IOSR Journal of Mathematics (IOSR-JM)

Volume 4 - Issue 6

Paper Type : Research Paper
Title : Exact Solutions of Space Dependent Korteweg-de Vries Equation by the Extended Unified Method
Country : Egypt
Authors : Hamdy I. Abdel-Gawad, Nasser S. Elazab, Mohamed Osman
: 10.9790/5728-0460106      logo
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Abstract: Recently the unified method for finding traveling wave solutions of non-linear evolution equations was proposed by one of the authors a. It was shown that, this method unifies all the methods being used to find these solutions. In this paper, we extend this method to find a class of formal exact solutions to Korteweg-de Vries (KdV) equation with space dependent coefficients. A new class of multiple-soliton or wave trains is obtained.
Keywords: Exact solution, Extended unified method, Korteweg-deVries equation, variable coefficients
[1] P. J. Olivier, Application of Lie Groups to Differential Equations. GTM, Vol. 107 ( Berlin, Springer) (1986).
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[3] R. Conte, Phys. Lett. A., 134, 100-104 (1988).
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[6] C. Rogers and W. F. Shadwick, BΓ€cklund Transformations (Academic, New York) (1982).
[7] K. M. Tamizhmani and M. Lakshamanan, J. Phys. A, Math. Gen. , 16 , 3773 (1983).
[8] Y. Xie, J. Phys. A Math. Gen., 37 5229 (2004).
[9] C. Rogers and Szereszewski, J. Phys. A Math. Theor. 42, 40-4015 (2009).
[10] E. Fan, and H. Zhang, Phys. Lett. A 245, 389-392 (1999)

Paper Type : Research Paper
Title : A Self-Starting Hybrid Linear Multistep Method for a Direct Solution of the General Second-Order Initial Value Problem
Country : Nigeria
Authors : Adamu Alhaji Mohammed, Amina Hamza and Umaru Mohammed
: 10.9790/5728-0460713      logo
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Abstract:A self- starting hybrid linear multistep method for direct solution of the general second-order initial value problem is considered. The continuous method is used to obtain Multiple Finite Difference Methods (MFDMs) (each of order 7) which are combined as simultaneous numerical integrators to provide a direct solution to IVPs over sub-intervals which do not overlap. The convergence of the MFDMs is discussed by conveniently representing the MFDMs as a block method and verifying that the block method is zero-stable and consistent. The superiority of the MFDMs over published work is established numerically.
Keywords: Multiple Finite Difference Methods, Second Order, Boundary Value Problem, Block Methods, Multistep Methods
[1] Awoyemi, D.O., 2003. A P-stable linear multistep method for solving general third order ordinary differential equations. Int. J.
Comput Math., 8: 985-991. DOI: 10.1080/0020716031000079572
[2] Awoyemi, D. and Idowu,O. 2005. A class hybrid collocation methods for third order of ordinary differential equations, Int. J.
Comput. Math., 82: 1287-1293. DOI: 10.1080/00207160500112902.
[3] Fatunla, S.O., (1994). A class of block methods for second order IVPs. Int. J. Comput. Math., 55: 119-133. DOI:
[4] Lambert, J.D., (1973). Computational Methods in Ordinary Differential Equations (John Willey and Sons, New York, USA., ISBN:
10: 0471511943, p: 294.)
[5] Adee, S.O., Onumanyi, P., Sirisena,U.W., and Yahaya,Y.A (2005). Note on starting numerov method more accurately by a hybrid
formula of order four for an initial value problem,. J.Computat. Applied Math., 175: 369-373. DOI: 10.1016/
[6] Jator, S.N(2007). A sixth order linear multistep method for the direct solution of y00 = f(x, y, y0), International Journal of Pure and
Applied Mathematics, 40, No. 4, 457-472.
[7] Jator, S.N and Li, J (2007) A self-starting linear multistep method for a direct solution of the general second order initial value
problem, International Journal of Computer Mathematics Vol. 86, No. 5, May 2009, 827–836
[8] Jator, S.N (2008) Multiple finite difference methods for solving third order ordinary differential equations, International Journal of
Pure and Applied Mathematics, 43, No. 2, 253 - 265.
[9] Mohammmed, U.,Jiya, M and Mohammed, A.A(2010). A class of six step block method for solution of general second order
ordinary differential equations, Pacific Journal of Science and Technology. 11(2):pp273-277.
[10] Mohammmed, U (2011). A class of implicit five step block method for general second order ordinary differential equations. Journal
of Nigerian Mathematical Society (JNMS). vol 30 p 25-39

