Runge Kutta Collocation Method for the Solution of First Order Ordinary Differential Equations
|
|
- Augusta Richard
- 5 years ago
- Views:
Transcription
1 Nonlinear Analysis and Differential Equations, Vol. 4, 2016, no. 1, HIKARI Ltd, Runge Kutta Collocation Method for the Solution of First Order Ordinary Differential Equations A. O. Adesanya 1 Department of Mathematics Modibbo Adama University of Technology Yola, Adamawa State, Nigeria A. U. Fotta Department of Mathematics Adamawa State Polytechnic Yola, Adamawa State, Nigeria R. O. Onsachi Department of Mathematics Modibbo Adama University of Technology Yola, Nigeria, Adamawa State, Nigeria Copyright c 2015 A. O. Adesanya, A. U. Fotta and R. O. Onsachi. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract We introduce one step continuous Runge Kutta collocation method with three free parameters for the solution of stiff first order ordinary differential equations. We adopt interpolation and collocation of the approximate solution at some selected grid points to give system of non linear equations. Using Crammer s rule to solve for the unknown parameters and substituting into the approximate solution gives the continuous method. To determine how best to fix the free parameters, we consider 1 Corresponding author
2 18 A. O. Adesanya, A. U. Fotta and R. O. Onsachi three cases; the Guass type, the equal interval, and the Radau type. Numerical results show that the farer the free parameters from the center, the better the results. Keywords: Runge Kutta collocation, interpolation, collocation, approximate solution, grid points, continuous method, stiff problems This paper considers solution to 1. Introduction y = f (x, y), y (x 0 ) = y 0 (1) where f is continuous and satisfies Lipschitz s condition, x 0 is the initial point and y 0 is the solution at x 0. Adoption of collocation and interpolation of power series approximate solution for the solution of initial value problems have been studied by [1 11], most of these methods fails when the problem is stiff or stiff oscillatory. Linear multistep method has been reported to be efficient and easier to implement for the solution of ordinary differential equations [12, 13]. [12 15] studied development of hybrid methods, they reported that hybrid methods are difficult to develop, but gives methods with good stability properties due to the reduction in the step length. [14 16] adopted approximate solution: n 1 y (x) = a j x j + a n e nx (2) j=0 where n is the numbers of interpolation and collocation points. They discovered that methods developed using (2) posses good stability properties which is good for stiff problems. In this paper, we introduce a new continuous Runge-Kutta collocation method with three free parameters using new approximate solution. The introduction of the free parameters and the approximate solution make this work different from the existing methods. Our interest also include investigation into how best to fix the free parameters, we consider three cases; Guass type, Radau type and equal interval methods. 2. Methods We consider an approximate solution N M y(x) = a n x n + a n e nx (3) n=0 N+1
3 Runge Kutta collocation method for the solution of first order ODE 19 The first derivative of (3) is y (x) = substituting (4)into (1)gives f (x, y) = N na n x n n=1 N na n x n n=1 M na n e nx (4) N+1 M na n e nx (5) we then construct a continuous approximation by imposing the following conditions N+1 Y (x n+j ) = y n+j, j = 0, 1,..., s (6) Y (x n+j ) = f n+j, j = 0, 1,..., r where s and r are the numbers of interpolation and collocation points respectively. we sought the solution of (1) on the partition Π N : x 0 < x 1 < x 2 <... < x N. over a constant stepsize h = x n+1 x n Interpolating (3) at x n+s, s = 0 and collocating (4) at points x n+r, gives a system of non linear equation XA = U (7) where A = [ a 0 a 1 a 2 a 3 a 4 a 5 a k ] T U = [ ] T y n f n f n+u f n+v f n+w f n+1 f n+k 1 x n x N n e (N+1)xn e Mxn 1 x n+1... x N n+1 e (N+1)x n+1 e Mx n x X = n+n... x N n+n e (N+1)x n+n e Mx n+n Nxn N 1 (N + 1) e (N+1)xn Me Mxn Nx N 1 n+n (N + 1) e (N+1)x n+1 Me Mx n Nx N 1 n+m (N + 1) e (N+1)x n+m Me Mx n+m Solving (7) for the a ks, gives an s-stage Runge-Kutta method s y n+1 = y n + h β i ϖ i i=0 ϖ i = f(x n + c i h, Y i ) s Y i = y n + h a ij ϖ j (8) j=0
4 20 A. O. Adesanya, A. U. Fotta and R. O. Onsachi where s is the internal stage, with the condition s c i = β i (9) To implement (8), [11] proposed a prediction equation in the form k Y i = y n + (c i ) i δ i f(x, y) δxi i=1 i=0 (xn,y n) Writing (5) compactly in a partitioned Butcher table gives c 1 a 11 a a 1s c [ ] 2 a 21 a a 2s.... c A b T = c s a s1 a s2... a ss b 1 b 2... b s (10) (11) [17] defined the stability function for the s-stage implicit R-K scheme as a rational function U 1 (z) given by U 1 (z) = 1 + zb T (I za) 1 e (12) where z = λh, I ia an identity matrix ans e is the s-vector e = [ ] T 2.1. Specification of the method. We consider X = 0 u < v < w < 1 A = [ a 0 a 1 a 2 a 3 a 4 a 5 ] T U = [ y n f n f n+u f n+v f n+w f n+1 ] T 1 x n e xn e 2xn e 3xn e 4xn 0 1 e xn 2e 2xn 3e 3xn 4e 4xn 0 1 e x n+u 2e 2x n+u 3e 3x n+u 4e 4x n+u 0 1 e x n+v 2e 2x n+v 3e 3x n+v 4e 4x n+v 0 1 e x n+w 2e 2x n+w 3e 3x n+w 4e 4x n+w 0 1 e x n+1 2e 2x n+1 3e 3x n+1 4e 4x n+1 Solving for the unknown constants, (8) gives y n+1 = y n + h(β 1 F 1 + β 2 F 2 + β 3 F 3 + β 4 F 4 + β 5 F 5 ) (13) where α 0 = 1 [ ] (5u + 5v + 5w 10uv 10uw 10vw + 30uvw 3) β 1 = 60uvw
