On novel nondifferentiable exact solutions to local fractional Gardner's equation using an effective technique

Ghanbari, Behzad

doi:10.1002/mma.7060

Cited by 129 publications

(18 citation statements)

References 41 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Recently, Ghanbari has proposed an efficient technique to examine the local partial differential equation to solve local fractional Gardner's equation with the following structure: 40

\frac{\partial^{ν} {normalΞ}_{ν} false(μ, η false)}{\partial η^{ν}} + δ_{2} {normalΞ}_{ν} false(μ, η false) \frac{\partial^{ν} {normalΞ}_{ν} false(μ, η false)}{\partial μ^{ν}} + δ_{3} {normalΞ}_{ν}^{2} false(μ, η false) \frac{\partial^{ν} {normalΞ}_{ν} false(μ, η false)}{\partial μ^{ν}} + δ_{4} \frac{\partial^{4 ν^{ϵ}} {normalΞ}_{ν} false(μ, η false)}{\partial μ^{4 ν}} = 0 .

…”

Section: The Main Methodsmentioning

confidence: 99%

“…The method can be assumed as a creative expansion of the well‐known generalized exponential rational function method (GERRM) 41–47 but with local derivatives. To exert the method in solving local fractional partial differential equations, the following process is required 40,48 : First, we consider the equation with partial derivatives of the following local type in terms of Λ ϵ ( ξ , τ ) as

scriptF false({normalΛ}_{ϵ} ((), ξ, τ), \frac{\partial^{ϵ} {normalΛ}_{ϵ} false(ξ, τ false)}{\partial τ^{ϵ}}, \frac{\partial^{ϵ} {normalΛ}_{ϵ} false(ξ, τ false)}{\partial ξ^{ϵ}}, \frac{\partial^{2 ϵ} {normalΛ}_{ϵ} false(ξ, τ false)}{\partial τ^{2 ϵ}}, \frac{\partial^{2 ϵ} {normalΛ}_{ϵ} false(ξ, τ false)}{\partial ξ^{2 ϵ}}, \dots false) = 0 .

With the aid of applying the transformations

{normalΛ}_{ϵ} ((), ξ, τ) = {normalΛ}_{ϵ} false(δ^{ϵ} false)

and

δ^{ϵ} = κ^{ϵ} ξ - ℏ^{ϵ} τ

in (), the following new differential equation with local derivative is then obtained:

scriptF false({normalΛ}_{ϵ} false(δ^{ϵ} false), - ℏ^{ϵ} \frac{d^{ϵ} {normalΛ}_{ϵ} false(δ^{ϵ} false)}{d δ^{ϵ}}, κ^{ϵ} d^{ϵ} {normalΛ}_{ϵ} false(<...>

…”

Section: The Main Methodsmentioning

confidence: 99%

See 1 more Smart Citation

Abundant exact solutions to a generalized nonlinear Schrödinger equation with local fractional derivative

Ghanbari

2021

Math Methods in App Sciences

Self Cite

132

View full text Add to dashboard Cite

The paper aims to employ a new effective methodology to build exact fractional solutions to the generalized nonlinear Schrödinger equation with a local fractional operator defined on Cantor sets. The equation contains group velocity dispersion and second‐order spatiotemporal dispersion coefficients. We obtain exact solutions of the equation via the generalized version of the exponential rational function method. This new version of the method uses a set of elementary functions that are adopted on the contour set. To study the dynamic behavior of the obtained results, extensive numerical simulations are provided. We observe that the employed method is simple but quite efficient for determining the exact solutions of the problem in the local sense. Moreover, they are practically compatible with solving various classes of nonlinear problems arising in mathematical physics. All computations are carried out using the Maple package.

show abstract

“…Recently, Ghanbari has proposed an efficient technique to examine the local partial differential equation to solve local fractional Gardner's equation with the following structure: 40

\frac{\partial^{ν} {normalΞ}_{ν} false(μ, η false)}{\partial η^{ν}} + δ_{2} {normalΞ}_{ν} false(μ, η false) \frac{\partial^{ν} {normalΞ}_{ν} false(μ, η false)}{\partial μ^{ν}} + δ_{3} {normalΞ}_{ν}^{2} false(μ, η false) \frac{\partial^{ν} {normalΞ}_{ν} false(μ, η false)}{\partial μ^{ν}} + δ_{4} \frac{\partial^{4 ν^{ϵ}} {normalΞ}_{ν} false(μ, η false)}{\partial μ^{4 ν}} = 0 .