Paper Type : Research Paper
Title : Determination of Thermal Stresses in A Thin Clamped Hollow Disk under Unsteady Temperature Field due To Point Heat Source
Country : India
Authors : G.D. Kedar, K.C. Deshmukh
: 10.9790/5728-0461419      logo
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Abstract:This paper deals with the determination of thermal stresses in a thin clamped hollow disk under unsteady temperature field due to point heat source situated at centre along radial and axial direction within it. A thin hollow disk is considered having arbitrary initial temperature and is subjected to arbitrary heat flux at the outer circular boundary; whereas inner boundary is at zero heat flux. Also, the upper and lower surfaces of the disk are at zero temperature. The inner and outer edges of the disk are clamped. The governing heat conduction equation has been solved by the method of integral transform technique. The results are obtained in a series form in terms of Bessel's functions. The results have been computed numerically and illustrated graphically.

Keywords: Heat Conduction, Point Heat Source, Thermal Stresses, clamped hollow disk, Unsteady Temperature

[1] Roy Choudhuri S.K., A note on the quasi-static stress in thin circular plate due to transient temperature applied along the
circumference of a circle over the upper face, Bull. Acad. Polon. Sci., Ser. Sci. Techn., 20, 21, (1972).
[2] Gogulwar V.S. and Deshmukh K.C., Thermal stresses in a thin circular plate with heat sources, Journal of Indian Academy of
Mathematics, 27 (1), 129-141, (2005).
[3] Kulkarni V. S., Deshmukh K. C. and Warbhe S. D., Quasi-Static thermal stresses due to heat generation in a thin hollow circular disk,
J. Thermal Stresses, 31(8), 698-705, (2008).
[4] Deshmukh K.C., Warbhe S.D., Kulkarni V.S., Non-homogeneous steady state heat conduction problem in a thin circular plate and
thermal stresses, Int. J. Thermophysics, 30, 1688-1696, (2009.)
[5] Ozisik M.N., Boundary value problems of heat conduction, International Textbook Company, Scranton, Pennsylvania, 148-163,
[6] Nowinski J.L., Theory of thermoelasticity with applications, Sijthoff International Publishers B.V. Alphen aan den Rijn, The
Netherlands, 407, (1978).

Paper Type : Research Paper
Title : On Generalized Half Canonical Cosine Transform
Country : India
Authors : A. S. Gudadhe and A.V. Joshi
: 10.9790/5728-0462025      logo
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Abstract: As generalization of the fractional Cosine transform (FRCT), the Canonical Cosine Transform (CCT) has been used in several areas, including optical analysis and signal processing. For practical purpose half canonical cosine transform is more useful. Hence in this paper we have proved some important results Differentiation property, Modulation property, Scaling property, Derivative property, Parseval's Identity for half canonical cosine transform (HCCT).
Keywords: Linear canonical transform, Fractional Fourier Transform.
[1] Akay O. and Bertels, (1998): Fractional Mellin Transformation: An extension of fractional frequency concept for scale, 8th IEEE,
Dig. Sign. Proc. Workshop, Bryce Canyan, Utah.
[2] Almeida, L.B., (1994): The fractional Fourier Transform and time- frequency representations, IEEE. Trans. on Sign. Proc., Vol. 42,
No.11, 3084-3091.
[3] A. S. Gudadhe and A.V. Joshi (August - 2012): Generalized Canonical Cosine Transform, International Journal of Engineering
Research & Technology (IJERT) Vol. 1 Issue 6.
[4] Moshinsky, M.(1971): Linear canonical transform and their unitary representation, Jour. Math, Phy.,Vol.12, No. 8 , P. 1772-1783.
[5] Namias V. (1980): The fractional order Fourier transform and its applications to quantum mechanics, Jour. Inst. Math's. App., Vol.
25, 241-265.
[6] Pei and Ding, (2002) : Eigenfunctions of Linear Canonical Transform Vol. 50, No.1.
[7] Pie and Ding, (2002): Fractional cosine, sine and Hartley Transforms, IEEE. Trans. On Sign. Proc. Vol. 50, No.7, 1661-1680.
[8] Sontakke, Gudadhe (2009): Convolution and Rayleigh's Theorem For Generalized Fractional Hartley Transform, EJPAM Vol. 2, No. 1, (162-170)

