Linearly compact scheme for 2D Sobolev equation with Burgers’ type nonlinearity

Zhang, Qifeng

doi:10.1007/s11075-022-01293-z

Cited by 18 publications

(3 citation statements)

References 39 publications

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“…So far, it is well known that the compact difference scheme has an impressive advantage due to its small stencil and better accuracy. Many related works can be refereed, such as [20] for the Cahn-Hilliard equation, [22,23] for nonlinear Burgers-type equations, and [24] for nonlinear time-fractional biharmonic problem. Huang and Gao [25] developed several compact difference schemes to solve the fourth-order parabolic equation with the third Dirichlet boundary conditions.…”

Section: Introductionmentioning

confidence: 99%

Pointwise error estimate of the compact difference methods for the fourth‐order parabolic equations with the third Neumann boundary conditions

Gao,

Huang,

Sun

2023

Math Methods in App Sciences

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Various boundary value conditions have been endowed for the fourth‐order differential equations. In the current work, a class of the fourth‐order parabolic equations with the third Neumann boundary conditions is concerned, where the values of the second and third spatial derivatives of the unknown function are given at the boundary. A novel average is defined to get the compact approximation near the boundary. Then a compact difference scheme is derived by using the weighted average and the method of order reduction. Due to the special and highly accurate discretization at the boundary, the related terms have to be handled skillfully during the analysis. By the energy method, the unique solvability, unconditional stability, and convergence of the derived compact difference scheme are strictly proved. The analytical difficulties caused by the boundary approximation are successfully overcome. As far as we know, this is the first time that the global pointwise fourth‐order convergence in space of the difference approach for this problem is achieved. In addition, the generalization to solve the case with a space‐dependent reaction coefficient is discussed. Finally, two numerical examples are computed to verify the accuracy of proposed numerical schemes.

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Section: Introductionmentioning

confidence: 99%

Pointwise error estimate of the compact difference methods for the fourth‐order parabolic equations with the third Neumann boundary conditions

Gao,

Huang,

Sun

2023

Math Methods in App Sciences

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“…Retrieving the analytical solutions for FDEs is sometimes difficult to achieve due to the computational complexities of fractional operators. In this respect, numerous methods have been constructed and motivated to investigate the approximate solution for FDEs, among which are the Sinc-collocation method [27], the predictor-corrector compact difference scheme [28], the variational iteration method [29], the homotopy analysis technique [30], the homotopy asymptotic method [31], the homotopy perturbation scheme [32], the differential transform method [33], the linearly compact scheme [34], the Adomian decomposition method [35] etc.…”

Section: Introductionmentioning

confidence: 99%

Computational Study for Fiber Bragg Gratings with Dispersive Reflectivity Using Fractional Derivative

Tariq¹,

Akram

Sadaf

et al. 2023

Fractal Fract

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In this paper, the new representations of optical wave solutions to fiber Bragg gratings with cubic–quartic dispersive reflectivity having the Kerr law of nonlinear refractive index structure are retrieved with high accuracy. The residual power series technique is used to derive power series solutions to this model. The fractional derivative is taken in Caputo’s sense. The residual power series technique (RPST) provides the approximate solutions in truncated series form for specified initial conditions. By using three test applications, the efficiency and validity of the employed technique are demonstrated. By considering the suitable values of parameters, the power series solutions are illustrated by sketching 2D, 3D, and contour profiles. The analysis of the obtained results reveals that the RPST is a significant addition to exploring the dynamics of sustainable and smooth optical wave propagation across long distances through optical fibers.

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“…In recent decades, fractional calculus has increased interest in a variety of science and engineering problems. Fractional partial differential equations (FPDEs) make it easy to solve numerous types of physical phenomena, notably elasticity, bloodstream fluid, solid geometry, optic fibers, processing of signals, radiation, hydrodynamics, medical science, and the process of diffusion [1][2][3][4][5] . The majority of FPDEs can not be solved precisely.…”

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confidence: 99%

Analytical study of one dimensional time fractional Schrödinger problems arising in quantum mechanics

Nadeem,

Alsayaad

2024

Sci Rep

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This work presents the analytical study of one dimensional time-fractional nonlinear Schrödinger equation arising in quantum mechanics. In present research, we establish an idea of the Sumudu transform residual power series method (ST-RPSM) to generate the numerical solution of nonlinear Schrödinger models with the fractional derivatives. The proposed idea is the composition of Sumudu transform (ST) and the residual power series method (RPSM). The fractional derivatives are taken in Caputo sense. The proposed technique is unique since it requires no assumptions or variable constraints. The ST-RPSM obtains its results through a series of successive iterations, and the resulting form rapidly converges to the exact solution. The results obtained via ST-RPSM show that this scheme is authentic, effective, and simple for nonlinear fractional models. Some graphical structures are displayed at different levels of fractional orders using Mathematica Software.

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Linearly compact scheme for 2D Sobolev equation with Burgers’ type nonlinearity

Cited by 18 publications

References 39 publications

Pointwise error estimate of the compact difference methods for the fourth‐order parabolic equations with the third Neumann boundary conditions

Pointwise error estimate of the compact difference methods for the fourth‐order parabolic equations with the third Neumann boundary conditions

Computational Study for Fiber Bragg Gratings with Dispersive Reflectivity Using Fractional Derivative

Analytical study of one dimensional time fractional Schrödinger problems arising in quantum mechanics

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