via Ponte Don Melillo, 84084 Fisciano (SA), Italy
E-mail: ccattani@unisa.it
ABSTRACT
In this paper nonperiodic harmonic wavelets, with band-limited Fourier transform, are considered. Their connection co-
efficients are explicitly computed at any order, thanks to a recursive formula. As application the Schr¨odinger equation is
solved at the lowest scale.
Key words: Harmonic Wavelets, connection coefficients, band-limited functions, Schr¨odinger equation
1. INTRODUCTION
The differential structure of (nonperiodic) harmonic wavelets [9] is investigated by explicitly giving the
connection coefficients (for the periodised harmonic wavelets see also [1],[3],[2],[4],[5],[8]). Because their
band-limited property in Fourier domain, these wavelets seems to be expedient for studying oscillations in
a small range time interval. So that the time interval is associated with a scale and the wave propagation
is investigated at each given scale, by showing the contribution of the wavelet details to the evolution.
The multiscale (or multiresolution) approach is a kind of approximation that at each scale increase the
“resolution” of the solution, so that at each scale more new details are added to the solution.
Harmonic wavelets are finitely defined orthogonal basis (derived from the Hardy-Littlewood basis),
infinitely differentiable functions. They are complex functions suitable for studying wave propagation.
The wave solution is searched as a truncated (to N) wavelet series. The wavelet coefficients of the series
are characterized by some linear differential equations. The construction of these equations follows from
the explicit computation of the connection coefficients. In wave propagation, very often, the approximate
solution is expressed in terms of functions which are significant only at a given resolution, and, some time,
2 Carlo Cattani
also the exact solution (like e.g. the D’Alemb ert solution of the wave equation) shows two fundamental
characteristic features of wavelets: the dilation (multiscale) and the translation properties.
We will see that the wavelet solution of the Schr¨odinger equation can be easily obtain as a wavelet
series, by using the connection coefficients.
The connection coefficients will be computed for any order of the derivative and a useful recursive
formula will be also given.
2. HARMONIC WAVELETS
Harmonic wavelets are the complex valued functions [1],[3],[4],[5],[8],[9]
ψ
n
k
(x) ≡ 2
n/2
e
4πi (2
n
x−k)
− e
2π i(2
n
x−k)
2πi(2
n
x − k)
, (2.1)
with n, k ∈ Z (see Fig. 1). Their Fourier transform [9] are the band-limited functions (see Fig. 2)
ψ
n
k
(ω) =
2
−n/2
2π
e
−iω k/2
n
χ(ω/2
n
) , (2.2)
where χ(ω) is the box function
χ(ω) ≡
1 , 2π ≤ ω < 4π
0 , elsewhere .
(2.3)
If we compare the harmonic wavelets (2.2) with the Morlet wavelets (see e.g. [6]),
ψ
n
k
(x) ≡ 2
−n/2
π
−1/4
e
2πif
0
(x−k/2
n
)
e
−x
2
/2
, f
0
≫ 0 ,
the Fourier transform of each harmonic wavelet is a band limited function (box function in the Fourier
domain) while the Fourier transform of the Morlet wavelets are shifted Gaussian. Harmonic wavelets are
closely related, in their definition, to the Shannon wavelets (see e.g. [7]), or sinc-function based wavelets,
which are band limited (box functions in the Fourier domain) too, but they are only real functions in the
x-domain.
If we assume as scalar product, of the functions f (x) , g (x),
⟨f, g⟩ ≡
∞
−∞
f (x) g (x)dx
= 2π
∞
−∞
f (ω) g (ω)dω = 2π
f, g
,
(2.4)
where the bar stands for the complex conjugate, we can easily check that (see also [9])
Harmonic Wavelet Solutions of the Schr¨odinger Equation 3
Theorem 2.1. Harmonic wavelets are orthonormal functions in the sense that
⟨ψ
n
k
(x) , ψ
m
h
(x)⟩ = δ
nm
δ
hk
δ
nm
being the Kroenecker symbol.
Proof.
⟨ψ
n
k
(x) , ψ
m
h
(x)⟩ = 2π
∞
−∞
2
−n/2
2π
e
−iω k/2
n
χ(ω/2
n
)
2
−m/2
2π
e
iωh/2
m
χ(ω/2
m
)dω
=
2
−(n+m)/2
2π
∞
−∞
e
−iωk/2
n
χ(ω/2
n
)e
iωh/2
m
χ(ω/2
m
)dω
which is zero for n ̸= m. For n = m it is
⟨ψ
n
k
(x) , ψ
n
h
(x)⟩ =
2
−n
2π
∞
−∞
e
−iω (h−k)/2
n
χ(ω/2
n
)dω
and, according to (2.3), by the change of variable ξ = ω/2
n
⟨ψ
n
k
(x) , ψ
n
h
(x)⟩ =
1
2π
4π
2π
e
−i(h−k)ξ
dξ .