Paper Type : Research Paper
Title : Static Deformation of a Uniform Half-Space with Rigid Boundary Due To a Vertical Dip-Slip Line Source
Country : India
Authors : Meenal Malik, Mahabir Singh and Jagdish Singh
: 10.9790/5728-0462637      logo
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Abstract: The Airy stress function for a vertical dip-slip line source buried in a homogeneous, isotropic, perfectly elastic half-space with rigid boundary is obtained. This Airy stress function is used to derive closedform analytical expressions for the stresses and displacements at an arbitrary point of the half-space caused by vertical dip-slip line source. The variation of the displacements and stress fields with distance from the fault and depth from the fault is studied numerically.
Keywords – Dip-slip faulting, Half-space, Rigid boundary, Static deformation
[1] Bonaccorso, A. and Davis, P., Dislocation Modeling of the 1989 Dike Intrusion into the Flank of Mt. Etna, Sicily, J. Geophys. Res.,
vol. 98(3), 1993, 4261-4268.
[2] Bonafede, M. and Danesi, S., Near-field Modifications of Stress Induced by Dyke Injection at Shallow Depth, Geophys. J. Int., vol.
130, 1997, 435-448.
[3] Bonafede, M. and Rivalta, E., The Tensile Dislocation Problem in a Layered Elastic Medium, Geophys. J. Int., vol. 136, 1999a, 341-
[4] Bonafede, M. and Rivalta, E., On Tensile Cracks Close to and Across the Interface Between Two Welded Elastic Half-spaces,
Geophys. J. Int., vol.138, 1999b, 410-434.
[5] Davis, P. M, Surface Deformation Associated with a Dipping Hydrofracture, Journal of Geophysical Research, vol. 88, 1983, 5826-
[6] Dundurs, J. and Hetenyi, M., Transmission of Force between Two Semi-infinite Solids, ASME, Journal of Applied Mechanics, vol.
32, 1965, 671-674.
[7] Freund, L.B. and Barnett, D.M, A Two-Dimensional Analysis of Surface Deformation due to Dip-Slip Faulting, Bull. Seismol. Soc.
Am.,vol. 66, 1976, 667-675.
[8] Heaton, T. H. and Heaton, R. E., Static Deformation From Point Forces and Point Force Couples Located in Welded Elastic
Poissonian Half-spaces:Implications for Seismic Moment Tensors, Bull. Seism. Soc. Am., vol. 79, 1989, 813-841.
[9] Jungels, P.H. and Frazier, G.A, Finite Element Analyses of the Residual Displacements for an Earthquake Rupture: Source
Parameters for the San Fernando earthquake, J. Geophys. Res., vol. 78, 1973, 5062-5083.
[10] Kumari, G., Singh, S. and Singh, K., Static Deformation of Two Welded Half-spaces Caused by a Point Dislocation Source, Phys.
Earth. Planet, Inter,vol. 73, 1992, 53-76.

