Numerical solutions of partial differential equations by discrete homotopy analysis method

Total variation (TV) minimization-based nonlinear models have been proven to be very useful and successful in image processing. A lot of effort has been devoted to overcome the nonlinearity of the model and at the same time to obtain fast numerical schemes. In this paper, we propose a restarted iterative homotopy analysis method (HAM) to improve the computational efficiency for the TV models and will show by experiments that this method demonstrates great potential for recovering the noise and with great speed in both image denoising and image segmentation models. The method modifies the existing HAM and makes it suitable to potentially solve other nonlinear partial differential equations arising from image processing models. In our examples, we will demonstrate the validity of a restarted HAM and that this method is efficient and robust even for images with large ratios of noise and with much less CPU time than other methods.

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“…The first attempt at a discrete version of the HAM was made recently by Zhu et al [55] for a diffusion equation.…”

Section: Review Of An Hammentioning

confidence: 99%

A restarted iterative homotopy analysis method for two nonlinear models from image processing

Ghanbari

Rada

Chen

2013

International Journal of Computer Mathematics

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“…Since HAM has many advantages in comparison to other analytical methods, it is employed to solve continuous problems. Hence after the discrete ADM method [ 36 ], the discrete homotopy analysis method (DHAM) was introduced in 2010 by Zhu et al [ 37 ]. This method can be applied to complex problems containing discontinuity in fluid characteristics and geometry of the problem.…”

Section: Introductionmentioning

confidence: 99%

Applying Discrete Homotopy Analysis Method for Solving Fractional Partial Differential Equations

Özpinar

2018

Entropy

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Abstract:In this paper we developed a space discrete version of the homotopy analysis method (DHAM) to find the solutions of linear and nonlinear fractional partial differential equations with time derivative α (0 < α ≤ 1). The DHAM contains the auxiliary parameter , which provides a simple way to guarantee the convergence region of solution series. The efficiency and accuracy of the proposed method is demonstrated by test problems with initial conditions. The results obtained are compared with the exact solutions when α = 1. It is shown they are in good agreement with each other.

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“…A number of techniques have been proposed for solving the inverse problems, such as the boundary element method (BEM) [2], the method of fundamental solutions [3,4], genetic algorithm [5], regularization method [6] and etc [7][8][9][10][11][12][13][14][15]. In this study, we use a modified VIM to construct a solution to an IHCP.…”

Section: Introductionmentioning

confidence: 99%

A modified VIM for solving an inverse heat conduction problem

Rostamian¹,

Shahrezaee²

2016

Mathematical Researches

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In this paper, we will use a modified variational iteration method (MVIM) for solving an inverse heat conduction problem (IHCP). The approximation of the temperature and the heat flux at 0 x  are considered. This method is based on the use of Lagrange multipliers for the identification of optimal values of parameters in a functional in Euclidean space. Applying this technique, a rapid convergent sequence to the exact solution is produced. Moreover, this method does not require any discretization, linearization or small perturbation, thus it can be considered as an efficient method to solve this problem. To show the strength and capability of this method, some examples are given.

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Numerical solutions of partial differential equations by discrete homotopy analysis method

Cited by 14 publications

References 21 publications

A restarted iterative homotopy analysis method for two nonlinear models from image processing

A restarted iterative homotopy analysis method for two nonlinear models from image processing

Applying Discrete Homotopy Analysis Method for Solving Fractional Partial Differential Equations

A modified VIM for solving an inverse heat conduction problem

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