For h = k (and n = m), is trivially
⟨ψ
n
k
(x) , ψ
n
k
(x)⟩ = 1 .
For h ̸= k, it is
4π
2π
e
−i(h−k)ξ
dξ =
i
(h − k)
e
−4iπ (h −k)
− e
−2iπ (h −k)
and since
e
±4iπ (h −k)
= e
±2iπ (h−k)
= 1 , (h − k ∈ Z) , (2.5)
the proof easily follows.
3. CONNECTION COEFFICIENTS
The first and second order derivatives of harmonic wavelets are
dψ
n
k
(x)
dx
=
2
−1+3n/2
i − 4 k π + 2
2+n
π x + e
2 i π (k−2
n
x)
−i + 2 k π − 2
1+n
π x

e
4 i π (k−2
n
x)
π (k − 2
n
x)
2
,
4 Carlo Cattani
-1 1
-1 1
-1 1
-1
1
-1 1
-1
1
Figure 1. Real (thick line) and imaginary (thin line) part
of the harmonic wavelets ψ
0
0
(x),ψ
1
0
(x), (first row) and ψ
2
0
(x)
ψ
3
0
(x) (second row).
and, respectively, (Fig. 1)
d
2
ψ
n
k
(x)
dx
2
=
2
n/2
4
n
i − 8 i k
2
π
2
+ 2
2+n
π x − i 2
3+2 n
π
2
x
2
+ 4 i k π
i + 2
2+n
π x

e
4 i π (k−2
n
x)
π (k − 2
n
x)
3
+
+
2
n/2
4
n

2 i k
2
π
2
+ 2 k π
1 − i 2
1+n
π x
+ i
−1 + i 2
1+n
π x + 2
1+2 n
π
2
x
2

e
2 i π (k−2
n
x)
π (k − 2
n
x)
3
.
They are infinitely differentiable functions, since it can be easily checked that
lim
x→k/2
n
d
p
ψ
n
k
(x)
dx
p
∝ 2
(p+2)n/2
i
n
π
n
< ∞ ,
but in the x-domain the computation of higher order derivatives implies cumbersome formulas. In the
Fourier domain, instead, the derivatives are simply given by
\
d
ℓ
dx
ℓ
ψ
n
k
(x) = (iω)
ℓ
ψ
n
k
(ω)
and according to (2.2)
\
d
ℓ
dx
ℓ
ψ
n
k
(x)
= (iω)
ℓ
2
−n/2
2π
e
−iωk/2
n
χ(ω/2
n
) (3.1)
The any order connection coefficients of the harmonic wavelets, are defined as
γ
(ℓ)nm
kh
≡
d
ℓ
dx
ℓ
ψ
n
k
(x) , ψ
m
h
(x)
. (3.2)
Harmonic Wavelet Solutions of the Schr¨odinger Equation 5
2
4
8 16 32
Figure 2. The Fourier transform of the harmonic wavelets
ψ
0
0
(x),ψ
1
0
(x), and ψ
2
0
(x) ψ
3
0
(x).
They can be easily computed by the following theorem (for the first and second order connection coeffi-
cients of periodic harmonic wavelets see also [3],[1],[4],[5],[8]).
Theorem 3.1. The connection coefficients (3.2) of the harmonic wavelets (2.1) are
γ
(ℓ)nm
kh
=
i
ℓ
2
nℓ
2π
δ
kh
ξ
ℓ+1
ℓ + 1
− (1 − δ
kh
)
ℓ
j=0
a
(ℓ)
j
ξ
ℓ−j
i
h − k
j+1
4π
2π
δ
nm
(3.3)
with
a
(0)
0
= 1 , a
(ℓ)
j
=
1 , j = ℓ
ℓ a
(ℓ−1)
j
, j = 0, . . . , ℓ − 1
(ℓ ≥ 1) (3.4)
and, as usual, [F (ξ)]
ξ
1
ξ
0
= F (ξ
1
) − F (ξ
0
).
