Reconstructive schemes for variational iteration method within Yang-Laplace transform with application to fractal heat conduction problem
TL;DR: In this article, a reconstructive scheme for variational iteration method using the Yang-Laplace transform is proposed and developed with the Yang Laplace transform and the identification of fractal Lagrange multiplier is investigated.
Abstract: A reconstructive scheme for variational iteration method using the
Yang-Laplace transform is proposed and developed with the Yang-Laplace
transform. The identification of fractal Lagrange multiplier is investigated
by the Yang-Laplace transform. The method is exemplified by a fractal heat
conduction equation with local fractional derivative. The results developed
are valid for a compact solution domain with high accuracy.
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TL;DR: In this paper, a tutorial review of fractal-Cantorian spacetime and fractional calculus is presented, starting with Leibniz's notation for derivative without limits which can be generalized to discontinous media like fractal derivative and q-derivative of quantum calculus.
Abstract: This tutorial review of fractal-Cantorian spacetime and fractional calculus begins with Leibniz's notation for derivative without limits which can be generalized to discontin- uous media like fractal derivative and q-derivative of quantum calculus. Fractal spacetime is used to elucidate some basic properties of fractal which is the foundation of fractional calculus, and El Naschie's mass-energy equation for the dark energy. The variational itera- tion method is used to introduce the definition of fractional derivatives. Fractal derivative is explained geometrically and q-derivative is motivated by quantum mechanics. Some effec- tive analytical approaches to fractional differential equations, e.g., the variational iteration method, the homotopy perturbation method, the exp-function method, the fractional com- plex transform, and Yang-Laplace transform, are outlined and the main solution processes are given.
386 citations
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TL;DR: In this paper, a coupling method of Sumudu transform and local fractional calculus is proposed to find the non-differentiable analytical solutions for initial value problems with LFT derivatives.
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TL;DR: In this paper, a comparison between the local fractional Adomian decomposition (LFAAD) and LFAFL decomposition was performed for solving the Laplace equation. But the results illustrate the significant features of the two methods which are both very effective and straightforward for solving differential equations with local fractionals derivative.
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68 citations
TL;DR: In this article, a unified theory of Maxwell's equations on the Cantor set with local fractional operators was proposed for the dynamics of cold dark matter in a fractal bounded domain.
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61 citations
References
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TL;DR: In this paper, a variational iteration method for non-linear problems is proposed, where the problems are initially approximated with possible unknowns and a correction functional is constructed by a general Lagrange multiplier, which can be identified optimally via the variational theory.
Abstract: In this paper, a new kind of analytical technique for a non-linear problem called the variational iteration method is described and used to give approximate solutions for some well-known non-linear problems. In this method, the problems are initially approximated with possible unknowns. Then a correction functional is constructed by a general Lagrange multiplier, which can be identified optimally via the variational theory. Being different from the other non-linear analytical methods, such as perturbation methods, this method does not depend on small parameters, such that it can find wide application in non-linear problems without linearization or small perturbations. Comparison with Adomian’s decomposition method reveals that the approximate solutions obtained by the proposed method converge to its exact solution faster than those of Adomian’s method.
2,371 citations
"Reconstructive schemes for variatio..." refers methods in this paper
...The variational iteration method (VIM) was first proposed by He [13] and was successfully applied to deal with heat conduction equations [14-17]....
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01 Jan 1985
TL;DR: Some applications of Volterraequations LinearVolterra equations of the second kind Nonlinear equations ofthe second kind Equations of the first kind Convolution equations The numerical solution of equations ofThe second kind.
Abstract: Some applications of Volterraequations Linear Volterra equations of the second kind Nonlinear equations of the second kind Equations of the first kind Convolution equations The numerical solution of equations of the second kind Product Integration methods for equations of the second kind Equations of the first kind with differentiable kernels Equations of the Abel type Integrodifferential equations Some computer programs Case studies.
844 citations
"Reconstructive schemes for variatio..." refers background in this paper
...RECONSTRUCTIVE SCHEMES FOR VARIATIONAL ITERATION METHOD WITHIN YANG-LAPLACE TRANSFORM WITH APPLICATION TO FRACTAL HEAT CONDUCTION PROBLEM by Chun-Feng LIU a*, Shan-Shan KONG b, and Shu-Juan YUAN c a College of Science, Hebei United University, Tangshan, China b College of Computer Science and Technology, Beijing University of Technology, Beijing, China c Qinggong College, Hebei United University, Tangshan, China Original scientific paper DOI: 10.2298/TSCI120826075L A reconstructive scheme for variational iteration method using the Yang-Laplace transform is proposed and developed with the Yang-Laplace transform....
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...…FOR VARIATIONAL ITERATION METHOD WITHIN YANG-LAPLACE TRANSFORM WITH APPLICATION TO FRACTAL HEAT CONDUCTION PROBLEM by Chun-Feng LIU a*, Shan-Shan KONG b, and Shu-Juan YUAN c a College of Science, Hebei United University, Tangshan, China b College of Computer Science and Technology, Beijing…...
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TL;DR: In this paper, the homotopy analysis method (HAM) is compared with the numerical and HPM in the heat transfer file and the auxiliary parameter ℏ, which provides a simple way to adjust and control the convergence region of solution series.
643 citations
"Reconstructive schemes for variatio..." refers methods in this paper
...…Beijing University of Technology, Beijing, China c Qinggong College, Hebei United University, Tangshan, China Original scientific paper DOI: 10.2298/TSCI120826075L A reconstructive scheme for variational iteration method using the Yang-Laplace transform is proposed and developed with…...
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