Fractional-order Bernoulli wavelets and their applications

Rahimkhani, Parisa; Ordokhani, Yadollah; Babolian, E.

doi:10.1016/j.apm.2016.04.026

Cited by 87 publications

(34 citation statements)

References 34 publications

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“…Definition The integral of order

ν \geq 0

(fractional) according to Riemann‐Liouville is given by

I^{normalν} f (x) = true{\begin{array}{l} left & \frac{1}{Γ (ν)} \int_{0}^{x} \frac{f false(s false)}{{false(x - s false)}^{1 - normalν}} d s = \frac{1}{Γ (ν)} x^{ν - 1} * f (x), & ν > 0, x > 0, \\ left & f (x), & ν = 0, \end{array}

where

x^{ν - 1} * f (x)

is the convolution product of

x^{ν - 1}

and

f (x)

.…”

Section: Preliminaries and Notationmentioning

confidence: 99%

“…Definition Caputo's fractional derivative of order

normalν

is defined as

D^{normalν} f (x) = \frac{1}{Γ (n - ν)} \int_{0}^{x} \frac{f^{false(n false)} false(s false)}{{false(x - s false)}^{normalν + 1 - n}} d s, n - 1 < ν \leq n, n \in ℕ .

…”

Section: Preliminaries and Notationmentioning

confidence: 99%

“…Wavelets constitute a family of functions constructed by performing translation and dilation on a single function

ψ (t),

where

ψ (t)

is a mother wavelet. When the dilation parameter

a

and the translation parameter

b

vary continuously, we have the following family of continuous wavelets

{normalψ}_{a, b} (t)

{normalψ}_{a, b} (t) = | a |^{- \frac{1}{2}} ψ (\frac{t - b}{a}), a, b \in ℝ, a \neq 0.

If we restrict the parameters

a

and

b

to discrete values as

a = a_{0}^{- k}

b = n b_{0} a_{0}^{- k}

a_{0} > 1

b_{0} > 0

, where

n

and

k

are positive integers, we have the following family of discrete wavelets

{normalψ}_{k, n} (t) = | a_{0} |^{k / 2} ψ (a_{0}^{k} t - n b_{0}),

where

{normalψ}_{k, n} (t)

…”

Section: Preliminaries and Notationmentioning

confidence: 99%

“…The Bernoulli wavelets

{normalψ}_{n, m} (t) = ψ (k, \hat{n}, m, t)

involve four arguments,

\hat{n} = n - 1

n = 1, 2, \dots, 2^{k - 1}, k

is assumed any positive integer,

m

is the degree of the Bernoulli polynomials and

t

is the normalized time. They are defined on the interval [0, 1) by

{normalψ}_{n, m} (t) = true{\begin{array}{l} left & 2^{\frac{k - 1}{2}} {\overset{β}{true}}_{m} (2^{k - 1} t - \hat{n}), & \frac{\overset{n}{true}}{2^{k - 1}} \leq t < \frac{\hat{n} + 1}{2^{k - 1}}, \\ left & 0, & otherwise, \end{array}

where

{\overset{β}{true}}_{m} (t) = true{\begin{array}{l} left & 1, & m = 0, \\ left & \frac{1}{\sqrt{\frac{{(- 1)}^{m - 1} {(m!)}^{2}}{(2 m)!} {normalβ}_{2 m}}} ... \end{array}

…”

Section: Preliminaries and Notationmentioning

confidence: 99%

See 3 more Smart Citations

A numerical scheme based on Bernoulli wavelets and collocation method for solving fractional partial differential equations with Dirichlet boundary conditions

Rahimkhani

Ordokhani

2018

Numerical Methods Partial

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In this paper, an efficient and accurate numerical method is presented for solving two types of fractional partial differential equations. The fractional derivative is described in the Caputo sense. Our approach is based on Bernoulli wavelets collocation techniques together with the fractional integral operator, described in the Riemann‐Liouville sense. The main characteristic behind this approach is to reduce such problems to those of solving systems of algebraic equations, which greatly simplifies the problem. By using Newton's iterative method, this system is solved and the solution of fractional partial differential equations is achieved. Some results concerning the error analysis are obtained. The validity and applicability of the method are demonstrated by solving four numerical examples. Numerical examples are presented in the form of tables and graphs to make comparisons with the results obtained by other methods and with the exact solutions much easier.

show abstract

“…Definition The integral of order

ν \geq 0

(fractional) according to Riemann‐Liouville is given by

I^{normalν} f (x) = true{\begin{array}{l} left & \frac{1}{Γ (ν)} \int_{0}^{x} \frac{f false(s false)}{{false(x - s false)}^{1 - normalν}} d s = \frac{1}{Γ (ν)} x^{ν - 1} * f (x), & ν > 0, x > 0, \\ left & f (x), & ν = 0, \end{array}

where

x^{ν - 1} * f (x)

is the convolution product of

x^{ν - 1}

and

f (x)

.…”

Section: Preliminaries and Notationmentioning

confidence: 99%

“…Definition Caputo's fractional derivative of order

normalν

is defined as

D^{normalν} f (x) = \frac{1}{Γ (n - ν)} \int_{0}^{x} \frac{f^{false(n false)} false(s false)}{{false(x - s false)}^{normalν + 1 - n}} d s, n - 1 < ν \leq n, n \in ℕ .