Proof. From their definition (3.2), taking into account equations (2.4)-(3.1), it is
γ
(ℓ)nm
kh
= 2 π
∞
−∞
(iω)
ℓ
2
−n/2
2π
e
−iωk/2
n
χ(ω/2
n
)
2
−m/2
2π
e
iω h/2
m
χ(ω/2
m
)dω
which is 0 when m ̸= n.
When m = n we have, with the change of variable ω/2
n
= ξ,
γ
(ℓ)nn
kh
=
i
ℓ
2
nℓ
2π
4π
2π
ξ
ℓ
e
iξ(h−k)
dξ .
So that, when k = h, we obtain
γ
(ℓ)nn
kk
=
i
ℓ
2
nℓ
2π(ℓ + 1)
ξ
ℓ+1
4π
2π
=
(iπ)
ℓ
2
ℓ(n+1)
ℓ + 1
2
ℓ+1
− 1
. (3.5)
6 Carlo Cattani
When k ̸= h, the connection coefficients for ℓ = 1, 2, 3, 4, are
γ
(1)nn
kh
=
i 2
n
2π
e
iξ(h−k)
1
(h − k)
2
−
iξ
(h − k)

4π
2π
γ
(2)nn
kh
=
−2
2n
2π
e
iξ(h−k)
2i
(h − k)
3
+
2ξ
(h − k)
2
−
iξ
2
(h − k)

4π
2π
γ
(3)nn
kh
=
−i 2
3n
2π
e
iξ(h−k)
−
6
(h − k)
4
+
6iξ
(h − k)
3
+
3ξ
2
(h − k)
2
−
iξ
3
(h − k)

4π
2π
γ
(4)nn
kh
=
2
4n
2π
e
i ξ (h−k)
−24 i
(h − k)
5
−
24 ξ
(h − k)
4
+
12 i ξ
2
(h − k)
3
+
4 ξ
3
(h − k)
2
−
i ξ
4
(h − k)

4π
2π
.
(3.6)
In general, since
4π
2π
ξ
ℓ
e
iξ(h−k)
dξ =
−e
iξ(h−k)
ℓ
j=0
a
(ℓ)
j
ξ
ℓ−j
i
h − k
j+1
4π
2π
,
with a
(ℓ)
j
given by (3.4), there follows, according to (2.5),
γ
(ℓ)nn
kh
= −
i
ℓ
2
nℓ
2π
ℓ
j=0
a
(ℓ)
j
ξ
ℓ−j
i
h − k
j+1
4π
2π
, (h ̸= k) .
3.1. Recursive equations for the connection coefficients
The connection coefficients (3.3) of different orders are not independent. In fact, they can be constructed
according to the following:
Theorem 3.2. The connection coefficients (3.3) are recursively given by
γ
(ℓ+1)nm
kh
=
δ
kh
(iπ) 2
n+1
(ℓ + 1)(2
ℓ+2
− 1)
(ℓ + 2)(2
ℓ+1
− 1)
+ (1 − δ
kh
)
−2
n
ℓ
(h − k)
γ
(ℓ)nm
kh
+
i
ℓ
π
ℓ−1
2
ℓ(n+1)+n−1
(h − k)
2
ℓ
− 1
(1 − δ
kh
) δ
nm
γ
(1)nm
kh
=
3 (iπ) 2
n
δ
kh
+ (1 − δ
kh
)
2
n
(h − k)
δ
nm
(3.7)
Proof. Assuming n = m, let us first consider the case h ̸= k. It can be easily seen by a direct computation
(but also from the explicit values of γ
(ℓ)nn
kh
in (3.6)) that the polynomials
ε
(ℓ)
hk
(ξ) ≡ −
ℓ
j=0
a
(ℓ)
j
ξ
ℓ−j
i
h − k
j+1
Harmonic Wavelet Solutions of the Schr¨odinger Equation 7
fulfill the recursive equations
ε
(ℓ+1)
hk
(ξ) =
i ℓ
(h − k)
ε
(ℓ)
hk
(ξ) − e
iξ(h−k)
i ξ
ℓ
(h − k)
ε
(1)
hk
(ξ) = e
iξ(h−k)
1
(h − k)
2
−
iξ
(h − k)
so that (when h ̸= k) it is
γ
(ℓ+1)nn
kh
=
i
(ℓ+1)
2
n(ℓ+1)
2π
ε
(ℓ+1)
hk
(ξ)
4π
2π
=
i
(ℓ+1)
2
n(ℓ+1)
2π
i ℓ
(h − k)
ε
(ℓ)
hk
(ξ) − e
iξ(h−k)
i ξ
ℓ
(h − k)
4π
2π
=
−2
n
ℓ
(h − k)
i
ℓ
2
nℓ
2π
ε
(ℓ)
hk
(ξ)
4π
2π
+
i
ℓ
2
n(ℓ+1)
2π
e
iξ(h−k)
ξ
ℓ
(h − k)
4π
2π
.