5 Runge Kutta collocation method for the solution of first order ODE 21 (5v + 5w 10vw 3) β 2 = 60u (u v) (u w) (u 1) (5u + 5w 10uw 3) β 3 = 60v (u v) (v w) (v 1) (5u + 5v 10uv 3) β 4 = 60w (u w) (v w) (w 1) (15u + 15v + 15w 20uv 20uw 20vw + 30uvw 12) β 5 = 60 (u 1) (v 1) (w 1) The internal stages are: Y 1 = y n (14) Y 2 = y n + h(γ 1 F 1 + γ 2 F 2 + γ 3 F 3 + γ 4 F 4 + γ 5 F 5 ) (15) γ 1 = (10u2 v 5u 3 v + 10u 2 w 5u 3 w 5u 3 + 3u u 2 vw 30uvw) 60vw γ 2 = u (20uv 15u2 w 15u 2 v + 20uw 30vw 15u u uvw) 60 (u v) (u w) (u 1) γ 3 = u2 (10uw 5u 2 w 5u 2 + 3u 3 ) 60v (u v) (v w) (v 1) γ 4 = u2 (10uv 5u 2 v 5u 2 + 3u 3 ) 60w (u w) (v w) (w 1) γ 5 = u2 (3u 3 5u 2 w 5u 2 v + 10uvw) 60 (u 1) (v 1) (w 1) Y 3 = y n + h (µ 1 F 1 + µ 2 F 2 + µ 3 F 3 + µ 4 F 4 + µ 5 F 5 ) (16) µ 1 = (10uv2 5uv v 2 w 5v 3 w 5v 3 + 3v uv 2 w 30uvw) 60uw µ 2 = v2 (10vw 5v 2 w 5v 2 + 3v 3 ) 60u (u v) (u w) (u 1) µ 3 = v (20uv 15v2 w 15uv 2 30uw + 20vw 15v v uvw) 60 (u v) (v w) (v 1) µ 4 = v2 (10uv 5uv 2 5v 2 + 3v 3 ) 60w (u w) (v w) (w 1) µ 5 = v2 (3v 3 5v 2 w 5uv uvw) 60 (u 1) (v 1) (w 1) Y 4 = y n + h(η 1 F 1 + η 2 F 2 + η 3 F 3 + η 4 F 4 + η 5 F 5 ) (17) η 1 = (10uw2 5uw vw 2 5vw 3 5w 3 + 3w uvw 2 30uvw) 60uv η 2 = w2 (10vw 5vw 2 5w 2 + 3w 3 ) 60u (u v) (u w) (u 1)
6 22 A. O. Adesanya, A. U. Fotta and R. O. Onsachi η 3 = w2 (10uw 5uw 2 5w 2 + 3w 3 ) 60v (u v) (v w) (v 1) η 4 = w (20uw 15vw2 30uv 15uw vw 15w w uvw) 60 (u w) (v w) (w 1) η 5 = w2 (3w 3 5vw 2 5uw uvw) 60 (u 1) (v 1) (w 1) Y 5 = y n + h(β 1 F 1 + β 2 F 2 + β 3 F 3 + β 4 F 4 + β 5 F 5 ) F 1 = f(x n, Y 1 ); F 2 = f(x n + uh, Y 2 ); F 3 = f(x n + vh, Y 3 ); F 4 = f(x n + wh, Y 4 ); F 5 = f(x n + h, Y 5 ) Eqn (11) reduces to 0 u v w γ 1 γ 2 γ 3 γ 4 γ 5 µ 1 µ 2 µ 3 µ 4 µ 5 η 1 η 2 η 3 η 4 η 5 β 1 β 2 β 3 β 4 β 5 β 1 β 2 β 3 β 4 β Cases. Three cases are considered: (i): the equal interval method: u = 1, v = 1, w = (ii): the Guass type: 3 3, v = 1, w = (iii): the Radau Type: u = 6 6, v = 1, w = Analysis of the basic properties of the method (18) 3.1. Order and truncation error of the method. We associate with (13) the linear operator N (y (x) ; h) specified by N (y (x) ; h) = y (x + h) β 1 F 1 β 2 F 2 β 3 F 3 β 4 F 4 β 5 F 5 (19) The method (19) is said to be of order p if N (y (x) ; h) = O (h p+1 ) and the local truncation error T n+1 at x n+1 is given by N (y (x n ) ; h) where y (x n ) is now the theoretical solution y n = y (x n ) = T n+1 = y(x n+1 ) y n+1 Expanding (19) in Taylor series about x n,equating the coefficient of h, the truncation error gives ( ) T n+1 = h6 3u + 3v + 3w 5uv + O ( h uw 5vw + 10uvw 2 7) (20) 3.2. Consistency. Method (13) is said to be consistent if y n+1 y n lim h 0 h It can be shown that (13) is consistent = y n
7 Runge Kutta collocation method for the solution of first order ODE Zero stability. Method (13) is said to be zero stable if lim h 0 y n+1 = y n It can be shown that (13) is zero stable 3.4. Convergence. (i): Method (13) is said to be convergent if lim y (x) y (x n) 0 h 0 where y (x) is the numerical result, y (x n ) is the exact solution which is y n. It can be shown that (13) is convergence (ii): Method (13) is said to be convergent iff it is consisitent and zero stable. It can be shown that our method is consistence and sero stable, hence convergent 3.5. Stability of the method. The stability matrix becomes, where N (x) = D (x) = u 1 (z) = N (x) D (x) 36h 2 z 2 + 8h 3 z 3 + h 4 z hz 18h 2 uz 2 18h 2 vz 2 6h 3 uz 3 18h 2 wz 2 6h 3 vz 3 h 4 uz 4 6h 3 wz 3 h 4 vz 4 h 4 wz 4 24huz 24hvz 24hwz + 6h 2 uvz 2 + 6h 2 uwz 2 +4h 3 uvz 3 + 6h 2 vwz 2 + 4h 3 uwz 3 + h 4 uvz 4 + 4h 3 vwz 3 +h 4 uwz 4 + h 4 vwz 4 2h 3 uvwz 3 h 4 uvwz Convergence implies: 24hz + 6h 2 uz 2 + 6h 2 vz 2 + 6h 2 wz 2 24huz 24hvz 24hwz + 6h 2 uvz 2 + 6h 2 uwz 2 2h 3 uvz 3 + 6h 2 vwz 2 2h 3 uwz 3 2h 3 vwz 3 2h 3 uvwz 3 + h 4 uvwz (w 1) (v 1) (u 1) lim R (z) h uvw which is true for all values of u, v, w. It implies that the method is bounded s = { R (z) < 1} which shows that the method is A-stable 1
8 24 A. O. Adesanya, A. U. Fotta and R. O. Onsachi 4. Numerical Examples The following are use in the tables Error = y (x) y (x n ) where y (x) is the exact result and y n (x) is the computed result Error 1 = Error in case (i) Error 2 = Error in case (ii) Error 3 = rror in case (iii) Problem 1: We consider the highly stiff ordinary differential equation: y = 10(y 1) 2, y(0) = 2, h = 0.01 Exact solution: 1 y(x) = x Source: [16] Table 1: Results of problem I x Exact Error 1 Error 2 Error ( 10) ( 10) ( 10) ( 10) ( 10) ( 11) ( 11) ( 11) ( 11) ( 11) ( 11) ( 11) ( 11) ( 11) ( 11) ( 11) ( 11) ( 11) ( 12) ( 12) ( 12) Problem II: We consider the highly stiff ordinary differential equation: y = xy, y(0) = 1, h = 0.01 Exact solution: 1 y(x) = x Source: [16] Table 2: Results of problem II x Exact Error 1 Error 2 Error ( 16) ( 16) ( 15) ( 15) ( 15) ( 16) ( 14) ( 14) ( 14) ( 14) ( 14) ( 14) ( 13) ( 13) ( 14) ( 13) ( 13) ( 13) ( 13) ( 13) ( 13) ( 13) ( 13) ( 13) ( 12) ( 12) ( 13) ( 12) ( 12) ( 12) Problem III: We consider the highly stiff ordinary differential equation:
9 Runge Kutta collocation method for the solution of first order ODE 25 y = 100xy 2, y(1) = 1 51, h = 0.25 Exact solution: 1 y(x) = x 2 Source: [16] Table 3: Results of problem III x Exact Error 1 Error 2 Error ( 08) ( 08) ( 08) ( 09) ( 09) ( 09) ( 10) ( 10) ( 10) ( 10) ( 10) ( 10) ( 10) ( 10) ( 10) ( 11) ( 11) ( 11) ( 11) ( 11) ( 11) 5. Discussion of Result We consider three numerical examples; case (iii) which is the Radau type gives the best results followed by Case (ii) which is the Gauss type. The implication is that the farer u and w from v, the better the results. 6. Conclusion We discuss development of Runge Kutta collocation method with five internal stages for the solution of first order initial value problems. Three free parameters are considered, further work can be done by assigning different values to the free parameters to substantiate our claims above. The comparison of our method with the existing methods is not shown but our method compete favourably with the existing methods. References [1] P. Onumanyi, D. O. Awoyemi, S. N. Jator, U. W. Sirisena, New linear multistep method with constant coefficients for first order initial value problems, J. of Nig. Maths. Soc., 13 (1994), [2] P. Onumayi, U. W. Sirisena, S. N. Jator, Continuous finite difference approximation for solving differential equations, Intern. J. of Comput. Maths., 72 (1999), [3] J. D. Lambert, Computational Methods in Ordinary Differential Equations, John Wiley, New York, [4] J. D. Lambert, Safe point methods for separable stiff systems of ordinary differential equations, UK, University of Dundee, Department of Mathematics report No NA/31, (1979). [5] A. A. James, A. O. Adesanya, A note on the construction of constant order predictor corrector algorithm for the solution of y = f (x, y), British Journal of Mathematics and Computer Science, 4 (2014), no. 6,
10 26 A. O. Adesanya, A. U. Fotta and R. O. Onsachi [6] U. W. Sirisena, An accurate implementation of the butcher hybrid formula for the initial value problems in ordinary differential equations, Nig. J. Maths. and Appls., 12 (1999), [7] S. N. Jator, A Sixth Order Linear Multistep Method for the Direct Solution of y = f(x, y, y ), IJPAM, 40 (2007), no. 4, [8] D. O. Awoyemi, A new sixth-order algorithm for general second order ordinary differential equations, Intern. J. Compu. Math., 77 (2001), [9] A. O. Adesanya, A. A. James, S. Joshua, Hybrid block predictor-hybrid block corrector for the solution of first order ordinary differential equations, Eng. Math. Lett., 2014 (2014), [10] A. O. Adesanya, M. O. Udoh, A. M. Alkali, A new block-predictor corrector algorithm for the solution of y = f(x, y, y, y ), American Journal of Computational Mathematics, 2 (2012), [11] A. O. Adesanya, M. R. Odekunle, M. O. Udoh, Four steps continuous method for the solution of y = f(x, y, y ), American Journal of Computational Mathematics, 3 (2013), [12] T. A. Anake, D. O. Awoyemi, A. O. Adesanya, One step implicit hybrid block method for the direct solution of general second order ordinary differential equations, Intern. J. Appl. Math., (2012). [13] M. K. Fasasi, A. O. Adesanya, S. O. Adee, Block numerical integrator for the solution of y = f(x, y, y, y ), International Journal of Pure and Apllied Mathematics, 92 (2014), [14] J. Sunday, M.R. Odekunle, A. O. Adesanya, A. A. James, Extended block integrator for first order stiff and oscillatory differential equations, American Journal of Computational and Applied Mathematics, 3 (2013), no. 6, [15] J. Sunday, A. O. Adesanya, M. R. Odekunle, A self-starting four step fifth order block integrator for stiff and oscillatory differential equations, J. Maths. and Compu. Sci., 4 (2014), no. 1, [16] A. A. Momoh, A. O. Adesanya, M. K. Fasasi, A. Tahir, A new numerical integrator for the solution of stiff first order ordinary differential equations, Eng. Math. Lett., 2014 (2014), [17] S. O. Fatunla, Applied Numerical Method for Initial Value Problems in Ordinary Differential Equations, Academic Press Cambridge, Received: August 18, 2015; Published: October 16, 2015
On Nonlinear Methods for Stiff and Singular First Order Initial Value Problems
Nonlinear Analysis and Differential Equations, Vol. 6, 08, no., 5-64 HIKARI Ltd, www.m-hikari.com https://doi.org/0.988/nade.08.8 On Nonlinear Methods for Stiff and Singular First Order Initial Value Problems
More informationContinuous Block Hybrid-Predictor-Corrector method for the solution of y = f (x, y, y )
International Journal of Mathematics and Soft Computing Vol., No. 0), 5-4. ISSN 49-8 Continuous Block Hybrid-Predictor-Corrector method for the solution of y = f x, y, y ) A.O. Adesanya, M.R. Odekunle
More informationZ. Omar. Department of Mathematics School of Quantitative Sciences College of Art and Sciences Univeristi Utara Malaysia, Malaysia. Ra ft.