Paper Type : Research Paper
Title : Buoyancy Effects on Laminar Mixed Convection in Vertical Channel With Dissipation
Country : India
Authors : T. Gopal Reddy, M. Rama Chandra Reddy, B. Rama Bhupal Reddy
: 10.9790/5728-0463845      logo
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Abstract: Deals with the buoyancy effects on laminar mixed convection in vertical channel is considered. Solutions of the governing parabolic equations are obtained by the use of an implicit finite difference technique coupled with a marching procedure. The velocity, the temperature and the pressure profiles are presented graphically for different values of governing parameters like Eckert Number Ek, buoyancy parameter Gr/Re and Prandtl number Pr and their behaviour discussed.
Keywords: Mixed convection, Vertical channel, Dissipation
[1] Aung. W. Handbook of Single-Phase Convective Heat Transfer, Wiley, New York, 1987.
[2] Aung. W., and Worku, G., ASME. J. Heat Transfer, Vol. 108, pp.299- 304, 1986.
[3] Aung. W., and Worku, G., ASME. J. Heat Transfer, Vol. 108, pp.485- 488, 1986.
[4] Barletta, A., Int. J. Heat Mass Transfer, Vol.45, pp.641-654, 200
[5] Barletta, A., Int. J. Heat Mass Transfer Vol.48, pp 2042-2049, 2005.
[6] Barletta, A. Zanchini, E., Int. Commun. Heat Mass Transfer Vol. 28, pp.1043-1052, 2001.
[7] Barletta, A., and Zanchini, E., Int. J. Heat Mass Transfer Vol.44, pp 4267-4275, 2001.
[8] Bodoia, J.R., and Osterle, J.F., ASME. J. Heat Transfer, Vol. 84, pp.40-44, 196
[9] Chamkha Ali, J., Int. J. Heat Mass Transfer, Vol.45, pp.2509-2525, 200
[10] Chen, Y.C., and Chung, J.N., ASME J. Heat Transfer, Vol.120, pp.127-131, 1998.

Paper Type : Research Paper
Title : Clique Dominating Sets of Euler Totient Cayley Graphs
Country : India
Authors : M. Manjuri, B. Maheswari
: 10.9790/5728-0464649      logo
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Abstract: Graph Theory has been realized as one of the most flourishing branches of modern Mathematics finding widest applications in all most all branches of Sciences, Social Sciences, Engineering, Computer Science, etc. Number Theory is one of the oldest branches of Mathematics, which inherited rich contributions from almost all greatest mathematicians, ancient and modern. Using the number theoretic function Euler totient function we have defined an Euler totient Cayley graph and in this paper we study the Clique domination parameters of Euler totient Cayley graphs.
Keywords: Cayley Graph, Clique, Complete graph, Dominating clique, Euler totient Cayley Graph
[1]. Nathanson and B.Melvyn, Connected components of arithmetic graphs, Monat.fur.Math, 29, 1980, 219 – 220.
[2]. L.Madhavi, Studies on domination parameters and enumeration of cycles in some Arithmetic Graphs, Ph. D. Thesis submitted to
S.V.University, Tirupati, India, 2002.
[3]. S.Uma Maheswari, Some Studies on the Product Graphs of Euler Totient Cayley Graphs and Arithmetic 𝑉𝑛 Graphs, Ph. D. Thesis submitted to S.P.Women's University, Tirupati, India, 2012.
[4]. S.Uma Maheswari, and B.Maheswari, Domination parameters of Euler Totient Cayley Graphs, Rev.Bull.Cal.Math.Soc. 19 (2),
2011, 207-214.
[5]. O.Ore, Theory of Graphs, Amer. Math. Soc. Colloq. Publ., 38, Providence, 1962.
[6]. C. Berge, The Theory of Graphs and its Applications, Methuen, London 1962.
[7]. M.B. Cozzens, and L.L. Kelleher, Dominating cliques in graphs, Discrete Math. 86, 1990, 101-116.
[8]. G. Bacso and Z. Tuza, Dominating cliques in P5 – free graphs, Period. Math. Hungar. 21, 1990, 303-308.