Since
e
iξ(h−k)
ξ
ℓ
(h − k)
4π
2π
= (2π)
ℓ
2
ℓ
− 1
/(h − k), and
e
iξ(h−k)
1
(h − k)
2
−
iξ
(h − k)

4π
2π
= −
2 i π
(h − k)
we finally obtain
γ
(ℓ+1)nn
kh
=
−2
n
ℓ
(h − k)
γ
(ℓ)nn
kh
+
i
ℓ
π
ℓ−1
2
ℓ(n+1)+n−1
(h − k)
2
ℓ
− 1
γ
(1)nn
kh
=
2
n
(h − k)
.
Analogously we can easily derive the recursive formula when k = h. From (3.3) it is
γ
(ℓ+1)nn
kk
=
(iπ)
ℓ+1
2
(ℓ+1)(n+1)
ℓ + 2
2
ℓ+2
− 1
from where
γ
(ℓ+1)nn
kk
= (iπ) 2
n+1
(ℓ + 1)(2
ℓ+2
− 1)
(ℓ + 2)(2
ℓ+1
− 1)
γ
(ℓ)nn
kk
γ
(1)nn
kk
= 3 (iπ) 2
n
.
There follows the explicit recursive formula (3.7) for any value of the indices.
In particular, taking into account that k = 0, . . . 2
n
−1, h = 0, . . . 2
m
−1 we have for the first order (ℓ = 1)
coefficients γ
′nm
kh
, from (3.3), or equivalently from (3.6)
1
and (3.5), at the lower scales 0 ≤ n = m ≤ 2:
γ
′00
00
= 3πi
8 Carlo Cattani
γ
′11
kh
=
6 i π −2
2 6 i π
γ
′22
kh
=
12 i π −4 −2 −4/3
4 12 i π −4 −2
2 4 12 i π −4
4/3 2 4 12 i π
γ
′33
kh
=
24 i π −8 −4 −
8
3
−2 −8/5 −4/3 −8/7
8 24 i π −8 −4 −8/3 −2 −8/5 −
4
3
4 8 24 i π −8 −4 −
8
3
−2 −8/5
8/3 4 8 24 i π −8 −4 −8/3 −2
2 8/3 4 8 24 i π −8 −4 −8/3
8/5 2 8/3 4 8 24 i π −8 −4
4/3 8/5 2 8/3 4 8 24 i π −8
8/7 4/3 8/5 2 8/3 4 8 24 i π
Analogously, we have, for the connection coefficients of the second derivative, at the lower scales
0 ≤ n = m ≤ 2:
γ
′′00
00
= −28π
2
/3
γ
′′11
kh
=
−112 π
2
3
−2
(
4 π +12 i π
2
)
π
−2
(
4 π −12 i π
2
)
π
−112 π
2
3
γ
′′22
kh
=
−448 π
2
3
−8
(
4 π +12 i π
2
)
π
−8
(
π +6 i π
2
)
π
−8
(
4 π
9
+4 i π
2
)
π
−8
(
4 π −12 i π
2
)
π
−448 π
2
3
−8
(
4 π +12 i π
2
)
π
−8
(
π +6 i π
2
)
π
−8
(
π −6 i π
2
)
π
−8
(
4 π −12 i π
2
)
π
−448 π
2
3
−8
(
4 π +12 i π
2
)
π
−8
(
4 π
9
−4 i π
2
)
π
−8
(
π −6 i π
2
)
π
−8
(
4 π −12 i π
2
)
π
−448 π
2
3
4. HARMONIC WAVELET SOLUTION OF THE SCHR
¨
ODINGER EQUATION
Let us consider the Schr¨odinger equation:
i
∂Ψ
∂t
= −
~
2m
∂
2
Ψ
∂x
2
, x ∈ R , t > 0 (4.1)
~ being the Planck constant. As initial condition we take
Ψ(x, 0) = W (x) . (4.2)
Harmonic Wavelet Solutions of the Schr¨odinger Equation 9
If we assume that the solution can be expressed as harmonic wavelet series
Ψ(x, t) =
N−1
n=0
2
n−1
−1
k=0
β
n
k
(t)ψ
n
k
(x)
by a substitution into (4.1), and using the connection coefficients, we have, at the resolution N, the
following Cauchy problem in the wavelet coefficients
i
dβ
n
k
(t)
dt
= −
~
2m
N−1
m=0
2
m−1
−1
h=0
β
m
h
(t)γ
(2)mn
hk
,
W (x) =
N−1
n=0
2
n−1
−1
k=0
β
n
k
(0)ψ
n
k
(x)
For instance, if we take as initial condition W (x) = ψ
0
0
(x) so that N = 1 and β
0
0
= 1, we have the
equation
i
dβ
0
0
(t)
dt
= −
~
2m
β
0
0
(t)γ
(2)00
00
,
W (x) = β
0
0
(0)ψ
0
0
(x)
that is,
i
dβ
0
0
(t)
dt
= −
~
2m
−28π
2
/3
β
0
0
(t) ,
β
0
0
(0) = 1
whose solution is
β
0
0
(t) = e
−14i~π
2
t
3m
.
