Lakshmi - Manoj Generalized Yang-Fourier Transforms to Heat-Conduction in a Semi-Infinite Fractal Bar

Lakshmi Narayan Mishra; Manoj Sharma; Vishnu Narayan Mishra

doi:doi:10.11648/j.pamj.20150402.15

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Lakshmi - Manoj Generalized Yang-Fourier Transforms to Heat-Conduction in a Semi-Infinite Fractal Bar

Lakshmi Narayan Mishra, Manoj Sharma, Vishnu Narayan Mishra

Published in Pure and Applied Mathematics Journal (Volume 4, Issue 2)

Received: 11 December 2014 Accepted: 13 December 2014 Published: 24 March 2015

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Abstract

In the present era, fractional calculus plays an important role in various fields. Fractional Calculus is a field of mathematic study that grows out of the traditional definitions of the calculus integral and derivative operators in much the same way fractional exponents is an outgrowth of exponents with integer value. Based on the wide applications in engineering and sciences such as physics, mechanics, chemistry, and biology, research on fractional ordinary or partial differential equations and other relative topics is active and extensive around the world. In the past few years, the increase of the subject is witnessed by hundreds of research papers, several monographs, and many international conferences.The purpose of present paper to solve 1-D fractal heat-conduction problem in a fractal semi-infinite bar has been developed by local fractional calculus employing the analytical Manoj Generalized Yang-Fourier transforms method.

Published in	Pure and Applied Mathematics Journal (Volume 4, Issue 2)
DOI	10.11648/j.pamj.20150402.15
Page(s)	57-61
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2015. Published by Science Publishing Group

Keywords

Fractal Bar, Heat-Conduction Equation, Lakshmi-Manoj Generalized Yang-Fourier Transforms, Yang-Fourier Transforms, Local Fractional Calculus