Paper Type : Research Paper
Title : A Note on Water Transport Phenomenon by Homotopy Analysis Method
Country : India
Authors : Twinkle Singh, R.K. Singh
: 10.9790/5728-0465053      logo
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Abstract: In this paper, an analytic technique, named the Homotopy Analysis method (HAM) has been applied for solving Richard's equation, which is converted into the Basic Burger's equitation, which shows the well-known equations, to desire the behaviour of the infiltration of unsaturated zones in soil as a porous medium.
Keywords: Homotopy analysis method, Water transport phenomenon
[1] Brook, R.H. and A. T. Corey, Hydraulic Properties of Porous Media, Hydrol Paper 3, Colorado State University, Fort Collins, 1964.
[2] Corey, A. T., Mechanics of Immiscible Fluids in Porous Media, Water Resources Publication, Highlands Ranch, CO, 1994, pp: 252.
[3] Davood, D. Ganji, M. Esmaeilpour and E. Moheseni, Application of the Homotopy Perturbation Method to Micropolor Flow in a
Porous Channel, 2009.
[4] Liao, S.J., The proposed homotopy analysis technique for the solution of nonlinear problems. PhD. Thesis, Shanghai Jiao Tong
University, 1992.
[5] Liao, S.J., An explicit, totally analytic approximation of Blasius viscous flow problems. Int. J. Nonlinear Mech., 1999, 34: 759-778.
[6] Liao, S.J., Beyond perturbation; Introduction to the Homotopy Analysis Method. Champan and Hall/CRC Press, Boca Raton,
[7] Liao, S.J., On the analytic solution of magneto hydrodynamic flows of non-Newtonian fluids over a stretching sheet. J. Fluid
Mech., 2003b, 488:189-212 DOI: 10.1017/S0022112003004865
[8] Liao, S.J.,On the homotopy analysis method for nonlinear problems. Applied Mathe. Comput., 147 (2004), pp. 499–
[9] Liao, S.J., A new branch of solutions of boundary-layer flows over a permeable stretching plate. Int. J. Heat Mass Transfer 48
(2005) 2529 2539
[10] M. Ayub, A. Rasheed and T. Hayat.Exact flow of a third grade fluid past a porous plate using homotopy analysis method,
November, Pages 2091-2103 International Journal of Engineering Science Volume 41, Issue 18

Paper Type : Research Paper
Title : Standard Linear Combination of Incline Vectors
Country : India
Authors : AR. Meenakshi, P. Shakila Banu
: 10.9790/5728-0465458      logo
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Abstract: We prove that every finite subspace generated by the linearly ordered idempotent elements in an incline has a unique standard basis. This leads to every finite subspace of a regular incline whose elements are all linearly ordered has a unique standard basis and thereby we disprove the result of Cao that is "Every subspace of a finite incline whose idempotent elements are linearly ordered has a unique standard basis". As an application we exhibit that under certain conditions each vector in a finitely generated subspace of a vector space has a unique decomposition as a linear combination of the standard basis vectors.
Keywords: incline, regular incline, distributive lattice, basis, standard basis, standard linear combination
[1] Z.Q. Cao , K.H, Kim, F.W. Roush, Incline algebra and applications, John Wiley and Sons, New York, 1984.
[2] K.H. Kim, F.W. Roush, Inclines and incline matrices: a survey, Linear algebra appl., 379,457-473(2004).
[3] K.H.Kim, F.W.Roush, Generalized fuzzy matrices, Fuzzy sets and systems, 4 , 293-315 (1980) .
[4] AR.Meenakshi, Fuzzy matrix Theory and its applications, MJP Publishers, Chennai, 2008.
[5] AR.Meenakshi, S. Anbalagan, On regular elements in an incline, Int J. Math. and Math. Sci. (2010) article ID 903063, 12 Pages.
[6] AR.Meenakshi, P.Shakila Banu, Incline relational equations (communicated).