Therefore at the level N = 1, we have the solution (see Figs. 3-4)
Ψ(x, t) = e
(−14i~π
2
t)/(3m)
ψ
0
0
(x) .
It can be easily checked that this solution coincide with the one obtained by the classical method of
similarity, but it is obtained in a conceptual scheme different in the sense that the solution can be
“improved” by increasing the resolution level, i.e. adding more details to the evolving initial profile.
CONCLUSION
In this paper, some differential properties of non-periodic harmonic wavelets are given and discussed.
The any order connection coefficients are explicitly defined by a recursive formula. As application the
multi-level solution of the Schr¨odinger equation is given rediscovering at the lowest level the classical
10 Carlo Cattani
solution (usually obtained by the method of similarity). This method could be easily used for a further
investigation of the multiscale properties of the Schr¨odinger equation in order to show the contribution
of the higher levels.
Figure 3. Real part of the harmonic solution at the level N = 0 of the
Schr¨odinger equation.
-2
1-1
2
-1
1
0.3
-1
1
Figure 4. Projection of the harmonic solution on the (x, Ψ) plane (left) and on the (t, Ψ) plane of
the real part of the harmonic solution, at the level N = 0, of the Schr¨odinger equation.
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Harmonic Wavelet Solutions of the Schr¨odinger Equation 11
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[3] C. Cattani. Wavelets in the propagation of waves in materials with microstructure. In: Proc. of the Intern. Conference
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... then, we can easily get Fourier transform for (15) and (16), i.e., Φ HW (ω) = ℵ (ω + 2π) ...
... to(21), (28), (42), (43), results the coefficients α HW , β HW and their complex conjugatesᾱ HW ,β HW (for more technical details of computation see[11] [16][15] ...
The 8 th INTERNATIONAL SYMPOSIUM ON ADVANCED TOPICS IN ELECTRICAL ENGINEERING Wavelet Solution of the Time Independent Schrödinger Equation for a Rectangular Potential Barrier
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  • May 2023
Electronics industry is a major user of Metal- Insulator-Metal (MIM) structures. When the tunneling phenomenon occurs at the metal-insulator contact barrier, as for the MIM structures, we are dealing with the Fowler-Nordheim field emission. In MIM structures we have to calculate the currents densities that go through. Unfortunately, the calculation methods currently used lead to values that get predictions of emission current density too low. The current density is calculated based on the transmission coefficient through the barrier, which is determined by solving the Schr¨odinger equation. One of the oldest and most efficient methods of solving the Schr¨odinger equation is the WKB method. This paper proposes the wavelet method for solving the 1D Schr¨odinger equation, that is: determining the wave function using the harmonic multiresolution analysis.
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... According to [27,28], the NETLM can be written in the form where 1 , 2 denote respectively to the space in the longitudinal and transvers directions, while V = V(x, y, t) denotes to the voltage in lines and , , w 0 ,w 0 are non-zero real values. In literature the above model illustrates the wave propagation on the network lines and in the last few decays it had been discussed through [46][47][48]. Now, let the linear transformation V( ) = V(x, y, t) where = k(x + y − ct) by which Eq. (1) will be converted to This article is arranged as follow: Sections 2, 3 contain the mathematical analysis for the PPAM and its applications respectively. ...
