iosr-Jm   Volume-2 ~ Issue-2

Abstract: This paper presents fast iterative algorithms for solution of PDEs arisen from minimization of multiplicative noise removal model [14]. This model may be regarded as an improved version of the Total Variation (TV) de-noising models. For the TV and the multiplicative noise removal models, their associated Euler-Lagrange equations are highly nonlinear Partial Differential Equations (PDEs). For this model a very slow explicit time marching method has been reported. The main contribution we present in this paper is the implementation of the fixed point, semi-implicit and additive operator splitting schemes which do not yield good results. Consequently a fast and efficient multi-grid method with AOS as smoother is developed. Numerical experiments are presented to show the good performance of the fast multi-grid algorithm.
Key words. Synthetic Aperture Radar (SAR), Total Variation (TV)-based noise reduction, AOS (Additive Operator Splitting), Multi-Grid (MG), BV-Bounded Variation.

[1] G. Aubert and J. F. Aujol, A variational approach to removing multiplicative noise,SIAM Journal on Applied Mathematics 68 (2008), no. 4, 925–946.
[2] G. Aubert and P. Kornprobst, Mathematical problems in image processing of applied mathematical sciences, Springer, Berlin, Germany 147 (2002).
[3] N. Badshah and K. Chen, Multigrid method for the chan-vese model in variational segmentation, Communications in Computational Physics 4 (2008), no. 2, 294–316.
[4] N. Badshah and K. Chen, On two multi-grid algorithms for modelling variational multi-phase image segmentation, IEEE transactions on image Processing 18 (2009),no. 5, 1097–1106.7 Conclusion 13
[5] C.B. Burkhardt, Speckle in ultrasound b-mode scans,, IEEE Trans. Ultrasonic, 25 (1978), no. 1, 1–6.
[6] V. Vaselles G. Sapiro C. Ballester, M. Bertalmio and J. Verera, Filling in by joing interpolation of vector fields and grey levels, IMA Technical Report, university of Minnesota 69 (2002), no. 7, 131–147.
[7] Y. Liping C. Sheng, Y. Xin and S. Kun, Total variation based speckle reduction using multigrid algorithm for ultrasound images,, Springer-Verlag Berlin Heidelberg 36 (2005), no. 17, 245–252.
[8] T. F. Chan and K. Chen, On a nonlinear multi-grid algorithm with primal relaxation for the image total variation minimization, SIAM J. Sci. Comput. 20 (2006), no. 13, 387–411.
[9] R. Deriche, Fast algorithms for low-level vision, IEEE Transactions pattern Anal. Mach. Intell. 12 (1990), no. 9, 78–87.
[10] X. Zeng F. Tian Z. Li G, Liu and K. Chaibou, Speckle reduction by adaptive window anisotropic diffusion, signal processing 89 (2009), no. 11, 233–243.

Abstract : In this paper, the challenging difficulties encountered in solving non-polynomial variable coefficients differential equations by the use ofChebyshev expansion method is resolved. In such problems, where f(x) is non-polynomial, there exists the need to express products like f(x)T (x) r in series of Chebyshev polynomials for easy comparison of both sides of the differential equation. Numerical experiments is carried out on notable functions and the results are presented.
Keyword: Chebyshev polynomials, taylor's series expansion, non-polynomial variable coeefficients, .

[1] Grewal,B.S.Numerical methods in Engineering and science, 7th ed. Kanna Publishers Delhi, 2005.
[2] Fox, L. and Parker,I. B.Chebyshev Polynomials in Numerical analysis,Oxford University press NY Toroto, 1968.
[3] Fox, L.The use and construction of Mathematical tables, Math. Tab. Phys. Lab. 1, London, H. M. Stat. office, 1956.
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[5] Aysegul, A.,Chebyshev Polynomials in Numerical approximations for PDEs with complicated condition,Num. Methods for PDEs
25(3), 2008, 610-621.
[6] Mason, J.C. and Handscomb, D.C.,Chebyshev Polynomials, Rhapman & Hall – CRC, Roca Raton, London, New York, Washington
D.C. 2003
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[8] Clenshaw, C. W. A note on the summation of Chebyshev series.Math. Tab Wash. 9, 1955, 119- 120