Paper Type : Research Paper
Title : Semi-Compatible Maps On Intuitionistic Fuzzy Metric Space
Country : India
Authors : Pradeep Kumar Dwivedi1 & Anil Rajput
: 10.9790/5728-0465964      logo
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Abstract: In this paper, we prove common flexed point theorems for semi-compatible mappings on intuitionistic fuzzy metric space with different some conditions of Park and Kim ([10], 2008). This research extended and generalized the results of Singh and Chauhan ([14], 2000). The concept of fuzzy set was developed extensively by many authors and used in various fields. Several authors have defined fuzzy metric space Kramosil and Michalek(([5],1975) etc.) with various methods to use this concept in analysis. Jungck (([3],1986), ([4],1988)) researched the more generalized concept compatibility than commutativity and weak commutativity in metric space and proved common fixed point theorems, and Singh and Chauhan ([14],2000) introduced the concept of compatibility in fuzzy metric space and studied common fixed point theorems for four compatible mappings.
[1] S. Banach; Theorie des operations linearires, Monografje Mathematyczne., Warsaw 1932.
[2] M. Grabiec; Fixed point in fuzzy metric spaces, Fuzzy Sets and Systems 27 (1988), 385-389.
[3] G. Jungck; Compatible mappings and common fixed points, Internat. J. Math. Math. Sci. 9 (1986), 779-791.
[4] G.Jungck, K.B. Moon and S. Park; Compatible mappings and common fixed point (2), Internat, J. Math. Math. Sci. 11 (1988), No. 2, 285-288.
[5] J. Kramosil and J. Michalek; Fuzzy metric and statistical metric spaces, Kybernetica 11 (1975), 326-334.
[6] J. H. Park; Intuitionistic fuzzy metric spaces, Chaos Solitons & Fractals 22 (2004), no. 5, 1039-1046.
[7] J.H. Park, J.S. Park, and Y.C. Kwun; A common fixed point theorem in the intuitionistic fuzzy metric space, Advances in Natural
Comput. Data Mining (Proc. 2nd ICNC and 3rd FSKD) (2006), 293-300.
[8] J.S. Park; On some results intuitionistic fuzzy metric space, J. Fixed Point Theory & Appl. 3 (2008), No. 1, 39-48.
[9] J.S. Park and S. Y. Kim; A fixed point Theorem in a fuzzy m3etric space, F. J.M.S. 1 (1999), No. 6, 927-934.
[10] J.S. Park and S. Y. Kim; Common fixed point theorem and example in intuitionistic fuzzy metric space, J.K.I.I.S. 18 (2008), no. 4,