New Perceptions for the Soliton Solutions to the Complex Wave Patterns Model Against its Numerical Solutions
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Full-text available
  • May 2024
  • INT J THEOR PHYS
The main target of this article is extracting new various types of the solitons emerging from the complex wave patterns which is famous model arising in telecommunications engineering line. These new visions of these solitons will be constructed in the framework of three impressive techniques that are used for the first time for this purpose. The three suggested different techniques are the Paul-Painlevé approach method (PPAM), the Ricatti-Bernolli Sub ODE method (RBSOM) and the variational iteration method VIM. The first and second techniques are the famous two ansatze methods while the third one is the famous numerical method. Many new different perceptions of the soliton solutions to the proposed model through these different algorithms that are implemented individually and parallel have been documented. Furthermore, the comparison that isn't only with the solutions achieved by these suggested techniques but also with that realized before via other authors who used various algorithms has been demonstrated.
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... In the¯eld of nonlinear science, the nonlinear Schr€ odinger equations (NL-SEs), with their diverse range of applications, are most frequently employed nonlinear models. [1][2][3] The most intriguing area of research in nonlinear wave propagation in optical¯bers right now is optical solitons because of their remarkable stability qualities. [4][5][6][7][8][9] The non-linearity principles in nonlinear optics explain how optical solitons appear in multiple dimensions. ...
Numerous optical soliton solutions of the Triki–Biswas model arising in optical fiber
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A model condition for the propagation of ultrashort pulses in optical fiber frameworks could be established through the extensive generalization of the nonlinear Schrödinger equation introduced by Triki and Biswas. This study uses two significant and effective trustworthy methods to examine the Triki–Biswas (TB) equation. Sardar Subequation Methods (SSM) and the generalized Kudryashov (KUD) method generate solutions for trigonometric, hyperbolic, exponential, and rational functions. These approaches are specifically designed for handling solitary wave patterns, dark-singular-mixed solitons, combined dark-bright solitons, singletons, bright solitons, exponential and rational functions, as well as periodic solitons. All of these solution classes contribute to the physical dynamics of outcomes. These findings represent an innovative extension of the soliton domain within the TB model and are being reported for the first time in our investigation. In addition, Mathematica simulations are used to display the 3D and 2D graphs to explain the identified solutions’ physical dynamics.
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... These techniques have made great strides in the fields of numerical analysis and approximation theory because of their low complexity, computational efficiency, and speedy convergence. Haar wavelets [16,19], Bernoulli wavelets [20], harmonic wavelets [21], ultraspherical wavelets [22], Legendre wavelets [20], Laguerre wavelets [23], Chebyshev wavelets [17], and Euler wavelets [24] are all examples of wavelet families that are useful in addressing a wide range of physical, engineering, and biological issues [25,26]. ...
Fibonacci wavelet method for time fractional convection–diffusion equations
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  • Nov 2023
  • MATH METHOD APPL SCI
This study concentrates on time fractional convection–diffusion equations (TFCDEs) with variable coefficients and their numerical solutions. Caputo derivative is used to calculate the time fractional order derivatives. In order to give an approximate solution to the TFCDE, an effective approach is proposed utilizing Fibonacci wavelet and block pulse functions. The Fibonacci wavelets operational matrices of fractional order integration are constructed. By combining the collocation technique, they are used to simplify the fractional model to a collection of algebraic equations. The suggested approach is quite practical for resolving issues of this nature. The comparison and analysis with other approaches demonstrate the effectiveness and precision of the suggested approach.
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... It would be very interesting to determine the analytical localized modes because this would provide an extended interpretation of system dynamics based upon existence modes [2][3][4]7,27] . There are several theoretical methods to study nonlinear localized excitations in the generalized higher-order nonlinear Schrödinger dynamical model with higher order dispersion and nonlinear terms and its applications [5,9,15,17] . ...
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  • Aug 2021
  • APPL MATH COMPUT
This work completes and extends the Ref. Tchakoutio Nguetcho et al. (2017), in which we have focused our attention only on the dynamic behavior of gap soliton solutions of the anharmonic Klein-Gordon model immersed in a parameterized on-site substrate potential. We expand our work now inside the permissible frequency band. These considerations have crucial effects on the response of nonlinear excitations that can propagate along this model. Moreover, working in the allowed frequency band is not only interesting from a physical point of view, it also provides an extraordinary mathematical model, a new class of differential equations possessing vital parameters and vertical singular straight lines. The dynamics around these singularities gives new informations of great interest, such as to better understand the break up (rupture) of a stretched polymer chain by pulling and its relation to soliton destruction and which until now had no mathematical explanations yet. The mathematical model of the differential equation we obtain corresponds, as an equivalent experimental model, to the nonlinear transmission electrical line introduced in 2009 in Ref. Yemélé and Kenmogné(2009), whose equations were corrected but not entirely resolved in 2016 in Ref. Yamgoué and Pelap(2016). Our dynamic study thus presents a theoretical prediction for their experimental set up. An extensive account of the bifurcation theory for modulated-wave propagation is given. By investigating the dynamical behavior and bifurcations of solutions of the planar dynamical systems, we not only make a total inventory of the solutions that can include such differential equations, but we derive a variety of exotic solutions corresponding to each phase trajectories as well as their different conditions of existence.
