• Title/Summary/Keyword: Functional iteration method

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ITERATION METHOD FOR CONSTRAINED OPTIMIZATION PROBLEMS GOVERNED BY PDE

  • Lee, Hyung-Chun
    • Communications of the Korean Mathematical Society
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    • v.13 no.1
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    • pp.195-209
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    • 1998
  • In this paper we present a new iteration method for solving optimization problems governed by partial differential equations. We generalize the existing methods such as simple gradient methods and pseudo-time methods to get an efficient iteration method. Numerical tests show that the convergence of the new iteration method is much faster than those of the pseudo-time methods especially when the parameter $\sigma$ in the cost functional is small.

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FUNCTIONAL ITERATIVE METHODS FOR SOLVING TWO-POINT BOUNDARY VALUE PROBLEMS

  • Lim, Hyo Jin;Kim, Kyoum Sun;Yun, Jae Heon
    • Journal of applied mathematics & informatics
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    • v.31 no.5_6
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    • pp.733-745
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    • 2013
  • In this paper, we first propose a new technique of the functional iterative methods VIM (Variational iteration method) and NHPM (New homotopy perturbation method) for solving two-point boundary value problems, and then we compare their numerical results with those of the finite difference method (FDM).

COMMON FIXED POINT THEOREMS FOR COMPLEX-VALUED MAPPINGS WITH APPLICATIONS

  • Maldar, Samet;Atalan, Yunus
    • Korean Journal of Mathematics
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    • v.30 no.2
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    • pp.205-229
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    • 2022
  • The aim of this paper is to obtain some results which belong to fixed point theory such as strong convergence, rate of convergence, stability, and data dependence by using the new Jungck-type iteration method for a mapping defined in complex-valued Banach spaces. In addition, some of these results are supported by nontrivial numerical examples. Finally, it is shown that the sequence obtained from the new iteration method converges to the solution of the functional integral equation in complex-valued Banach spaces. The results obtained in this paper may be interpreted as a generalization and improvement of the previously known results.

IMPROVED GENERALIZED M-ITERATION FOR QUASI-NONEXPANSIVE MULTIVALUED MAPPINGS WITH APPLICATION IN REAL HILBERT SPACES

  • Akutsah, Francis;Narain, Ojen Kumar;Kim, Jong Kyu
    • Nonlinear Functional Analysis and Applications
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    • v.27 no.1
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    • pp.59-82
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    • 2022
  • In this paper, we present a modified (improved) generalized M-iteration with the inertial technique for three quasi-nonexpansive multivalued mappings in a real Hilbert space. In addition, we obtain a weak convergence result under suitable conditions and the strong convergence result is achieved using the hybrid projection method with our modified generalized M-iteration. Finally, we apply our convergence results to certain optimization problem, and present some numerical experiments to show the efficiency and applicability of the proposed method in comparison with other improved iterative methods (modified SP-iterative scheme) in the literature. The results obtained in this paper extend, generalize and improve several results in this direction.

AN EFFICIENT THIRD ORDER MANN-LIKE FIXED POINT SCHEME

  • Pravin, Singh;Virath, Singh;Shivani, Singh
    • Nonlinear Functional Analysis and Applications
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    • v.27 no.4
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    • pp.785-795
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    • 2022
  • In this paper, we introduce a Mann-like three step iteration method and show that it can be used to approximate the fixed point of a weak contraction mapping. Furthermore, we prove that this scheme is equivalent to the Mann iterative scheme. A comparison is made with the other third order iterative methods. Results are presented in a table to support our conclusion.

NUMERICAL SOLUTIONS OF NONLINEAR VOLTERRA-FREDHOLM INTEGRO-DIFFERENTIAL EQUATIONS BY USING MADM AND VIM

  • Abed, Ayoob M.;Younis, Muhammed F.;Hamoud, Ahmed A.
    • Nonlinear Functional Analysis and Applications
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    • v.27 no.1
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    • pp.189-201
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    • 2022
  • The aim of the current work is to investigate the numerical study of a nonlinear Volterra-Fredholm integro-differential equation with initial conditions. Our approximation techniques modified adomian decomposition method (MADM) and variational iteration method (VIM) are based on the product integration methods in conjunction with iterative schemes. The convergence of the proposed methods have been proved. We conclude the paper with numerical examples to illustrate the effectiveness of our methods.

Numerical Study on the Reflection of a Solitary Wave by a Vertical Wall Using the Improved Boussinesq Equation with Stokes Damping (고립파의 수직 벽면 반사와 Stokes 감쇠에 관한 개선된 부시네스크 방정식을 이용한 수치해석 연구)

  • Park, Jinsoo;Jang, Taek Soo
    • Journal of the Society of Naval Architects of Korea
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    • v.59 no.2
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    • pp.64-71
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    • 2022
  • In this paper, we simulate the collision of a solitary wave on a vertical wall in a uniform water channel and investigate the effect of damping on the amplitude attenuation. In order to take into account the damping effect, we introduce the Stokes damping whose dissipation is dependent on the velocity of wave motion on the surface of a thin layer of oil. That is, we use the improved Boussinesq equation with Stokes damping to describe the damped wave motion. Our work mainly focuses on the amplitude attenuation of a propagating solitary wave, which may depend on the Stokes damping together with the initial position and initial amplitude of the wave. We utilize the method of images and a powerful numerical tool (functional iteration method) for solving the improved Boussinesq equation, yielding an effective numerical simulation. This enables us to find the amplitudes of the incident wave and reflected one, whose ratio is a measure of the (wave) amplitude attenuation. Accordingly, we have shown that the reflection of a solitary wave by a vertical wall is dependent on not only the initial amplitude and position of a solitary but the Stokes damping.

A VISCOSITY TYPE PROJECTION METHOD FOR SOLVING PSEUDOMONOTONE VARIATIONAL INEQUALITIES

  • Muangchoo, Kanikar
    • Nonlinear Functional Analysis and Applications
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    • v.26 no.2
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    • pp.347-371
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    • 2021
  • A plethora of applications from mathematical programmings, such as minimax, mathematical programming, penalization and fixed point problems can be framed as variational inequality problems. Most of the methods that used to solve such problems involve iterative methods, that is why, in this paper, we introduce a new extragradient-like method to solve pseudomonotone variational inequalities in a real Hilbert space. The proposed method has the advantage of a variable step size rule that is updated for each iteration based on previous iterations. The main advantage of this method is that it operates without the previous knowledge of the Lipschitz constants of an operator. A strong convergence theorem for the proposed method is proved by letting the mild conditions on an operator 𝒢. Numerical experiments have been studied in order to validate the numerical performance of the proposed method and to compare it with existing methods.

LOCAL CONVERGENCE OF FUNCTIONAL ITERATIONS FOR SOLVING A QUADRATIC MATRIX EQUATION

  • Kim, Hyun-Min;Kim, Young-Jin;Seo, Jong-Hyeon
    • Bulletin of the Korean Mathematical Society
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    • v.54 no.1
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    • pp.199-214
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    • 2017
  • We consider fixed-point iterations constructed by simple transforming from a quadratic matrix equation to equivalent fixed-point equations and assume that the iterations are well-defined at some solutions. In that case, we suggest real valued functions. These functions provide radii at the solution, which guarantee the local convergence and the uniqueness of the solutions. Moreover, these radii obtained by simple calculations of some constants. We get the constants by arbitrary matrix norm for coefficient matrices and solution. In numerical experiments, the examples show that the functions give suitable boundaries which guarantee the local convergence and the uniqueness of the solutions for the given equations.