Solutions to Uncertain Volterra Integral Equations by Fitted Reproducing Kernel Hilbert Space Method

Gumah, Ghaleb; Moaddy, K.; Al‐Smadi, Mohammed; Hashim, Ishak

doi:10.1155/2016/2920463

Cited by 32 publications

(20 citation statements)

References 30 publications

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“…Over the past decades, various techniques have been proposed to deal with the uncertain integral equations. Based on this trend, the iterative procedure based on quadrature formula is described in [27,37], while the successive approximations procedure is presented in [19,20]. The numericalanalytical procedures to solve fuzzy integral equations including fuzzy Laplace transform method [36], homotopy analysis method [39], fuzzy differential transform method [32], Adomian decomposition method [38], finite differences method [15] and variational iteration method [34] are also applied.…”

Section: Introductionmentioning

confidence: 99%

Reliable numerical algorithm for handling fuzzy integral equations of second kind in Hilbert spaces

Al‐Smadi

2019

Filomat

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Integral equations under uncertainty are utilized to describe different formulations of physical phenomena in nature. This paper aims to obtain analytical and approximate solutions for a class of integral equations under uncertainty. The scheme presented here is based upon the reproducing kernel theory and the fuzzy real-valued mappings. The solution methodology transforms the linear fuzzy integral equation to crisp linear system of integral equations. Several reproducing kernel spaces are defined to investigate the approximate solutions, convergence and the error estimate in terms of uniform continuity. An iterative procedure has been given based on generating the orthonormal bases that rely on Gram-Schmidt process. Effectiveness of the proposed method is demonstrated using numerical experiments. The gained results reveal that the reproducing kernel is a systematic technique in obtaining a feasible solution for many fuzzy problems.

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Section: Introductionmentioning

confidence: 99%

Reliable numerical algorithm for handling fuzzy integral equations of second kind in Hilbert spaces

Al‐Smadi

2019

Filomat

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“…Therefore, it is necessary to obtain some mathematical tools to understand the complex structure of uncertainty models [1][2][3][4][5]. On the other hand, the theory of fractional calculus, which is a generalization of classical calculus, deals with the discussion of the integrals and derivatives of noninteger order, has a long history, and dates back to the seventeenth century [6][7][8][9][10]. Different forms of fractional operators are introduced to study FDEs such as Riemann-Liouville, Grunwald-Letnikov, and Caputo.…”

Section: Introductionmentioning

confidence: 99%

Computational Optimization of Residual Power Series Algorithm for Certain Classes of Fuzzy Fractional Differential Equations

Alaroud

Al‐Smadi

Ahmad

et al. 2018

International Journal of Differential Equations

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This paper aims to present a novel optimization technique, the residual power series (RPS), for handling certain classes of fuzzy fractional differential equations of order 1 < ≤ 2 under strongly generalized differentiability. The proposed technique relies on generalized Taylor formula under Caputo sense aiming at extracting a supportive analytical solution in convergent series form. The RPS algorithm is significant and straightforward tool for creating a fractional power series solution without linearization, limitation on the problem's nature, sort of classification, or perturbation. Some illustrative examples are provided to demonstrate the feasibility of the RPS scheme. The results obtained show that the scheme is simple and reliable and there is good agreement with exact solution.

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“…The objective of the present paper is to use the HAM and Laplace transform to provide optimal solutions for a fractional order differential system model of human T-cell lymphotropic virus I (HTLV-I) infection of CD4 + T-cells. However, other category of methods to handle large amount of fractional problem can be found in [24][25][26][27][28][29].…”

Section: Introductionmentioning

confidence: 99%

Simulation as an alternative to linear programming of human T-cell lymphotropic virus I (HTLV-I) model infection of CD4+ T-cells

Alabedalhadi¹

2018

imf

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Simulation is a tradition of operating a process or system in the real world. The act of simulating something first requires the development of a model; this model represents the main characteristics of the selected behaviors, processes, and functions. The model represents the system itself, while the simulation represents the operation of the system over time. In this paper, the optimal solutions of fractional human T-cell lymphotropic virus model infection of CD4+ T-cells will selected by using an efficient technique, called the LHAM, which is a series solution method based on the HAM and Laplace transform in obtaining the solutions for a wide class of problem. The Pade approximation technique to enlarge region of convergence for the solutions. Results obtained using the shame presented here are in good agreement with the numerical results obtained before. Our work confirms the efficiency of LHAM as a tool for solving linear and nonlinear fractional differential equations. The numerical method proposed in this thesis can be utilized to solve other problems in field of nonlinear fractional differential equations.

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Solutions to Uncertain Volterra Integral Equations by Fitted Reproducing Kernel Hilbert Space Method

Cited by 32 publications

References 30 publications

Reliable numerical algorithm for handling fuzzy integral equations of second kind in Hilbert spaces

Reliable numerical algorithm for handling fuzzy integral equations of second kind in Hilbert spaces

Computational Optimization of Residual Power Series Algorithm for Certain Classes of Fuzzy Fractional Differential Equations

Simulation as an alternative to linear programming of human T-cell lymphotropic virus I (HTLV-I) model infection of CD4+ T-cells

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