Abstract : We explore the use of Legendre polynomials of the first kind in solving constant coefficients , nonhomogenous differential equations. To achieve this, trial solution is formulated with the use of Legendre polynomials as basis functions. We thereafter apply direct and indirect comparison techniques to reduce the entire problem whether initial or boundary value problems into a system of algebraic equations. Numerical examples are given to illustrate the efficiency and good performance of these methods.
Keywords: Algebraic equations, Direct comparison, Indirect comparison , Legendre polynomials.

[1] Mason, J.C. and Handscomb, D.C. (2003): Chebyshev Polynomials, Rhapman & Hall – CRC, Roca Raton, London, New York, Washington D.C
[2] Canuto, C. And Quateroni, A., Hussaini, M.Y. and Zang, T.A.; Spectral Methods; Fundamentals in single domains (2006): Springer- Verlag Berlin Heidelberg
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[5] Taiwo, O.A. and Olagunju, A. S.: Chebyshev Methods for the Numerical Solution of 4th order Differential Equations, Pioneer Journal of Mathematics and Mathematical science 3(1), 73-82. (2011)
[6] Kreyszig, E. Advanced Engineering mathematics, 8th ed. Wiley, (1999).
[7] Olagunju, A.S. Chebyshev- Collocation Approximation Methods for Numerical Solution of Boundary Value Problems. (2012), a Doctoral thesis, University of Ilorin, Nigeria.
[8] Boyd, J.P (2000): Chebyshev and Fourier Spectral Methods, 2nd ed. Dover, New York.
[9] Arfken, G., "Legendre functions of the second kind", Mathematical methods for physicists, 3rd ed. Orlando, FL: Academic press, pp. 701-707, (1985)
[10] David, S.B. Finite Element Analysis, from concepts to applications, AT&T Bell Laboratory, Whippany, New Jersey, (1987)

Abstract: In this paper, it is shown that neither of the iterative methods always converges. That is, it is possible to apply the Jacobi method or the Gauss-Seidel method to a system of linear equations and obtain a divergent sequence of approximations. In such cases, it is said that the method diverges.So for convergence, the Diagonal Dominance of the matrix is necessary condition before applying any iterative methods. Moreover, also discussed about the error reduction factor in each iteration in Jacobi and Gauss-Seidel method.
Keywords: Jacobi Method, Gauss-Seidel Method, Convergence and Divergence, Diagonal Dominance, Reduction of Error.

[1] Datta B N.Numerical Linear Algebra and Applications. Brooks/Cole Publishing Company,1995.
[2] Samuel D. Conte/Carl de Boor Elementary Numerical analysis An Algorithmic Approach, McGRAW-HILL INTERNATIONAL EDITION, Mathematics and statistics Series .
[3] Li W. A note on the preconditioned Gauss Seidel (GS) method for linear system. J. Comput. Appl.Math., 2005, 182: 81-90.
[4] Saad Y. Iterative Methods for Sparse Linear Systems. PWS Press, New York, 1995

Abstract: This paper is on the development of a new scheme for the solution of initial value problems in ordinary differential equations. We present some basic concepts and fundamental theories which are very vital to the development of the scheme. The new scheme is of order seven and its derivation is based on the representation of the theoretical solution by perturbed polynomial function of degree four. This scheme is suitable for problems associated with the systems of ordinary differential equations having oscillatory or exponential solution.
Key words: Convergence, Exponential Solution, Ordinary Differential Equation,