References

[1]	Kilbas, A.A., Srivastava H M., Trujillo, J J.; Theory and Applications of Fractional Differential Equations, Elsevier Science, Amsterdam, 2006.
[2]	Mainardi, F., Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press, London, 2010.
[3]	Podlubny, I., Fractional Differential Equations, Academic Press, New York, USA, 1999.
[4]	Klafter, J., et al., (Eds.), Fractional Dynamics in Physics: Recent Advances, World Scientific, Singapore, 2012.
[5]	Zaslavsky, G.M., Hamiltonian Chaos and Fractional Dynamics, Oxford University Press, Oxford, UK, 2005.
[6]	West, B, M. Bologna, and P. Grigolini, Physics of Fractal Operators, Springer, New York, USA, 2003.
[7]	Carpinteri, A., Mainardi, F., (Eds.), Fractals and Fractional Calculus in Continuum Mechanics, Springer, Wiena, 1997.
[8]	Baleanu, D., Golmankhaneh AK,; Golmankhaneh AK, Fractional Calculus Models and Numerical Methods, Complexity, Nonlinearity and Chaos, World Scientific, Singapore, 2012.
[9]	Hristov, J., Heat-Balance Integral to Fractional (Half-Time) Heat Diffusion Sub-Model, Thermal Science, 14 (2010), 2, pp. 291-316.
[10]	Hristov, J., Integral-Balance Solution to the Stokes' First Problem of a Viscoelastic Generalized Second Grade Fluid, Thermal Science, 16 (2012), 2, pp. 395-410.
[11]	Hristov, J., Transient Flow of a Generalized Second Grade Fluid Due to a Constant Surface Shear Stress: An Approximate Integral-Balance Solution, Int. Rev. Chem. Eng., 3 (2011), 6, pp. 802-809.
[12]	Jafari, H., Tajadodi, H; A Modified Variational Iteration Method for Solving Fractional Riccati Differential Equation by Adomian Polynomials, Fractional Calculus and Applied Analysis, 16 (2013), 1, pp. 109-122.
[13]	Wu, G. C., Baleanu, D., Variational Iteration Method for Fractional Calculus-a Universal Approach by Laplace Transform, Advances in Difference Equations, 2013 (2013), 1, pp. 1-9.
[14]	Ates, I., Yildirim, A., Application of Variational Iteration Method to Fractional Initial-Value Problems, Int. J. Nonl. Sci. Num. Sim., 10 (2009), 7, pp. 877-884.
[15]	Duan, J. S., et al., Solutions of the Initial Value Problem for Nonlinear Fractional Ordinary Differential Equations by the Rach-Adomian-Meyers Modified Decomposition Method, Appl. Math. Comput., 218 (2012), 17, pp. 8370-8392.
[16]	Momani, S., Yildirim, A., Analytical Approximate Solutions of the Fractional Convection-Diffusion Equation with Nonlinear Source Term by He's Homotopy Perturbation Method, Int. J. Comp. Math. Sci., 87 (2010), 5, pp. 1057-1065.
[17]	Guo, S., Mei, L, Li, Y., Fractional Variational Homotopy Perturbation Iteration Method and Its Application to a Fractional Diffusion Equation, Appl. Math. Comput., 219 (2013), 11, pp. 5909-5917.
[18]	Sun, H. G., et al., A Semi-Discrete Finite Element Method for a Class of Time-Fractional Diffusion Equations, Phil. Trans. Royal Soc. A: 371 (2013), 1990, pp. 1471-2962.
[19]	Jafari, H., et al., Fractional Subequation Method for Cahn-Hilliard and Klein-Gordon Equations, Abstract and Applied Analysis, 2013 (2013), Article ID 587179.
[20]	Luchko,Y., Kiryakova, V., The Mellin Integral Transform in Fractional Calculus, Fractional Calculus and Applied Analysis, 16 (2013), 2, pp .405-430.
[21]	Abbasbandy, S., Hashemi, M. S., On Convergence of Homotopy Analysis Method and its Application to Fractional integro-Differential Equations, Quaestiones Mathematicae, 36 (2013), 1, pp. 93-105.
[22]	Hashim, I., et al., Homotopy Analysis Method for Fractional IVPs, Comm. Nonl. Sci. Num. Sim., 14 (2009), 3, pp. 674-684.
[23]	Li, C., Zeng, F., The Finite Difference Methods for Fractional Ordinary Differential Equations, Num. Func. Anal. Optim., 34 (2013), 2, pp. 149-179.
[24]	Demir, A., et al., Analysis of Fractional Partial Differential Equations by Taylor Series Expansion, Boundary Value Problems, 2013 (2013), 1, pp. 68-80.
[25]	Kolwankar, K. M., Gangal, A. D., Local Fractional Fokker-Planck Equation, Physical Review Letters, 80 (1998), 2, pp. 214-217.
[26]	Chen, W., Time-Space Fabric Underlying Anomalous Diffusion, Chaos, Solitons & Fractals, 28 (2006), 4, pp. 923-929.
[27]	Fan, J., He, J.- H., Fractal Derivative Model for Air Permeability in Hierarchic Porous Media, Abstract and Applied Analysis, 2012 (2012), Article ID 354701.
[28]	Jumarie, G., Probability Calculus of Fractional Order and Fractional Taylor's Series Application to Fokker-Planck Equation and Information of Non-Random Functions, Chaos, Solitons & Fractals, 40 (2009), 3, pp. 1428-1448.
[29]	Carpinteri, A., Sapora, A., Diffusion Problems in Fractal Media Defined on Cantor Sets, ZAMM, 90 (2010), 3, pp. 203-210.
[30]	Yang, X. J., Local Fractional Functional Analysis and Its Applications, Asian Academic publisher Limited, Hong Kong, 2011.
[31]	Yang, X. J., Local Fractional Integral Transforms, Progress in Nonlinear Science, 4 (2011).
[32]	Yang, X. J., Advanced Local Fractional Calculus and Its Applications, World Science Publisher, New York, USA, 2012.
[33]	Su,W. H., et al., Fractional Complex Transform Method for Wave Equations on Cantor Sets within Local Fractional Differential Operator, Advances in Difference Equations, 2013 (2013), 1, pp. 97-103.
[34]	Yang Xiaojun, Hu, M. S., Baleanu, D., One-Phase Problems for Discontinuous Heat Transfer in Fractal Media, Mathematical Problems in Engineering, 2013 (2013), Article ID 358473, 2013.
[35]	Yang, X, J., Baleanu, D., Fractal Heat Conduction Problem Solved by Local Fractional Variation Iteration Method, Thermal Science, 17 (2013), 2, pp. 625-628.
[36]	Yang, Y., J., Dumitru Baleanu, Xiao-Jun Yang., A Local Fractional Variational Iteration Method for Laplace Equation within Local Fractional Operators, Abstract and Applied Analysis, 2013 (2013), Article ID 202650.
[37]	Su,W. H., et al., Damped Wave Equation and Dissipative Wave Equation in Fractal Strings within the Local Fractional Variational Iteration Method, Fixed Point Theory and Applications, 2013 (2013), 1, pp. 89-102.
[38]	Hu, M. S., et al., Local Fractional Fourier Series with Application to wave Equation in Fractal Vibrating String, Abstract and Applied Analysis, 2012 (2012), Article ID 567401.
[39]	Zhong, W. P., et al., Applications of Yang-Fourier Transform to Local Fractional Equations with Local Fractional Derivative and Local Fractional Integral, Advanced Materials Research, 461 (2012), pp. 306-310.
[40]	He, J.-H., Asymptotic Methods for Solitary Solutions and Compactions, Abstract and Applied Analysis, 2012 (2012), Article ID 916793.
[41]	Yang, Ai-M, Zhang, Y-Z, Long, Y, the Yang-Fourier transforms to heat-conduction in a semi-infinite fractal bar, Thermal Science, 2013, vol. 17, no. 3, pp.707-713.
[42]	Deepmala, A study on fixed point theorems for nonlinear contractions and its applications [Ph.D. thesis], Pt. Ravishankar Shukla University, Raipur, India, 2014.
[43]	Deepmala, “Existence theorems for solvability of a functional equation arising in dynamic programming,” International Journal ofMathematics andMathematical Sciences, vol. 2014,Article ID706585, 9 pages, 2014.
[44]	V. N. Mishra, K. Khatri, L. N. Mishra, and Deepmala, “Inverse result in simultaneous approximation by Baskakov-Durrmeyer Stancu operators,” Journal of Inequalities and Applications, vol. 2013, article 586, 2013.
[45]	V.N. Mishra, Some Problems on Approximations of Functions in Banach Spaces, Ph.D. Thesis (2007), Indian Institute of Technology, Roorkee - 247 667, Uttarakhand, India.
[46]	Deepmala and H. K. Pathak, A study on some problems on existence of solutions for nonlinear functional-integral equations, Acta Mathematica Scientia, 33 B(5) (2013), 1305–1313.