International Journal of Mathematical Analysis Vol. 9, 015, no. 46, 57-7 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.1988/ijma.015.57181 Developing a Single Step Hybrid Block Method with Generalized
More informationOne-Step Hybrid Block Method with One Generalized Off-Step Points for Direct Solution of Second Order Ordinary Differential Equations
Applied Mathematical Sciences, Vol. 10, 2016, no. 29, 142-142 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2016.6127 One-Step Hybrid Block Method with One Generalized Off-Step Points for
More informationOne Step Continuous Hybrid Block Method for the Solution of
ISSN 4-86 (Paper) ISSN -9 (Online) Vol.4 No. 4 One Step Continuous Hybrid Block Method for the Solution of y = f ( x y y y ). K. M. Fasasi * A. O. Adesanya S. O. Adee Department of Mathematics Modibbo
More informationBlock Algorithm for General Third Order Ordinary Differential Equation
ICASTOR Journal of Mathematical Sciences Vol. 7, No. 2 (2013) 127-136 Block Algorithm for General Third Order Ordinary Differential Equation T. A. Anake 1, A. O. Adesanya 2, G. J. Oghonyon 1 & M.C. Agarana
More informationModified block method for the direct solution of second order ordinary differential equations
c Copyright, Darbose International Journal of Applied Mathematics and Computation Volume 3(3),pp 181 188, 2011 http://ijamc.psit.in Modified block method for the direct solution of second order ordinary
More informationCONVERGENCE PROPERTIES OF THE MODIFIED NUMEROV RELATED BLOCK METHODS
Vol 4 No 1 January 213 CONVERGENCE PROPERTIES OF THE MODIFIED NUMEROV RELATED BLOCK METHODS YS, Awari 1 and AA, Abada 2 1 Department of Mathematics & 2 Department of Statistics, BinghamUniversity, Karu,
More informationSolution of First Order Initial Value Problem by Sixth Order Predictor Corrector Method
Global Journal of Pure and Applied Mathematics. ISSN 0973-1768 Volume 13, Number 6 (2017), pp. 2277 2290 Research India Publications http://www.ripublication.com/gjpam.htm Solution of First Order Initial
More informationA CLASS OF CONTINUOUS HYBRID LINEAR MULTISTEP METHODS FOR STIFF IVPs IN ODEs
ANALELE ŞTIINŢIFICE ALE UNIVERSITĂŢII AL.I. CUZA DIN IAŞI (S.N.) MATEMATICĂ, Tomul LVIII, 0, f. A CLASS OF CONTINUOUS HYBRID LINEAR MULTISTEP METHODS FOR STIFF IVPs IN ODEs BY R.I. OKUONGHAE Abstract.
More informationTwo Step Hybrid Block Method with Two Generalized Off-step Points for Solving Second Ordinary Order Differential Equations Directly
Global Journal of Pure and Applied Matematics. ISSN 0973-768 Volume 2, Number 2 (206), pp. 59-535 Researc India Publications ttp://www.ripublication.com/gjpam.tm Two Step Hybrid Block Metod wit Two Generalized
More informationA Zero-Stable Block Method for the Solution of Third Order Ordinary Differential Equations.
A Zero-Stable Block Method for the Solution of Third Order Ordinary Differential Equations. K. Rauf, Ph.D. 1 ; S.A. Aniki (Ph.D. in view) 2* ; S. Ibrahim (Ph.D. in view) 2 ; and J.O. Omolehin, Ph.D. 3
More informationAdebayo O. Adeniran1, Saheed O. Akindeinde2 and Babatunde S. Ogundare2 * Contents. 1. Introduction
Malaya Journal of Matematik Vol. No. 73-73 8 https://doi.org/.37/mjm/ An accurate five-step trigonometrically-fitted numerical scheme for approximating solutions of second order ordinary differential equations
More information-Stable Second Derivative Block Multistep Formula for Stiff Initial Value Problems
IAENG International Journal of Applied Mathematics, :3, IJAM 3_7 -Stable Second Derivative Bloc Multistep Formula for Stiff Initial Value Problems (Advance online publication: 3 August ) IAENG International
More informationA Family of Block Methods Derived from TOM and BDF Pairs for Stiff Ordinary Differential Equations
American Journal of Mathematics and Statistics 214, 4(2): 121-13 DOI: 1.5923/j.ajms.21442.8 A Family of Bloc Methods Derived from TOM and BDF Ajie I. J. 1,*, Ihile M. N. O. 2, Onumanyi P. 1 1 National
More informationA Family of L(α) stable Block Methods for Stiff Ordinary Differential Equations
American Journal of Computational and Applied Mathematics 214, 4(1): 24-31 DOI: 1.5923/.acam.21441.4 A Family of L(α) stable Bloc Methods for Stiff Ordinary Differential Equations Aie I. J. 1,*, Ihile
More informationUniform Order Legendre Approach for Continuous Hybrid Block Methods for the Solution of First Order Ordinary Differential Equations
IOSR Journal of Mathematics (IOSR-JM) e-issn: 8-58, p-issn: 319-65X. Volume 11, Issue 1 Ver. IV (Jan - Feb. 015), PP 09-14 www.iosrjournals.org Uniform Order Legendre Approach for Continuous Hybrid Block
More informationResearch Article Diagonally Implicit Block Backward Differentiation Formulas for Solving Ordinary Differential Equations
International Mathematics and Mathematical Sciences Volume 212, Article ID 767328, 8 pages doi:1.1155/212/767328 Research Article Diagonally Implicit Block Backward Differentiation Formulas for Solving
More informationFour Point Gauss Quadrature Runge Kuta Method Of Order 8 For Ordinary Differential Equations
International journal of scientific and technical research in engineering (IJSTRE) www.ijstre.com Volume Issue ǁ July 206. Four Point Gauss Quadrature Runge Kuta Method Of Order 8 For Ordinary Differential
More informationA Study on Linear and Nonlinear Stiff Problems. Using Single-Term Haar Wavelet Series Technique
Int. Journal of Math. Analysis, Vol. 7, 3, no. 53, 65-636 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/.988/ijma.3.3894 A Study on Linear and Nonlinear Stiff Problems Using Single-Term Haar Wavelet Series