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... In the last several decades, many experts have directed their studies to the deeper physical meanings of real-world problems (Cattani 2012(Cattani ,2003a(Cattani ,2003bSingh et al. 2020;Zhang et al. 2015;Wang et al. 2017;Ciancio et al. 2008;Yokus et al. 2018;Cordero et al. 2019;Paez et al. 2019;Eskitascioglu et al. 2019;Ciancio 2007;Durur et al. 2020;Cattani and Pierro 2013;Zamir et al. 2020;Yokus and Bulut 2018;Bulut et al. 2016;Gao et al. 2020a,b,c;Ozer 2020;Baskonus and Gómez-Aguilar 2019;Guirao et al. 2020). In this sense, one of the most studied properties such problems is soliton theory. ...
Investigations of the complex wave patterns to the generalized Calogero–Bogoyavlenskii–Schiff equation
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Full-text available
  • May 2021
  • SOFT COMPUT
This paper focuses on the generalized Calogero–Bogoyavlenskii–Schiff equation to extract new complex solutions by using two analytical methods, namely, Bernoulli sub-equation function method, Modified exponential function method. For better understanding of physical meanings of solutions, simulations are reported by using a computational package program. Moreover, strain conditions for validity of complex solutions are also archived. Finally, a conclusion part completes the paper by mentioning the novelties of paper.
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Shannon Wavelets Theory
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  • Jan 2009
  • MATH PROBL ENG
Shannon wavelets are studied together with their differential properties (known as connection coefficients). It is shown that the Shannon sampling theorem can be considered in a more general approach suitable for analyzing functions ranging in multifrequency bands. This generalization coincides with the Shannon wavelet reconstruction of L 2 ( ℝ ) functions. The differential properties of Shannon wavelets are also studied through the connection coefficients. It is shown that Shannon wavelets are C ∞ -functions and their any order derivatives can be analytically defined by some kind of a finite hypergeometric series. These coefficients make it possible to define the wavelet reconstruction of the derivatives of the C ℓ -functions.
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A New Survey to the Nonlinear Electrical Transmission Line Model
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  • Nov 2021
In this paper, we find new travelling wave solutions to the nonlinear electrical transmission line model. Exponential function method is applied. This model is used to the voltage behaviors in electrical transmission lines. Under a suitable choice of control parameters, we plot the figures along with contour surfaces of new solutions by using computational programs such as Mathematica, Maple and Matlab. Some information about the behavior of voltage on the electrical lines is extracted. The hyperbolic, periodic or singular properties are reported.
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Computational Investigation of Stefan Blowing Effect on Flow of Second-Grade Fluid Over a Curved Stretching Sheet
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  • Jun 2021
Non-Newtonian fluids have extensive range of applications in the field of industries like plastics processing, manufacturing of electronic devices, lubrication flows, medicine and medical equipment. Stimulated from these applications, a theoretical analysis is carried out to scrutinize the flow of a second-grade liquid over a curved stretching sheet with the impact of Stefan blowing condition, thermophoresis and Brownian motion. The modelled governing equations for momentum, thermal and concentration are deduced to a system of ordinary differential equations by introducing suitable similarity transformations. These reduced equations are solved using Runge–Kutta–Fehlberg fourth fifth order method (RKF-45) by adopting shooting technique. The solutions for the flow, heat and mass transference features are found numerically and presented with the help of graphical illustrations. Results reveal that, curvature and Stefan blowing parameters have propensity to rise the heat transfer. Further, second grade fluid shows high rate of mass and heat transfer features when related to Newtonian fluid for upsurge in values of Brownian motion parameter.
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Multiscale analysis of wave propagation in composite materials
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Full-text available
  • Jan 2003
The multiscale solution of the Klein‐Gordon equations in the linear theory of (two‐phase) materials with microstructure is defined by using a family of wavelets based on the harmonic wavelets. The connection coefficients are explicitly computed and characterized by a set of differential equations. Thus the propagation is considered as a superposition of wavelets at different scale of approximation, depending both on the physical parameters and on the connection coefficients of each scale. The coarse level concerns with the basic harmonic trend while the small details, arising at more refined levels, describe small oscillations around the harmonic zero‐scale approximation.