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APA Style

Lakshmi Narayan Mishra, Manoj Sharma, Vishnu Narayan Mishra. (2015). Lakshmi - Manoj Generalized Yang-Fourier Transforms to Heat-Conduction in a Semi-Infinite Fractal Bar. Pure and Applied Mathematics Journal, 4(2), 57-61. https://doi.org/10.11648/j.pamj.20150402.15

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ACS Style

Lakshmi Narayan Mishra; Manoj Sharma; Vishnu Narayan Mishra. Lakshmi - Manoj Generalized Yang-Fourier Transforms to Heat-Conduction in a Semi-Infinite Fractal Bar. Pure Appl. Math. J. 2015, 4(2), 57-61. doi: 10.11648/j.pamj.20150402.15

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AMA Style

Lakshmi Narayan Mishra, Manoj Sharma, Vishnu Narayan Mishra. Lakshmi - Manoj Generalized Yang-Fourier Transforms to Heat-Conduction in a Semi-Infinite Fractal Bar. Pure Appl Math J. 2015;4(2):57-61. doi: 10.11648/j.pamj.20150402.15

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@article{10.11648/j.pamj.20150402.15,
  author = {Lakshmi Narayan Mishra and Manoj Sharma and Vishnu Narayan Mishra},
  title = {Lakshmi - Manoj Generalized Yang-Fourier Transforms to Heat-Conduction in a Semi-Infinite Fractal Bar},
  journal = {Pure and Applied Mathematics Journal},
  volume = {4},
  number = {2},
  pages = {57-61},
  doi = {10.11648/j.pamj.20150402.15},
  url = {https://doi.org/10.11648/j.pamj.20150402.15},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20150402.15},
  abstract = {In the present era, fractional calculus plays an important role in various fields. Fractional Calculus is a field of mathematic study that grows out of the traditional definitions of the calculus integral and derivative operators in much the same way fractional exponents is an outgrowth of exponents with integer value. Based on the wide applications in engineering and sciences such as physics, mechanics, chemistry, and biology, research on fractional ordinary or partial differential equations and other relative topics is active and extensive around the world. In the past few years, the increase of the subject is witnessed by hundreds of research papers, several monographs, and many international conferences.The purpose of present paper to solve 1-D fractal heat-conduction problem in a fractal semi-infinite bar has been developed by local fractional calculus employing the analytical Manoj Generalized Yang-Fourier transforms method.},
 year = {2015}
}

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TY  - JOUR
T1  - Lakshmi - Manoj Generalized Yang-Fourier Transforms to Heat-Conduction in a Semi-Infinite Fractal Bar
AU  - Lakshmi Narayan Mishra
AU  - Manoj Sharma
AU  - Vishnu Narayan Mishra
Y1  - 2015/03/24
PY  - 2015
N1  - https://doi.org/10.11648/j.pamj.20150402.15
DO  - 10.11648/j.pamj.20150402.15
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 57
EP  - 61
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20150402.15
AB  - In the present era, fractional calculus plays an important role in various fields. Fractional Calculus is a field of mathematic study that grows out of the traditional definitions of the calculus integral and derivative operators in much the same way fractional exponents is an outgrowth of exponents with integer value. Based on the wide applications in engineering and sciences such as physics, mechanics, chemistry, and biology, research on fractional ordinary or partial differential equations and other relative topics is active and extensive around the world. In the past few years, the increase of the subject is witnessed by hundreds of research papers, several monographs, and many international conferences.The purpose of present paper to solve 1-D fractal heat-conduction problem in a fractal semi-infinite bar has been developed by local fractional calculus employing the analytical Manoj Generalized Yang-Fourier transforms method.
VL  - 4
IS  - 2
ER  -