More informationResearch Article P-Stable Higher Derivative Methods with Minimal Phase-Lag for Solving Second Order Differential Equations
Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2011, Article ID 407151, 15 pages doi:10.1155/2011/407151 Research Article P-Stable Higher Derivative Methods with Minimal Phase-Lag
More informationarxiv: v1 [math.na] 31 Oct 2016
RKFD Methods - a short review Maciej Jaromin November, 206 arxiv:60.09739v [math.na] 3 Oct 206 Abstract In this paper, a recently published method [Hussain, Ismail, Senua, Solving directly special fourthorder
More informationConstruction and Implementation of Optimal 8 Step Linear Multistep
Page 138 ORIGINAL RESEARCH Construction and Implementation of Optimal 8 Step Linear Multistep method Bakre Omolara Fatimah, Awe Gbemisola Sikirat and Akanbi Moses Adebowale Department of Mathematics, Lagos
More informationA NEW INTEGRATOR FOR SPECIAL THIRD ORDER DIFFERENTIAL EQUATIONS WITH APPLICATION TO THIN FILM FLOW PROBLEM
Indian J. Pure Appl. Math., 491): 151-167, March 218 c Indian National Science Academy DOI: 1.17/s13226-18-259-6 A NEW INTEGRATOR FOR SPECIAL THIRD ORDER DIFFERENTIAL EQUATIONS WITH APPLICATION TO THIN
More informationFourth Order RK-Method
Fourth Order RK-Method The most commonly used method is Runge-Kutta fourth order method. The fourth order RK-method is y i+1 = y i + 1 6 (k 1 + 2k 2 + 2k 3 + k 4 ), Ordinary Differential Equations (ODE)
More informationResearch Article An Accurate Block Hybrid Collocation Method for Third Order Ordinary Differential Equations
Applied Mathematics, Article ID 549597, 9 pages http://dx.doi.org/1.1155/14/549597 Research Article An Accurate Bloc Hybrid Collocation Method for Third Order Ordinary Differential Equations Lee Ken Yap,
More informationA Note on Multiplicity Weight of Nodes of Two Point Taylor Expansion
Applied Mathematical Sciences, Vol, 207, no 6, 307-3032 HIKARI Ltd, wwwm-hikaricom https://doiorg/02988/ams2077302 A Note on Multiplicity Weight of Nodes of Two Point Taylor Expansion Koichiro Shimada
More informationThe Generalized Viscosity Implicit Rules of Asymptotically Nonexpansive Mappings in Hilbert Spaces
Applied Mathematical Sciences, Vol. 11, 2017, no. 12, 549-560 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2017.718 The Generalized Viscosity Implicit Rules of Asymptotically Nonexpansive
More informationOn the Computational Procedure of Solving Boundary Value Problems of Class M Using the Power Series Method
Australian Journal of Basic and Applied Sciences, 4(6): 1007-1014, 2010 ISSN 1991-8178 On the Computational Procedure of Solving Boundary Value Problems of Class M Using the Power Series Method 1 2 E.A.
More informationRemarks on Fuglede-Putnam Theorem for Normal Operators Modulo the Hilbert-Schmidt Class
International Mathematical Forum, Vol. 9, 2014, no. 29, 1389-1396 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/imf.2014.47141 Remarks on Fuglede-Putnam Theorem for Normal Operators Modulo the
More informationAN OVERVIEW. Numerical Methods for ODE Initial Value Problems. 1. One-step methods (Taylor series, Runge-Kutta)
AN OVERVIEW Numerical Methods for ODE Initial Value Problems 1. One-step methods (Taylor series, Runge-Kutta) 2. Multistep methods (Predictor-Corrector, Adams methods) Both of these types of methods are
More informationGeneralized Boolean and Boolean-Like Rings
International Journal of Algebra, Vol. 7, 2013, no. 9, 429-438 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ija.2013.2894 Generalized Boolean and Boolean-Like Rings Hazar Abu Khuzam Department
More informationRemark on the Sensitivity of Simulated Solutions of the Nonlinear Dynamical System to the Used Numerical Method
International Journal of Mathematical Analysis Vol. 9, 2015, no. 55, 2749-2754 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2015.59236 Remark on the Sensitivity of Simulated Solutions of
More informationSecure Weakly Connected Domination in the Join of Graphs
International Journal of Mathematical Analysis Vol. 9, 2015, no. 14, 697-702 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2015.519 Secure Weakly Connected Domination in the Join of Graphs
More informationAn Accurate Self-Starting Initial Value Solvers for. Second Order Ordinary Differential Equations
International Journal of Contemporar Matematical Sciences Vol. 9, 04, no. 5, 77-76 HIKARI Ltd, www.m-iari.com ttp://dx.doi.org/0.988/icms.04.4554 An Accurate Self-Starting Initial Value Solvers for Second
More informationA Class of an Implicit Stage-two Rational Runge-Kutta Method for Solution of Ordinary Differential Equations
Journal of Applied Mathematics & Bioinformatics, vol.2, no.3, 2012, 17-31 ISSN: 1792-6602 (print), 1792-6939 (online) Scienpress Ltd, 2012 A Class of an Implicit Stage-two Rational Runge-Kutta Method for
More informationDisconvergent and Divergent Fuzzy Sequences
International Mathematical Forum, Vol. 9, 2014, no. 33, 1625-1630 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/imf.2014.49167 Disconvergent and Divergent Fuzzy Sequences M. Muthukumari Research
More informationPAijpam.eu NEW SELF-STARTING APPROACH FOR SOLVING SPECIAL THIRD ORDER INITIAL VALUE PROBLEMS
International Journal of Pure and Applied Mathematics Volume 118 No. 3 218, 511-517 ISSN: 1311-88 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu doi: 1.12732/ijpam.v118i3.2
More informationInvestigation on the Most Efficient Ways to Solve the Implicit Equations for Gauss Methods in the Constant Stepsize Setting
