An implementation of Haar wavelet based method for numerical treatment of time-fractional Schrödinger and coupled Schrödinger systems

Khan, Nasir; Hameed, Tooba

doi:10.1109/jas.2016.7510193

Cited by 17 publications

(7 citation statements)

References 31 publications

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“…Equation (5) gives only second-order approximation of u xx . For the compact higher-order operator, we use the same steps/equations as done in the book of Ames 37 :…”

Section: Compact Finite Difference Schemementioning

confidence: 99%

“…Recently, FDEs are used in image processing 1 and dynamical systems in mathematical biology 2–3 . Fractional Schrödinger equation is used for many quantum processes in physics 4–5 . Fractional diffusion equation is used to describe phenomena of anomalous diffusion in transport processes, for predicting market behavior, 6–7 speculative option valuation, 8 and modeling for cancer tumor.…”

Section: Introductionmentioning

confidence: 99%

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A higher‐order unconditionally stable scheme for the solution of fractional diffusion equation

Ghaffar

Ullah

Badshah

et al. 2020

Math Methods in App Sciences

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In this paper, a higher‐order compact finite difference scheme with multigrid algorithm is applied for solving one‐dimensional time fractional diffusion equation. The second‐order derivative with respect to space is approximated by higher‐order compact difference scheme. Then, Grünwald–Letnikov approximation is used for the Riemann–Liouville time derivative to get an implicit scheme. The scheme is based on a heptadiagonal matrix with eighth‐order accurate local truncation error. Fourier analysis is used to analyze the stability of higher‐order compact finite difference scheme. Matrix analysis is used to show that the scheme is convergent with the accuracy of eighth‐order in space. Numerical experiments confirm our theoretical analysis and demonstrate the performance and accuracy of our proposed scheme.

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“…Equation (5) gives only second-order approximation of u xx . For the compact higher-order operator, we use the same steps/equations as done in the book of Ames 37 :…”

Section: Compact Finite Difference Schemementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A higher‐order unconditionally stable scheme for the solution of fractional diffusion equation

Ghaffar

Ullah

Badshah

et al. 2020

Math Methods in App Sciences

Self Cite

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“…The academic workers and scientists are very much interested in the study of such types of models. Some interesting studies of such models are given by Arqub and Shawagfeh, 7 Garg and Manohar, 8 Khan et al, 9,10 and several others. [11][12][13] In fractional calculus, many definitions of fractional derivatives and integrals are given.…”

Section: Introductionmentioning

confidence: 99%

Numerical investigation with stability analysis of time‐fractional Korteweg‐de Vries equations

Ullah

Butt

Buhader

2020

Math Methods in App Sciences

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In this paper, existing classical Korteweg‐de Vries (KdV) equations are converted into the corresponding time‐fractional KdV equations by using Caputo‐Fabrizio fractional derivative and then solved with appropriate initial conditions by implementing semi‐numerical technique, that is, Laplace transform together with an iterative scheme. The obtained solutions are novel, and previous literature lacks such derivations. The stability of implemented technique is analyzed by applying Banach contraction principle and S‐stable mapping. Efficiency of Caputo‐Fabrizio fractional derivative is exhibited through graphical illustrations, and fractional results are drafted in tabular form for specific values of fractional parameter to validate the numerical investigation.

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“…In the last few decades, the utilization of fractional order derivative has gained substantial importance in various fields as it leads to a significant transformation of the system [13][14][15]. Copious non-linear phenomena have been demonstrated in diffusion and reaction-diffusion equations by persuading the fractional order derivative.…”

Section: Introductionmentioning

confidence: 99%

The reaction dimerization: A resourceful slant applied on the fractional partial differential equation

et al. 2019

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The paper presents an analysis to investigate the time-fractional coupled diffusion equation with non-linear reaction subject to piecewise initial conditions. The governance system is determined analytically, and numerically adopting an artificial neural network taking the Caputo fractional derivative. In this scheme, the differential equations are transformed into an optimization problem, and minimize the problem by using sophisticated computer simulation Nelder-Mead algorithm. Finally, an experiment has been executed in the application of o-phenylenedioxy-dimethylsilane reaction dimerization which is a trial solution of resources as coefficients. The obtained solution is analyzed by tabulation of numerical values and plotting time based graphs. The novelty of the solution will be provided by comparison with the previous work in the literature as a special case. The numerical outcomes demonstrate the momentous features, efficiency and reliability of the approach.

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An implementation of Haar wavelet based method for numerical treatment of time-fractional Schrödinger and coupled Schrödinger systems

Cited by 17 publications

References 31 publications

A higher‐order unconditionally stable scheme for the solution of fractional diffusion equation

A higher‐order unconditionally stable scheme for the solution of fractional diffusion equation

Numerical investigation with stability analysis of time‐fractional Korteweg‐de Vries equations

The reaction dimerization: A resourceful slant applied on the fractional partial differential equation

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