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Author Information

Lakshmi Narayan Mishra

Department of Mathematics, National Institute of Technology, Silchar - 788 010, District - Cachar (Assam), India
Manoj Sharma

Department of Mathematics, RJIT, BSF Academy, Tekanpur, Gwalior (M.P.), India
Vishnu Narayan Mishra

Applied Mathematics and Humanities Department, Sardar Vallabhbhai National Institute of Technology, Ichchhanath Mahadev Dumas Road, Surat – 395 007 (Gujarat), India

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Plain Text BibTeX RIS

APA Style

Lakshmi Narayan Mishra, Manoj Sharma, Vishnu Narayan Mishra. (2015). Lakshmi - Manoj Generalized Yang-Fourier Transforms to Heat-Conduction in a Semi-Infinite Fractal Bar. Pure and Applied Mathematics Journal, 4(2), 57-61. https://doi.org/10.11648/j.pamj.20150402.15

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ACS Style

Lakshmi Narayan Mishra; Manoj Sharma; Vishnu Narayan Mishra. Lakshmi - Manoj Generalized Yang-Fourier Transforms to Heat-Conduction in a Semi-Infinite Fractal Bar. Pure Appl. Math. J. 2015, 4(2), 57-61. doi: 10.11648/j.pamj.20150402.15

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AMA Style

Lakshmi Narayan Mishra, Manoj Sharma, Vishnu Narayan Mishra. Lakshmi - Manoj Generalized Yang-Fourier Transforms to Heat-Conduction in a Semi-Infinite Fractal Bar. Pure Appl Math J. 2015;4(2):57-61. doi: 10.11648/j.pamj.20150402.15

Copy | Download

@article{10.11648/j.pamj.20150402.15,
  author = {Lakshmi Narayan Mishra and Manoj Sharma and Vishnu Narayan Mishra},
  title = {Lakshmi - Manoj Generalized Yang-Fourier Transforms to Heat-Conduction in a Semi-Infinite Fractal Bar},
  journal = {Pure and Applied Mathematics Journal},
  volume = {4},
  number = {2},
  pages = {57-61},
  doi = {10.11648/j.pamj.20150402.15},
  url = {https://doi.org/10.11648/j.pamj.20150402.15},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20150402.15},
  abstract = {In the present era, fractional calculus plays an important role in various fields. Fractional Calculus is a field of mathematic study that grows out of the traditional definitions of the calculus integral and derivative operators in much the same way fractional exponents is an outgrowth of exponents with integer value. Based on the wide applications in engineering and sciences such as physics, mechanics, chemistry, and biology, research on fractional ordinary or partial differential equations and other relative topics is active and extensive around the world. In the past few years, the increase of the subject is witnessed by hundreds of research papers, several monographs, and many international conferences.The purpose of present paper to solve 1-D fractal heat-conduction problem in a fractal semi-infinite bar has been developed by local fractional calculus employing the analytical Manoj Generalized Yang-Fourier transforms method.},
 year = {2015}
}

Copy | Download

TY  - JOUR
T1  - Lakshmi - Manoj Generalized Yang-Fourier Transforms to Heat-Conduction in a Semi-Infinite Fractal Bar
AU  - Lakshmi Narayan Mishra
AU  - Manoj Sharma
AU  - Vishnu Narayan Mishra
Y1  - 2015/03/24
PY  - 2015
N1  - https://doi.org/10.11648/j.pamj.20150402.15
DO  - 10.11648/j.pamj.20150402.15
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 57
EP  - 61
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20150402.15
AB  - In the present era, fractional calculus plays an important role in various fields. Fractional Calculus is a field of mathematic study that grows out of the traditional definitions of the calculus integral and derivative operators in much the same way fractional exponents is an outgrowth of exponents with integer value. Based on the wide applications in engineering and sciences such as physics, mechanics, chemistry, and biology, research on fractional ordinary or partial differential equations and other relative topics is active and extensive around the world. In the past few years, the increase of the subject is witnessed by hundreds of research papers, several monographs, and many international conferences.The purpose of present paper to solve 1-D fractal heat-conduction problem in a fractal semi-infinite bar has been developed by local fractional calculus employing the analytical Manoj Generalized Yang-Fourier transforms method.
VL  - 4
IS  - 2
ER  -

Copy | Download