Adaptation of Caputo residual power series scheme in solving nonlinear time fractional Schrödinger equations

Kopçasız, Bahadır; Yaşar, Emrullah

doi:10.1016/j.ijleo.2023.171254

Cited by 7 publications

(1 citation statement)

References 34 publications

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“…Besides the time-fractional diffusion equation, lots of scholars have developed many schemes to cope with the time fractional Burgers' equation, like the operator splitting approach and artificial boundary method [19], the nonuniform Alikhanov formula of the Caputo time fractional derivative and Fourier spectral approximation in space [20], the L1 scheme and the local discontinuous Galerkin method [21], the Lucas polynomials coupled with finite difference method [22], the fourth-order compact difference scheme [23], the L1 implicit difference scheme based on non-uniform meshes [24], a secondorder energy stable and nonuniform time-stepping scheme [25], a collocation approach with trigonometric tension B-splines [26], the cubic B-spline functions and θ-weighted scheme [27], the local projection stabilization virtual element method [28], a compact difference scheme [29], the Caputo-Katugampola fractional derivative by extending the Laplace transform [30], and the tailored finite point method based on exponential basis [31]. For the time-fractional Schrödinger equations, lots of researchers have proposed many algorithms, like the conformable natural transform and the homotopy perturbation method [32], the conformable fractional derivatives modified Khater technique and the Adomian decomposition method [33], the Laplace Adomian decomposition method and the modified generalized Mittag-Leffler function method [34], a Caputo residual power series scheme [35], and the extended Kudryashov method [36].…”

Section: Introductionmentioning

confidence: 99%

A Symmetry of Boundary Functions Method for Solving the Backward Time-Fractional Diffusion Problems

Liu,

Kuo,

Chang

2024

Symmetry

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In the paper, we develop three new methods for estimating unknown initial temperature in a backward time-fractional diffusion problem, which is transformed to a space-dependent inverse source problem for a new variable in the first method. Then, the initial temperature can be recovered by solving a second-order boundary value problem. The boundary functions and a unique zero element constitute a group symmetry. We derive energetic boundary functions in the symmetry group as the bases to retrieve the source term as an unknown function of space and time. In the second method, the solution bases are energetic boundary functions, and then by collocating the governing equation we obtain the expansion coefficients for retrieving the entire solution and initial temperature. For the first two methods, boundary fluxes are over-specified to retrieve the initial condition. In the third method, we give two boundary conditions and a final time temperature to construct the bases in another symmetry group; the governing equation is collocated to a linear system to obtain the whole solution (initial temperature involved). These three methods are assessed and compared by numerical experiments.

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Section: Introductionmentioning

confidence: 99%

A Symmetry of Boundary Functions Method for Solving the Backward Time-Fractional Diffusion Problems

Liu,

Kuo,

Chang

2024

Symmetry

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Exploration of interactional phenomena and multi‐wave solutions of the fractional‐order dual‐mode nonlinear Schrödinger equation

Kopçasız,

Yaşar

2023

Math Methods in App Sciences

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This work concerns the fractional‐order dual‐mode nonlinear Schrödinger equation (FDMNLSE), which portrays the augmentation or absorption of dual waves. This model dissects the concurrent generation of two characteristic waves dealing with dual modes and introduces three physical parameters: nonlinearity, phase velocity, and dispersive factor. In the context of photonics, NLSE models the propagation of soliton pulses over intercontinental distances. Throughout this work, the fractional derivative is given in terms of time and space conformable sense. We analyze the multi‐waves method, homoclinic breather approach, and interactional solution with the double ‐functions procedure, and their applications for this equation are obtained using logarithmic transformation. The multi‐wave method is a well‐known phenomenon in nonlinear science that describes the interaction of three waves that satisfy certain resonance conditions. A breather wave is a localized and oscillatory solution that maintains its shape over time. Finally, we will discuss the dynamics of our newly obtained solutions with the help of graphs by assigning appropriate values to the parameters. The proposed methods are straight and aggressive, so the approved form can be extended for more nonlinear models. The findings are exceptional in comparison to previous findings in the literature. These outcomes may have significance for additional investigation of such frameworks to handle the nonlinear issues in applied sciences. The obtained results help us understand fluid propagation and incompressible fluids.

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M-truncated fractional form of the perturbed Chen–Lee–Liu equation: optical solitons, bifurcation, sensitivity analysis, and chaotic behaviors

Kopçasız,

Yaşar

2024

Opt Quant Electron

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This investigation discusses the modified M-truncated form of the perturbed Chen–Lee–Liu (pCLL) dynamical equation. The pCLL equation is a generalization of the original CLL equation, which describes the propagation of optical solitons in optical fibers. The pCLL equation includes additional terms that account for various influences such as chromatic dispersion, nonlinear dispersion, inter-modal dispersion, and self-steepening. A new version of the generalized exponential rational function method is utilized to obtain multifarious types of soliton solutions. Moreover, the planar dynamical system of the concerned equation is created using a Hamiltonian transformation, all probable phase portraits are given, and sensitive inspection is applied to check the sensitivity of the considered equation. Furthermore, after adding a perturbed term, chaotic and quasi-periodic behaviors have been observed for different values of parameters, and multistability is reported at the end. Numerical simulations of the solutions are added to the analytical results to better understand the dynamic behavior of these solutions. The study’s findings could be extremely useful in solving additional nonlinear partial differential equations.

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Adaptation of Caputo residual power series scheme in solving nonlinear time fractional Schrödinger equations

Cited by 7 publications

References 34 publications

A Symmetry of Boundary Functions Method for Solving the Backward Time-Fractional Diffusion Problems

A Symmetry of Boundary Functions Method for Solving the Backward Time-Fractional Diffusion Problems

Exploration of interactional phenomena and multi‐wave solutions of the fractional‐order dual‐mode nonlinear Schrödinger equation

M-truncated fractional form of the perturbed Chen–Lee–Liu equation: optical solitons, bifurcation, sensitivity analysis, and chaotic behaviors

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