…”

Section: Preliminaries and Notationmentioning

confidence: 99%

“…Wavelets constitute a family of functions constructed by performing translation and dilation on a single function

ψ (t),

where

ψ (t)

is a mother wavelet. When the dilation parameter

a

and the translation parameter

b

vary continuously, we have the following family of continuous wavelets

{normalψ}_{a, b} (t)

{normalψ}_{a, b} (t) = | a |^{- \frac{1}{2}} ψ (\frac{t - b}{a}), a, b \in ℝ, a \neq 0.

If we restrict the parameters

a

and

b

to discrete values as

a = a_{0}^{- k}

b = n b_{0} a_{0}^{- k}

a_{0} > 1

b_{0} > 0

, where

n

and

k

are positive integers, we have the following family of discrete wavelets

{normalψ}_{k, n} (t) = | a_{0} |^{k / 2} ψ (a_{0}^{k} t - n b_{0}),

where

{normalψ}_{k, n} (t)

…”

Section: Preliminaries and Notationmentioning

confidence: 99%

“…The Bernoulli wavelets

{normalψ}_{n, m} (t) = ψ (k, \hat{n}, m, t)

involve four arguments,

\hat{n} = n - 1

n = 1, 2, \dots, 2^{k - 1}, k

is assumed any positive integer,

m

is the degree of the Bernoulli polynomials and

t

is the normalized time. They are defined on the interval [0, 1) by

{normalψ}_{n, m} (t) = true{\begin{array}{l} left & 2^{\frac{k - 1}{2}} {\overset{β}{true}}_{m} (2^{k - 1} t - \hat{n}), & \frac{\overset{n}{true}}{2^{k - 1}} \leq t < \frac{\hat{n} + 1}{2^{k - 1}}, \\ left & 0, & otherwise, \end{array}

where

{\overset{β}{true}}_{m} (t) = true{\begin{array}{l} left & 1, & m = 0, \\ left & \frac{1}{\sqrt{\frac{{(- 1)}^{m - 1} {(m!)}^{2}}{(2 m)!} {normalβ}_{2 m}}} ... \end{array}

…”

Section: Preliminaries and Notationmentioning

confidence: 99%

See 2 more Smart Citations

A numerical scheme based on Bernoulli wavelets and collocation method for solving fractional partial differential equations with Dirichlet boundary conditions

Rahimkhani

Ordokhani

2018

Numerical Methods Partial

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show abstract

“…Also, we use a set of 2D‐FOBPs for solving nonlinear fractional partial Volterra integro‐differential equations. The authors presented the fractional‐order Bernoulli wavelets (FBWs) and the fractional‐order Bernoulli functions (FBFs), by writing t → t α , α > 0 in the standard Bernoulli wavelets and Bernoulli polynomials, to find numerical solution of fractional differential equations . Yuzbasi constructed the truncated fractional Bernstein series for solving the fractional Riccati‐type differential equations.…”

Section: Introductionmentioning

confidence: 99%

Fractional‐order orthogonal Bernstein polynomials for numerical solution of nonlinear fractional partial Volterra integro‐differential equations

Mirzaee

Alipour

2019

Math Methods in App Sciences

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In this paper, a new two‐dimensional fractional polynomials based on the orthonormal Bernstein polynomials has been introduced to provide an approximate solution of nonlinear fractional partial Volterra integro‐differential equations. For this aim, the fractional‐order orthogonal Bernstein polynomials (FOBPs) are constructed, and its operational matrices of integration, fractional‐order integration, and derivative in the Caputo sense and product operational matrix are derived. These operational matrices are utilized to reduce the under study problem to a nonlinear system of algebraic equations. Using the approximation of FOBPs, the convergence analysis and error estimate associated to the proposed problem have been investigated. Finally, several examples are included to clarify the validity, efficiency, and applicability of the proposed technique via FOBPs approximation.

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An efficient wavelet method for the time‐fractional Black–Scholes equations

Yuttanan,

Razzaghi,

2024

Math Methods in App Sciences

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A European option is one of the common types of options in financial markets, which can be modeled by a time‐fractional parabolic PDE, known as the time‐fractional Black–Scholes equation (BSE). In this article, we propose an effective numerical scheme by applying Müntz–Legendre wavelets (MLW) for the solution of the given BSE. Different from classical wavelets (such as Legendre and Chebyshev), the MLW have an extra parameter representing the fractional order. Therefore, they provide more reliable results for certain fractional calculus problems. The convergence analysis of the method is provided in detail. Several test examples are given to illustrate the advantages of MLW over other classical wavelets and the high accuracy of this technique compared to existing methods in the literature.

show abstract

Fractional-order Bernoulli wavelets and their applications

Cited by 87 publications

References 34 publications

A numerical scheme based on Bernoulli wavelets and collocation method for solving fractional partial differential equations with Dirichlet boundary conditions

A numerical scheme based on Bernoulli wavelets and collocation method for solving fractional partial differential equations with Dirichlet boundary conditions

Fractional‐order orthogonal Bernstein polynomials for numerical solution of nonlinear fractional partial Volterra integro‐differential equations

An efficient wavelet method for the time‐fractional Black–Scholes equations

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