Applied Mathematical Sciences, Vol. 12, 2018, no. 2, 93-103 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2018.711340 Investigation on the Most Efficient Ways to Solve the Implicit Equations
More informationOn One Justification on the Use of Hybrids for the Solution of First Order Initial Value Problems of Ordinary Differential Equations
Pure and Applied Matematics Journal 7; 6(5: 74 ttp://wwwsciencepublisinggroupcom/j/pamj doi: 648/jpamj765 ISSN: 6979 (Print; ISSN: 698 (Online On One Justiication on te Use o Hybrids or te Solution o First
More informationA Note on Trapezoidal Methods for the Solution of Initial Value Problems. By A. R. Gourlay*
MATHEMATICS OF COMPUTATION, VOLUME 24, NUMBER 111, JULY, 1970 A Note on Trapezoidal Methods for the Solution of Initial Value Problems By A. R. Gourlay* Abstract. The trapezoidal rule for the numerical
More informationSolving Homogeneous Systems with Sub-matrices
Pure Mathematical Sciences, Vol 7, 218, no 1, 11-18 HIKARI Ltd, wwwm-hikaricom https://doiorg/112988/pms218843 Solving Homogeneous Systems with Sub-matrices Massoud Malek Mathematics, California State
More informationReformulation of Block Implicit Linear Multistep Method into Runge Kutta Type Method for Initial Value Problem
International Journal of Science and Technology Volume 4 No. 4, April, 05 Reformulation of Block Implicit Linear Multitep Method into Runge Kutta Type Method for Initial Value Problem Muhammad R., Y. A
More informationACG M and ACG H Functions
International Journal of Mathematical Analysis Vol. 8, 2014, no. 51, 2539-2545 HIKARI Ltd, www.m-hiari.com http://dx.doi.org/10.12988/ijma.2014.410302 ACG M and ACG H Functions Julius V. Benitez Department
More informationApplied Math for Engineers
Applied Math for Engineers Ming Zhong Lecture 15 March 28, 2018 Ming Zhong (JHU) AMS Spring 2018 1 / 28 Recap Table of Contents 1 Recap 2 Numerical ODEs: Single Step Methods 3 Multistep Methods 4 Method
More informationA New Embedded Phase-Fitted Modified Runge-Kutta Method for the Numerical Solution of Oscillatory Problems
Applied Mathematical Sciences, Vol. 1, 16, no. 44, 157-178 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/1.1988/ams.16.64146 A New Embedded Phase-Fitted Modified Runge-Kutta Method for the Numerical Solution
More informationRestrained Weakly Connected Independent Domination in the Corona and Composition of Graphs
Applied Mathematical Sciences, Vol. 9, 2015, no. 20, 973-978 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2015.4121046 Restrained Weakly Connected Independent Domination in the Corona and
More informationBasins of Attraction for Optimal Third Order Methods for Multiple Roots
Applied Mathematical Sciences, Vol., 6, no., 58-59 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/.988/ams.6.65 Basins of Attraction for Optimal Third Order Methods for Multiple Roots Young Hee Geum Department
More informationInternational Mathematical Forum, Vol. 9, 2014, no. 36, HIKARI Ltd,
International Mathematical Forum, Vol. 9, 2014, no. 36, 1751-1756 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/imf.2014.411187 Generalized Filters S. Palaniammal Department of Mathematics Thiruvalluvar
More informationSecure Connected Domination in a Graph
International Journal of Mathematical Analysis Vol. 8, 2014, no. 42, 2065-2074 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2014.47221 Secure Connected Domination in a Graph Amerkhan G.
More informationA Direct Proof of Caristi s Fixed Point Theorem
Applied Mathematical Sciences, Vol. 10, 2016, no. 46, 2289-2294 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2016.66190 A Direct Proof of Caristi s Fixed Point Theorem Wei-Shih Du Department
More informationA Generalization of p-rings
International Journal of Algebra, Vol. 9, 2015, no. 8, 395-401 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ija.2015.5848 A Generalization of p-rings Adil Yaqub Department of Mathematics University
More informationDevelopment of a New One-Step Scheme for the Solution of Initial Value Problem (IVP) in Ordinary Differential Equations
International Journal of Theoretical and Applied Mathematics 2017; 3(2): 58-63 http://www.sciencepublishinggroup.com/j/ijtam doi: 10.11648/j.ijtam.20170302.12 Development of a New One-Step Scheme for the
More informationChapter 6: Rational Expr., Eq., and Functions Lecture notes Math 1010
Section 6.1: Rational Expressions and Functions Definition of a rational expression Let u and v be polynomials. The algebraic expression u v is a rational expression. The domain of this rational expression
More informationInner Variation and the SLi-Functions
International Journal of Mathematical Analysis Vol. 9, 2015, no. 3, 141-150 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2015.411343 Inner Variation and the SLi-Functions Julius V. Benitez
More informationNumerical solution of stiff ODEs using second derivative general linear methods
Numerical solution of stiff ODEs using second derivative general linear methods A. Abdi and G. Hojjati University of Tabriz, Tabriz, Iran SciCADE 2011, Toronto 0 5 10 15 20 25 30 35 40 1 Contents 1 Introduction
More informationImprovements in Newton-Rapshon Method for Nonlinear Equations Using Modified Adomian Decomposition Method
International Journal of Mathematical Analysis Vol. 9, 2015, no. 39, 1919-1928 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2015.54124 Improvements in Newton-Rapshon Method for Nonlinear
More informationAnother Sixth-Order Iterative Method Free from Derivative for Solving Multiple Roots of a Nonlinear Equation
Applied Mathematical Sciences, Vol. 11, 2017, no. 43, 2121-2129 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2017.76208 Another Sixth-Order Iterative Method Free from Derivative for Solving