[1] Fatunla S. O. (1980): "Numerical Integrators for Stiff and Highly Oscillatory Differential Equations", Mathematics of Computation 34, 373-390.
[2] Gautschi W. (1961): "Numerical Integration of Ordinary Differential Equations based on Trigonometric Polynomials", Numerische Mathematik 3, 381-397.
[3] Ibijola E. A. (1997): New Schemes for Numerical Integration of Special Initial Value Problems in Ordinary Differential Equations Ph.D. Thesis, University of Benin, Nigeria.
[4] Ibijola E. A and Ogunrinde R. B. (2010): "On a New Numerical Scheme for the Solution of Initial Value Problems",Accepted Australian Journal of Basic and Applied Sciences.
[5] Joseph-Loius Lagrange. (1813): "On Root finding and Polynomial Interpolation in Ordinary Differential Equations".
[6] Ogunrinde R. B. (2009): On a New Numerical Scheme for the Solution of Initial Value Problem in Ordinary Differential Equations.
[7] Wallace C. S and Gupta G. K. (1973): "General Linear Multistep Methods to Solve Ordinary Differential Equations". The Australian Computer Journal, Vol. 5, 62-69.

Abstract: In this study, an EOQ (Economic Order Quantity) inventory mathematical model is constructed for a deteriorating item having time dependent demand when delay in payment is permissible. The deterioration rate is assumed to be a constant and the time varying demand rate is taken to be an exponential declining function of time. Mathematical models are also derived under two different circumstances, that is, Case I: The credit period is less than or equal to the cycle time for setting the account and Case II: The credit period is greater than the cycle time for setting the account. Numerical examples are provided to illustrate the model and the sensitivity analysis is also studied.
Keywords:– Deterioration Exponential declining demand, Inventory, Permissible delay in payment.

[1] H. M. Wagner, T. M. Whitin, Dynamic version of the economic lot size model, Management Science, 5, 1958, 89-96.
[2] P. Ghare, G. Schrader, A model for exponential decaying inventories, Journal of Industrial Engineering, 14, 1963, 238-243.
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[4] Aggarwal, S.P., A note on an order-level model for a system with constant rate of deterioration, Opsearch, 15, 1978, 184-187.
[5] R. Covert, G. C. Philip, An EOQ model for items with Weibull distribution, AIIE Transactions, 5, 1973, 323-326.
[6] Philip, G.C., A generalized EOQ model for items with Weibull distribution, AIIETransactions, 6, 1974, 159-162.
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Abstract: One of the momentous equations in financial mathematics is the Black-Scholes equation, a partial differential equation that governs the value of financial derivatives, such as options. In this paper, we attempt to show the application of Stochastic Process. We have shown how geometric Brownian motion & Ito's Lemma overlaps on Option Pricing.
Key Words: - Geometric Brownian motion, Ito's Lemma, Black-Scholes Equation

1] A. Malliaris. Ito's calculus in financial decision making. SIAM Review, 25(4):481-496, 1983.
[2] K. Ito. On stochastic differential equations. Memoirs, American Mathematical Society, (4):1-51, 1951.
[3] I. Gikhman and A. Skorokhod. Introduction to the Theory of Random Processes. W. B. Saunders Company, 1969.
[4] F. Black and M. Scholes. The pricing of options and corporate liabilities. The Journal of Political Economy, 81(3):637-654, 1973.
[5] R. Almgren. Financial derivatives and partial differential equations. The American Mathematical Monthly, 109(1):1-12, 2002.
[6] J. C. Hull & A. White, Journal of Finance, 42, (1987), 281.
[7] J. C. Hull, Options, Futures & Other Derivatives, Prentice Hall, (1997).
[8] R. C. Merton, Journal of Financial Economics, (1976), 125.
[9] Paul Wilmott, Quantitative Finance, John Wiley, Chichester, (2000).
[10] A. Dragulescu, & M. Yakovenko, Statistical Mechanics of money, income and wealth: A Short Survey, arXiv: cond-mat/0211175 v1, 9 Nov 2002.

Abstract: In this paper, we introduce and investigate the notions of Ig*-closed sets and Ig*-continuous maps in ideal topological spaces. Also we introduce the notion of G-I-LC*-sets and G-I-LC*-continuous maps to obtain decompositions of -continuity. Further, we introduce the notions of weakly GLC*-sets, rg*-closed sets and weakly GLC*-continuous maps, rg*-continuous maps in topological spaces to obtain decompositions of continuity.
Keywords: G-I-LC*-sets, weakly G-I-LC*-sets, Ig*-closed sets, Irg*-closed sets.

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