Embedding (3 + 1)-dimensional diffusion, telegraph, and Burgers’ equations into fractal 2D and 3D spaces: An analytical study

Alquran, Marwan; Jaradat, Imad; Abdel-Muhsen, Ruwa

doi:10.1016/j.jksus.2018.05.024

Cited by 11 publications

(4 citation statements)

References 36 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The following theorem shows the mixed Caputo fractional derivatives of analytic functions in fractional sense of γ γ γ -Maclaurin [53,54].…”

Section: Theorem 23 the γ γ γ -Maclaurin (21) Converges Absolutely mentioning

confidence: 99%

“…Now, as our goal is to furnish an analytical solution of higher-dimensional FPDEs, we take into account the Caputo fractional derivative, which is defined for an appropriate function as follows: where is the Caputo fractional derivative order. Accordingly, immediate computations lead to The following theorem shows the mixed Caputo fractional derivatives of analytic functions in fractional sense of γ̅ -Maclaurin [ 53 , 54 ].…”

Section: Analytic Solution Ansatz Of Higher-dimensional Fpdesmentioning

confidence: 99%

See 1 more Smart Citation

Higher-dimensional physical models with multimemory indices: analytic solution and convergence analysis

et al. 2020

Self Cite

View full text Add to dashboard Cite

The purpose of this work is to analytically simulate the mutual impact for the existence of both temporal and spatial Caputo fractional derivative parameters in higher-dimensional physical models. For this purpose, we employ the γ γ γ-Maclaurin series along with an amendment of the power series technique. To supplement our idea, we present the necessary convergence analysis regarding the γ γ γ-Maclaurin series. As for the application side, we solved versions of the higher-dimensional heat and wave models with spatial and temporal Caputo fractional derivatives in terms of a rapidly convergent γ γ γ-Maclaurin series. The method performed extremely well, and the projections of the obtained solutions into the integer space are compatible with solutions available in the literature. Finally, the graphical analysis showed a possibility that the Caputo fractional derivatives reflect some memory characteristics.

show abstract

“…The following theorem shows the mixed Caputo fractional derivatives of analytic functions in fractional sense of γ γ γ -Maclaurin [53,54].…”

Section: Theorem 23 the γ γ γ -Maclaurin (21) Converges Absolutely mentioning

confidence: 99%

Section: Analytic Solution Ansatz Of Higher-dimensional Fpdesmentioning

confidence: 99%

Higher-dimensional physical models with multimemory indices: analytic solution and convergence analysis

et al. 2020

Self Cite

View full text Add to dashboard Cite

show abstract

“…The subject has gained special importance in the last two or three decades [1][2][3]. Many phenomena in engineering and other sciences have been characterized mathematically by fractional derivatives [4][5][6][7]. These representations have offered good results in the modeling of real-world problems [8][9][10].…”

Section: Introductionmentioning

confidence: 99%

Analytical Solution of One-Dimensional Nonlinear Conformable Fractional Telegraph Equation by Reduced Differential Transform Method

Deresse

2022

Advances in Mathematical Physics

View full text Add to dashboard Cite

In this paper, the reduced differential transform method (RDTM) is successfully implemented to obtain the analytical solution of the space-time conformable fractional telegraph equation subject to the appropriate initial conditions. The fractional-order derivative will be in the conformable (CFD) sense. Some properties which help us to solve the governing problem using the suggested approach are proven. The proposed method yields an approximate solution in the form of an infinite series that converges to a closed-form solution, which is in many cases the exact solution. This method has the advantage of producing an analytical solution by only using the appropriate initial conditions without requiring any discretization, transformation, or restrictive assumptions. Four test modeling problems from mathematical physics, conformable fractional telegraph equations in which we already knew their exact solution using other numerical methods, were taken to show the liability, accuracy, convergence, and efficiency of the proposed method, and the solution behavior of each illustrative example is presented using tabulated numerical values and two- and three-dimensional graphs. The results show that the RDTM gave solutions that coincide with the exact solutions and the numerical solutions that are available in the literature. Also, the obtained results reveal that the introduced method is easily applicable and it saves a lot of computational work in solving conformable fractional telegraph equations, and it may also find wide application in other complicated fractional partial differential equations that originate in the areas of engineering and science.

show abstract

“…It has been demonstrated in certain circumstances that the nonlocality feature of the fractional derivatives makes them appropriate for describing the memory and hereditary features of various materials. Thus, the fractional derivative order α can be physically described as an index of memory [7][8][9][10][11][12][13].…”

Section: Introductionmentioning

confidence: 99%

Ternary-fractional differential transform schema: theory and application

et al. 2019

Self Cite

View full text Add to dashboard Cite

In this article, we propose a novel fractional generalization of the three-dimensional differential transform method, namely the ternary-fractional differential transform method, that extends its applicability to encompass initial value problems in the fractal 3D space. Several illustrative applications, including the Schrödinger, wave, Klein-Gordon, telegraph, and Burgers' models that are fully embedded in the fractal 3D space, are considered to demonstrate the superiority of the proposed method compared with other generalized methods in the literature. The obtained solution is expressed in a form of an α α α-fractional power series, with easily computed coefficients, that converges rapidly to its closed-form solution. Moreover, the projection of the solutions into the integer 3D space corresponds with the solutions of the classical copies for these models. This reveals that the suggested technique is effective and accurate for handling many other linear and nonlinear models in the fractal 3D space. Thus, research on this trend is worth tracking.

show abstract

Embedding (3 + 1)-dimensional diffusion, telegraph, and Burgers’ equations into fractal 2D and 3D spaces: An analytical study

Cited by 11 publications

References 36 publications

Higher-dimensional physical models with multimemory indices: analytic solution and convergence analysis

Higher-dimensional physical models with multimemory indices: analytic solution and convergence analysis

Analytical Solution of One-Dimensional Nonlinear Conformable Fractional Telegraph Equation by Reduced Differential Transform Method

Ternary-fractional differential transform schema: theory and application

Contact Info

Product

Resources

About

Embedding (3 + 1)-dimensional diffusion, telegraph, and Burgers’ equations into fractal 2D and 3D spaces: An analytical study

Cited by 11 publications

References 36 publications

Higher-dimensional physical models with multimemory indices: analytic solution and convergence analysis

Higher-dimensional physical models with multimemory indices: analytic solution and convergence analysis

Analytical Solution of One-Dimensional Nonlinear Conformable Fractional Telegraph Equation by Reduced Differential Transform Method

Ternary-fractional differential transform schema: theory and application

Contact Info

Product

Resources

About

Embedding (3 + 1)-dimensional diffusion, telegraph, and Burgers’ equations into fractal 2D and 3D spaces: An analytical study