More informationOn Optimality Conditions for Pseudoconvex Programming in Terms of Dini Subdifferentials
Int. Journal of Math. Analysis, Vol. 7, 2013, no. 18, 891-898 HIKARI Ltd, www.m-hikari.com On Optimality Conditions for Pseudoconvex Programming in Terms of Dini Subdifferentials Jaddar Abdessamad Mohamed
More informationSymmetric Properties for Carlitz s Type (h, q)-twisted Tangent Polynomials Using Twisted (h, q)-tangent Zeta Function
International Journal of Algebra, Vol 11, 2017, no 6, 255-263 HIKARI Ltd, wwwm-hiaricom https://doiorg/1012988/ija20177728 Symmetric Properties for Carlitz s Type h, -Twisted Tangent Polynomials Using
More informationLinear Multistep Methods
Linear Multistep Methods Linear Multistep Methods (LMM) A LMM has the form α j x i+j = h β j f i+j, α k = 1 i 0 for the approximate solution of the IVP x = f (t, x), x(a) = x a. We approximate x(t) on
More informationBlock Milne s Implementation For Solving Fourth Order Ordinary Differential Equations
Engineering Technology & Applied Science Research Vol 8 No 3 2018 2943-2948 2943 Block Milne s Implementation For Solving Fourth Order Ordinary Differential Equations J G Oghonyon Department of Mathematics
More informationA Note of the Strong Convergence of the Mann Iteration for Demicontractive Mappings
Applied Mathematical Sciences, Vol. 10, 2016, no. 6, 255-261 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2016.511700 A Note of the Strong Convergence of the Mann Iteration for Demicontractive
More informationAug Vol. 5. No. 03 International Journal of Engineering and Applied Sciences EAAS & ARF. All rights reserved
FORMULATION OF PREDICTOR-CORRECTOR METHODS FROM 2-STEP HYBRID ADAMS METHODS FOR THE SOLUTION OF INITIAL VALUE PROBLEMS OF ORDINARY DIFFERENTIAL EQUATIONS 1 ABUBAKAR M. BAKOJI, 2 ALI M. BUKAR, 3 MUKTAR
More informationA Family of Optimal Multipoint Root-Finding Methods Based on the Interpolating Polynomials
Applied Mathematical Sciences, Vol. 8, 2014, no. 35, 1723-1730 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.4127 A Family of Optimal Multipoint Root-Finding Methods Based on the Interpolating
More informationOn Numerical Solutions of Systems of. Ordinary Differential Equations. by Numerical-Analytical Method
Applied Mathematical Sciences, Vol. 8, 2014, no. 164, 8199-8207 HIKARI Ltd, www.m-hiari.com http://dx.doi.org/10.12988/ams.2014.410807 On Numerical Solutions of Systems of Ordinary Differential Equations
More informationA Numerical-Computational Technique for Solving. Transformed Cauchy-Euler Equidimensional. Equations of Homogeneous Type
Advanced Studies in Theoretical Physics Vol. 9, 015, no., 85-9 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.1988/astp.015.41160 A Numerical-Computational Technique for Solving Transformed Cauchy-Euler
More informationFuzzy Sequences in Metric Spaces
Int. Journal of Math. Analysis, Vol. 8, 2014, no. 15, 699-706 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2014.4262 Fuzzy Sequences in Metric Spaces M. Muthukumari Research scholar, V.O.C.
More informationFifth-Order Improved Runge-Kutta Method With Reduced Number of Function Evaluations
Australian Journal of Basic and Applied Sciences, 6(3): 9-5, 22 ISSN 99-88 Fifth-Order Improved Runge-Kutta Method With Reduced Number of Function Evaluations Faranak Rabiei, Fudziah Ismail Department
More informationLinearization of Two Dimensional Complex-Linearizable Systems of Second Order Ordinary Differential Equations
Applied Mathematical Sciences, Vol. 9, 2015, no. 58, 2889-2900 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2015.4121002 Linearization of Two Dimensional Complex-Linearizable Systems of
More informationThe Expansion of the Confluent Hypergeometric Function on the Positive Real Axis
Applied Mathematical Sciences, Vol. 12, 2018, no. 1, 19-26 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2018.712351 The Expansion of the Confluent Hypergeometric Function on the Positive Real
More informationIntegrating ODE's in the Complex Plane-Pole Vaulting
MATHEMATICS OF COMPUTATION, VOLUME 35, NUMBER 152 OCTOBER 1980, PAGES 1181-1189 Integrating ODE's in the Complex Plane-Pole Vaulting By George F. Corliss Abstract. Most existing algorithms for solving
More informationAn Embedded Fourth Order Method for Solving Structurally Partitioned Systems of Ordinary Differential Equations
Applied Mathematical Sciences, Vol. 9, 0, no. 97, 484-48 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/0.988/ams.0.98 An Embedded Fourth Order Method for Solving Structurally Partitioned Systems of Ordinary
More informationGeneralized RK Integrators for Solving Ordinary Differential Equations: A Survey & Comparison Study
Global Journal of Pure and Applied Mathematics. ISSN 7-78 Volume, Number 7 (7), pp. Research India Publications http://www.ripublication.com/gjpam.htm Generalized RK Integrators for Solving Ordinary Differential
More informationDIRECTLY SOLVING SECOND ORDER LINEAR BOUNDARY VALUE PROBLEMS OF ORDINARY DIFFERENTIAL EQUATIONS. Ra ft Abdelrahim 1, Z. Omar 2
International Journal of Pure and Applied Matematics Volume 6 No. 6, -9 ISSN: - (printed version); ISSN: -95 (on-line version) url: ttp://www.ijpam.eu doi:.7/ijpam.v6i. PAijpam.eu DIRECTLY SOLVING SECOND
More informationHyers-Ulam-Rassias Stability of a Quadratic-Additive Type Functional Equation on a Restricted Domain
Int. Journal of Math. Analysis, Vol. 7, 013, no. 55, 745-75 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.1988/ijma.013.394 Hyers-Ulam-Rassias Stability of a Quadratic-Additive Type Functional Equation