Paper Type : Research Paper
Title : On Certain Class of Analytic Functions Involving Linear Operators
Country : India
Authors : Chena Ram, Garima Agarwal
: 10.9790/5728-0466572      logo
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Abstract: Invoking the Hadamard product (or convolution) , a class of univalent functions has been introduced. In the present paper we obtain necessary and sufficient conditions and some important properties for the analytic functions for its belongingness to certain class of functions. The distortion inequalities, closer theorems, radii of close-to-convexity, radii of starlikeness and radii of convexity are obtained for the same class of functions. Some properties involving Hadamard product are also obtained
Keywords: Univalent function; Hadamard product; Starlike function; convex function; Generalized hypergeometric function; Linear operator; Fractional differential and integral operators.
[1] A.A. Attiya and M.K.Aouf, A study on certain class of analytic function define by Ruscheweyh derivative, J. Soochow Journal of
mathematics, 33(2)(2007), 273-289.
[2] M. K. Aouf, H. E. Darwish and A. A. Attiya, Generalization of certain subclasses of analyticfunctions with negative coefficients,
Studia Univ. Babe-Bolyai Math., 45(1) (2000), 11-22.
[3] M. K. Aouf, H. E. Darwish and A. A. Attiya, On certain subclasses of analytic functions with negative coefficients, Southeast Asian
Bull. Math., 29(1)(2005), 1-16.
[4] M. K. Aouf, H. M. Hossen and A. Y. Lashin, On certain families of analytic functions with negative coefficients, Indian J. Pure
Appl. Math., 31(8) (2000), 999-1015.
[5] S. B. Joshi, An application of fractional calculus operator to a subclass of analytic functions with negative coefficients, J. Indian
Acad. Math., 25(2) (2003), 277-286.
[6] Chena Ram and Garima, On a class of meromorphically multivalent functions involving generalized hypergeometric functions, J.
Raj. Acad. of Phy. Sci., 11(3), 2012
[7] G. S. Salagean, Integral properties of certain classes of analytic functions with negativecoefficients, Int. J. Math. Math. Sci., 2005(1)
(2005), 125-131.
[8] A. Schild and H. Silverman, Convolutions of univalent functions with negative coefficients, Ann. Univ. Mariae Curie-Sklodowska
Sect. A., 29(1975), 99-106.
[9] Virchenko, N., Kalla, S.L. and Al-Zamel, A. (2001). Some results on a generalized hypergeometric function, Integral Transforms
and special Functions 12(1), 89-100.

Paper Type : Research Paper
Title : Fourier series And Fourier Transform
Country : India
Authors : Mr. Karan Asher
: 10.9790/5728-0467376      logo
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Abstract: An introduction to Fourier Series and Fourier Transform is the topic of this paper. It deals with what a Fourier Series means and what it represents. The general form of a Fourier Series with a provision for specific substitution has also been mentioned. The paper also includes a brief overview of Fourier Transform. The use of Fourier Transform to convert a time domain function into a frequency domain equivalent has also been shown. A method of converting the continuous Fourier Transform into a discrete form and thus obtaining the Discrete Fourier Transform has also been discussed. A few practical life application of Fourier analysis have been stated.

Paper Type : Research Paper
Title : Planar Near-Rings And Coding Theory
Country : India
Authors : Anil Kumar Kashyap, Madan Mohan Singh
: 10.9790/5728-0467780      logo
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Abstract: The purpose of this paper is to discuss the importance of algebraic coding theory and to investigate the special case in which BIB designs and codes are constructed from planar near-rings. Application of planar near-rings to binary codes were first explored by Modisett [12] and by Fuchs, Hofer and Pilz [14].
Keywords: planar near-ring, incidence structure, tactical configuration, BIBD, binary codes, block code.
[1] Aichinger, E.: "Planar rings", Results in Mathematics 30 (1996), 10–15.
[2] Abbasi, S.J. and Iqbal, K. : "On Units in Near Rings", TECHNOLOGY FORCES (Technol. forces): PAF -KIET Journal of
Engineering and Sciences Volume 02, Number 01, January- June 2008.
[3] Beidar, K. I., Fong, Yuen, and KE, Wen Fong "On finite circular planar near-rings", J. Algebra 85 (1996), 688–709.
[4] Blake, I. F. and Mullin, R. C.: "The Mathematical Theory of Coding", New York: Academic, 1975 .
[5] Clay, J. R.: "Generating balanced incomplete block designs from planar near-rings", J. Algebra 22 (1972), 319–331.
[6] Clay, J. R.: "Generating balanced incomplete block designs from planar nearrings", Oberwolfach, 1972.
[7] Claude Shannon: "The Mathematical Theory of Communications", Bell system,Technical journal,1948.
[8] Clay, J. R.: "Near-ring: Geneses and application", Oxford Univ. Press Inc. Oxford, 1992.
[9] Eggetsberger, Roland: "Codes from some residue class ring generated finite planar near-rings",Institutsber. No. 467,1993, Univ. Linz,
[10] Ferrero, G.:"Stems planari e BIB - disegni", Riv. Mat, Univ, Panna., Vol.11, pp 79-96, 1970.

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