…”

Section: The Main Methodsmentioning

confidence: 99%

scriptF false({normalΛ}_{ϵ} ((), ξ, τ), \frac{\partial^{ϵ} {normalΛ}_{ϵ} false(ξ, τ false)}{\partial τ^{ϵ}}, \frac{\partial^{ϵ} {normalΛ}_{ϵ} false(ξ, τ false)}{\partial ξ^{ϵ}}, \frac{\partial^{2 ϵ} {normalΛ}_{ϵ} false(ξ, τ false)}{\partial τ^{2 ϵ}}, \frac{\partial^{2 ϵ} {normalΛ}_{ϵ} false(ξ, τ false)}{\partial ξ^{2 ϵ}}, \dots false) = 0 .

With the aid of applying the transformations

{normalΛ}_{ϵ} ((), ξ, τ) = {normalΛ}_{ϵ} false(δ^{ϵ} false)

and

δ^{ϵ} = κ^{ϵ} ξ - ℏ^{ϵ} τ

in (), the following new differential equation with local derivative is then obtained:

scriptF false({normalΛ}_{ϵ} false(δ^{ϵ} false), - ℏ^{ϵ} \frac{d^{ϵ} {normalΛ}_{ϵ} false(δ^{ϵ} false)}{d δ^{ϵ}}, κ^{ϵ} d^{ϵ} {normalΛ}_{ϵ} false(<...>

…”

Section: The Main Methodsmentioning

confidence: 99%

Abundant exact solutions to a generalized nonlinear Schrödinger equation with local fractional derivative

Ghanbari

2021

Math Methods in App Sciences

Self Cite

132

View full text Add to dashboard Cite

show abstract

“…In 2018, an integration method called the generalized exponential rational function method (GERFM) was introduced by Ghanbari et al to solve the resonance nonlinear Schrödinger equation [66]. Following their work, the technique has been used successfully many times to handle other partial equations [67][68][69][70][71][72][73][74][75][76][77][78][79][80][81][82]. In this part, we outline the main steps of GERFM as follows.…”

Section: The Generalized Exponential Rational Function Methodsmentioning

confidence: 99%

Analytical solitons for the space-time conformable differential equations using two efficient techniques

Neirameh

Parvaneh

2021

Adv Differ Equ

View full text Add to dashboard Cite

Exact solutions to nonlinear differential equations play an undeniable role in various branches of science. These solutions are often used as reliable tools in describing the various quantitative and qualitative features of nonlinear phenomena observed in many fields of mathematical physics and nonlinear sciences. In this paper, the generalized exponential rational function method and the extended sinh-Gordon equation expansion method are applied to obtain approximate analytical solutions to the space-time conformable coupled Cahn–Allen equation, the space-time conformable coupled Burgers equation, and the space-time conformable Fokas equation. Novel approximate exact solutions are obtained. The conformable derivative is considered to obtain the approximate analytical solutions under constraint conditions. Numerical simulations obtained by the proposed methods indicate that the approaches are very effective. Both techniques employed in this paper have the potential to be used in solving other models in mathematics and physics.

show abstract

“…The mathematical formulations that involve a combination of differential equations and fractional derivations are widespread in many engineering and scientific disciplines 1–6 . In all these circumstances, the main difficulty in dealing with these models often involves how to find a suitable numerical approximation for their existing fractional operator.…”

Section: Introductionmentioning

confidence: 99%

Chaotic behaviors of the prevalence of an infectious disease in a prey and predator system using fractional derivatives

Ghanbari

2021

Math Methods in App Sciences

Self Cite

View full text Add to dashboard Cite

The risk of spread infectious diseases in the environment is always one of the main threats to the life of living organisms. This point can be assumed as a clear proof of the importance of studying such problems from various aspects such as computational mathematical models. In this contribution, we examine a mathematical model to investigate the prevalence of an infectious disease in a prey and predator system, including three subpopulations through a fractional system of nonlinear equations. The model characterizes a possible interaction between predator and prey where infectious disease outbreaks in a community. Further, the prey population is divided into two: susceptible and the infected population. This model involves the Caputo‐Fabrizio derivative. One of the basic features of this type of derivative is the use of a non‐singular (exponential) kernel, which increases its ability to describe phenomena compared to other existing operators. To the best of the author's knowledge, the use of fractional derivative operators for this model has not yet been investigated. Therefore, the presented results can be considered as new and interesting results for this model. In some of the acquired simulations, the chaotic behaviors are clearly detectable.

show abstract

On novel nondifferentiable exact solutions to local fractional Gardner's equation using an effective technique

Cited by 129 publications

References 41 publications

Abundant exact solutions to a generalized nonlinear Schrödinger equation with local fractional derivative

Abundant exact solutions to a generalized nonlinear Schrödinger equation with local fractional derivative

Analytical solitons for the space-time conformable differential equations using two efficient techniques

Chaotic behaviors of the prevalence of an infectious disease in a prey and predator system using fractional derivatives

Contact Info

Product

Resources

About