An Efficient Alternating Direction Explicit Method for Solving a Nonlinear Partial Differential Equation

Pourghanbar, Somayeh; Manafian, Jalil; Ranjbar, Mojtaba; Aliyeva, Aynura; Gasimov, Yusif S.

doi:10.1155/2020/9647416

Cited by 29 publications

(19 citation statements)

References 27 publications

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“…In order to verify the effect of the proposed method in image enhancement and denoising, the synthetic images and brain images with different variances of Gaussian white noise and salt and pepper noise were selected for enhancement and denoising. The original vibration model, L. Alvarez method model, and the method in this paper are used to process the noise image, and the results are compared [ 20 , 21 ].…”

Section: Interpretation Of Resultsmentioning

confidence: 99%

Construction of Biologic Microscopic Image Segmentation Model Based on Smoothing of Fourth-Order Partial Differential Equation

2022

Scanning

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In order to solve the problem of microscopic image noise, a biological microscopic image segmentation model based on the smoothing of the fourth-order partial differential equation was proposed. Based on the functional description of image smoothness by directional curvature mode value, a fourth-order PDE image denoising model is derived, which can effectively reduce noise while preserving edges. The result of this method is piecewise linear image, and the gradient at the edge of the target has a step. Using the feature of noise reduction, a new geodesic active contour model is proposed. The experiment result shows that when the variance of Gaussian white noise is 15, the enhancement and denoising effects of the proposed method are 80.35% and 69.84 higher than those of the original vibration filtering method and L. Alvarez method. In terms of time, the proposed method is 1.3075 seconds slower than the original vibration filtering method and 17.5754 seconds faster than the L. Alvarez method. When the variance of Gaussian white noise is 25, the enhancement and denoising effects of the proposed method are 97.79% and 81.16 higher than those of the original vibration filtering method and L. Alvarez method. In terms of time, the proposed method is 1.3246 seconds slower than the original vibration filtering method and 17.5796 seconds faster than the L. Alvarez method. Conclusion. The new model is not only stable but also has strong ability of contour extraction and fast convergence.

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Section: Interpretation Of Resultsmentioning

confidence: 99%

Construction of Biologic Microscopic Image Segmentation Model Based on Smoothing of Fourth-Order Partial Differential Equation

2022

Scanning

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“…Without loss of accuracy, pLTSEM with 2 cores reduces nearly 18 % of the total CPU time of LTSEM, and pLTSEM with 4 cores reduces nearly 36 %. Besides, Table 6 lists the errors, the mass and energy differences, and the total CPU time of LTSEM and pLTSEM at several final times with τ � 1/2 6…”

Section: Numerical Examplesmentioning

confidence: 99%

“…Since in general, nding the analytical solutions of the two-dimensional Schrödinger equation can be a hard task, one has to make use of numerical procedures and extensive numerical methods haven been studied in recent years [3]. Concerning the time discretization, di erent directions can be investigated, for example, the nite di erence scheme [4][5][6], the Crank-Nicolson (CN) scheme [7][8][9], the timesplitting scheme [10,11], the Runge-Kutta (RK) scheme [1,2,[12][13][14], and the integrating factor method [15]. Among these schemes, the CN scheme is implicit and unconditionally stable.…”

Section: Introductionmentioning

confidence: 99%

An Efficient Parallel Laplace Transform Spectral Element Method for the Two-Dimensional Schrödinger Equation

Shao

2022

Mathematical Problems in Engineering

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In this paper, a parallel Laplace transform spectral element method for the linear Schrödinger equation and the cubic nonlinear Schrödinger equation is introduced. The main advantage of this new scheme is that it has the high accuracy of spectral methods and the parallel efficiency of the Laplace transform method. The key idea is twofold. First, Laplace transform is utilized to eliminate time dependency and the subsequent boundary value problems are discretized by using spectral element method. Second, a parallel Laplace transform method is developed to improve the computational efficiency. For the cubic nonlinear Schrödinger equation, the increment linearization technique is employed to deal with the nonlinear terms. Numerical experiments are addressed to demonstrate the accuracy and effectiveness of the proposed method. Compared with the Crank–Nicolson scheme, the CPU time is greatly saved by the parallel Laplace transform for both linear and nonlinear problems.

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“…One can observe that the trend toward increasing parallelism in high-performance computing is reinforced, since unfortunately the CPU clock frequencies nowadays increase much slower than a few decades ago [17,18]. That is one of the reasons why we believe that the importance of easily parallelizable explicit and unconditionally stable methods is going to increase, even if currently not too many scholars work with them (see [19][20][21][22][23][24][25][26] for examples).…”

Section: Introductionmentioning

confidence: 99%

A Comparative Study of Explicit and Stable Time Integration Schemes for Heat Conduction in an Insulated Wall

2022

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Calculating heat transfer in building components is an important and nontrivial task. Thus, in this work, we extensively examined 13 numerical methods to solve the linear heat conduction equation in building walls. Eight of the used methods are recently invented explicit algorithms which are unconditionally stable. First, we performed verification tests in a 2D case by comparing them to analytical solutions, using equidistant and non-equidistant grids. Then we tested them on real-life applications in the case of one-layer (brick) and two-layer (brick and insulator) walls to determine how the errors depend on the real properties of the materials, the mesh type, and the time step size. We applied space-dependent boundary conditions on the brick side and time-dependent boundary conditions on the insulation side. The results show that the best algorithm is usually the original odd-even hopscotch method for uniform cases and the leapfrog-hopscotch algorithm for non-uniform cases.

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An Efficient Alternating Direction Explicit Method for Solving a Nonlinear Partial Differential Equation

Cited by 29 publications

References 27 publications

Construction of Biologic Microscopic Image Segmentation Model Based on Smoothing of Fourth-Order Partial Differential Equation

Construction of Biologic Microscopic Image Segmentation Model Based on Smoothing of Fourth-Order Partial Differential Equation

An Efficient Parallel Laplace Transform Spectral Element Method for the Two-Dimensional Schrödinger Equation

A Comparative Study of Explicit and Stable Time Integration Schemes for Heat Conduction in an Insulated Wall

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