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The Wavelet-Based Technique in Dispersive Wave Propagation
Article
Full-text available
  • Jan 2003
The paper expounds a wavelet-based technique. The techniques is applied to the dispersive wave equation by describing wave propagation by a composition of fundamental (harmonic) wavelets. The linear Klein–Gordon equation is analyzed and associated approximate wavelet solutions are considered for fixed resolution levels. Discretized wavelet families are computed explicitly using a basis of time harmonic wavelets. Some applications at various low resolution levels show that the technique proposed provides new opportunities for wave propagation analysis
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Harmonic Wavelet Analysis
Article
A new harmonic wavelet is suggested. Unlike wavelets generated by discrete dilation equations, whose shape cannot be expressed in functional form, harmonic wavelets have the simple structure w(x) = {exp(i4π x)-exp(i2π x)}/i2π x. This function w(x) is concentrated locally around x = 0, and is orthogonal to its own unit translations and octave dilations. Its frequency spectrum is confined exactly to an octave band so that it is compact in the frequency domain (rather than in the x domain). An efficient implementation of a discrete transform using this wavelet is based on the fast Fourier transform (FFT). Fourier coefficients are processed in octave bands to generate wavelet coefficients by an orthogonal transformation which is implemented by the FFT. The same process works backwards for the inverse transform.
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Galerkin modeling of the Burgers equation using harmonic wavelets
Article
  • Nov 1997
  • PHYS LETT A
In this paper we present a wavelet Galerkin approximation of the one-dimensional Burgers equation using complex harmonic wavelets. We derive exact expressions for the connection coefficients for the linear dissipation and nonlinear advection terms. A large reduction in the number of space-scale nonlinear modes is obtained by exploiting the band-limited spectra and localisation properties of harmonic wavelets. We discuss the advantages and disadvantages of implementing harmonic wavelets as opposed to the more standard choice of wavelets basis, highlighting the relevance to shell models of turbulence.
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Ten Lectures On Wavelets
Book
  • Jan 1992
Scitation is the online home of leading journals and conference proceedings from AIP Publishing and AIP Member Societies
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Wavelets in the propagation of waves in materials with microstructure
Conference Paper
  • Sep 2003
The analysis of evolution differential equations (parabolic-hyperbolic) might be considered within the framework of the (harmonic) wavelet theory. In fact, the multiresolution analysis of wavelets seems to be a suitable scheme for the investigation of phenomena, which appears at different scales of approximation. Very often, the approximate solution is expressed in terms of functions which are significant only at a given resolution, and, some time, also the exact solution (like e.g. the D'Alembert solution of the wave equation) shows two characteristic features of wavelets: the dilation (multiscale) and the translation properties. However, from physical point of view, the wavelets still have little interpretations, especially concerning the small details expressing small deviations nearby the steady solution, since there are only a few examples of physical propagation of wavelets. In particular, the wavelet solutions of the dispersive (Klein-Gordon) wave equation show that the multiresolution approach is a kind of approximation that at each (scale) step increases the "resolution" of the solution. Thus it seems interesting to investigate this multilevel process that, at each scale (level), adds some more details to the solution. As application, the wavelet solution of the Klein Gordon equations for materials with microstructure, is defined as follows: the dispersive wave solution of the propagation equation is interpreted as a superposition of "small" waves on a basic wave. So that the wave propagation will be investigated at each given resolution, by showing that the "minor" details of the solution, neglectable at the initial time, have a significant influence on the solution on a long (time) range.
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Atti Accademia Peloritana dei Pericolanti, Classe I, Scienze Fis
  • Jan 2002
  • 1-20
  • C Cattani
  • A Ciancio
C. Cattani and A. Ciancio. Wavelet analysis of linear transverse acoustic waves. Atti Accademia Peloritana dei Pericolanti, Classe I, Scienze Fis.Mat. e Nat., 80, 1-20, 2002.
On the propagation of transverse acoustic waves in the form of harmonic wavelets in isotropic media
  • Jan 2003
  • 249-262
  • V Ciancio
  • C Cattani
V. Ciancio and C. Cattani. On the propagation of transverse acoustic waves in the form of harmonic wavelets in isotropic media. In: Grigorios Tsagas(Ed.), Proc. of the Conference on Applied Geometry-General Relativity and the Workshop of Global Analysis, Differential Geometry and Lie Algebras, Bucharest, Romania, 2003, BSG Proceedings 9, 249-262, 2003.