More informationSecure Weakly Convex Domination in Graphs
Applied Mathematical Sciences, Vol 9, 2015, no 3, 143-147 HIKARI Ltd, wwwm-hikaricom http://dxdoiorg/1012988/ams2015411992 Secure Weakly Convex Domination in Graphs Rene E Leonida Mathematics Department
More informationAn Efficient Multiscale Runge-Kutta Galerkin Method for Generalized Burgers-Huxley Equation
Applied Mathematical Sciences, Vol. 11, 2017, no. 30, 1467-1479 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2017.7141 An Efficient Multiscale Runge-Kutta Galerkin Method for Generalized Burgers-Huxley
More informationDistribution Solutions of Some PDEs Related to the Wave Equation and the Diamond Operator
Applied Mathematical Sciences, Vol. 7, 013, no. 111, 5515-554 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.1988/ams.013.3844 Distribution Solutions of Some PDEs Related to the Wave Equation and the
More informationPowerPoints organized by Dr. Michael R. Gustafson II, Duke University
Part 6 Chapter 20 Initial-Value Problems PowerPoints organized by Dr. Michael R. Gustafson II, Duke University All images copyright The McGraw-Hill Companies, Inc. Permission required for reproduction
More informationBounded Subsets of the Zygmund F -Algebra
International Journal of Mathematical Analysis Vol. 12, 2018, no. 9, 425-431 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ijma.2018.8752 Bounded Subsets of the Zygmund F -Algebra Yasuo Iida Department
More informationA FAMILY OF EXPONENTIALLY FITTED MULTIDERIVATIVE METHOD FOR STIFF DIFFERENTIAL EQUATIONS
A FAMILY OF EXPONENTIALLY FITTED MULTIDERIVATIVE METHOD FOR STIFF DIFFERENTIAL EQUATIONS ABSTRACT. ABHULIMENC.E * AND UKPEBOR L.A Department Of Mathematics, Ambrose Alli University, Ekpoma, Nigeria. In
More informationSixth-Order and Fourth-Order Hybrid Boundary Value Methods for Systems of Boundary Value Problems
Sith-Order and Fourth-Order Hybrid Boundary Value Methods for Systems of Boundary Value Problems GRACE O. AKILABI Department of Mathematics Covenant University, Canaanland, Ota, Ogun State IGERIA grace.akinlabi@covenantuniversity.edu.ng
More informationNumerical Methods for Differential Equations
Numerical Methods for Differential Equations Chapter 2: Runge Kutta and Linear Multistep methods Gustaf Söderlind and Carmen Arévalo Numerical Analysis, Lund University Textbooks: A First Course in the
More informationA Class of Multi-Scales Nonlinear Difference Equations
Applied Mathematical Sciences, Vol. 12, 2018, no. 19, 911-919 HIKARI Ltd, www.m-hiari.com https://doi.org/10.12988/ams.2018.8799 A Class of Multi-Scales Nonlinear Difference Equations Tahia Zerizer Mathematics
More informationA NOTE ON EXPLICIT THREE-DERIVATIVE RUNGE-KUTTA METHODS (ThDRK)
BULLETIN OF THE INTERNATIONAL MATHEMATICAL VIRTUAL INSTITUTE ISSN 303-4874 (p), ISSN (o) 303-4955 www.imvibl.org / JOURNALS / BULLETIN Vol. 5(015), 65-7 Former BULLETIN OF THE SOCIETY OF MATHEMATICIANS
More informationMathematics for chemical engineers. Numerical solution of ordinary differential equations
Mathematics for chemical engineers Drahoslava Janovská Numerical solution of ordinary differential equations Initial value problem Winter Semester 2015-2016 Outline 1 Introduction 2 One step methods Euler
More informationReview Higher Order methods Multistep methods Summary HIGHER ORDER METHODS. P.V. Johnson. School of Mathematics. Semester
HIGHER ORDER METHODS School of Mathematics Semester 1 2008 OUTLINE 1 REVIEW 2 HIGHER ORDER METHODS 3 MULTISTEP METHODS 4 SUMMARY OUTLINE 1 REVIEW 2 HIGHER ORDER METHODS 3 MULTISTEP METHODS 4 SUMMARY OUTLINE
More informations-generalized Fibonacci Numbers: Some Identities, a Generating Function and Pythagorean Triples
International Journal of Mathematical Analysis Vol. 8, 2014, no. 36, 1757-1766 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2014.47203 s-generalized Fibonacci Numbers: Some Identities,
More informationOn Symmetric Property for q-genocchi Polynomials and Zeta Function
Int Journal of Math Analysis, Vol 8, 2014, no 1, 9-16 HIKARI Ltd, wwwm-hiaricom http://dxdoiorg/1012988/ijma2014311275 On Symmetric Property for -Genocchi Polynomials and Zeta Function J Y Kang Department
More informationAdvanced Studies in Theoretical Physics Vol. 8, 2014, no. 22, HIKARI Ltd,
Advanced Studies in Theoretical Physics Vol. 8, 204, no. 22, 977-982 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/0.2988/astp.204.499 Some Identities of Symmetry for the Higher-order Carlitz Bernoulli
More informationFixed Point Theorems for Modular Contraction Mappings on Modulared Spaces
Int. Journal of Math. Analysis, Vol. 7, 2013, no. 20, 965-972 HIKARI Ltd, www.m-hikari.com Fixed Point Theorems for Modular Contraction Mappings on Modulared Spaces Mariatul Kiftiah Dept. of Math., Tanjungpura
More informationMorera s Theorem for Functions of a Hyperbolic Variable
Int. Journal of Math. Analysis, Vol. 7, 2013, no. 32, 1595-1600 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2013.212354 Morera s Theorem for Functions of a Hyperbolic Variable Kristin
More informationSolving PDEs with PGI CUDA Fortran Part 4: Initial value problems for ordinary differential equations
Solving PDEs with PGI CUDA Fortran Part 4: Initial value problems for ordinary differential equations Outline ODEs and initial conditions. Explicit and implicit Euler methods. Runge-Kutta methods. Multistep
More informationSOME MULTI-STEP ITERATIVE METHODS FOR SOLVING NONLINEAR EQUATIONS
Open J. Math. Sci., Vol. 1(017, No. 1, pp. 5-33 ISSN 53-01 Website: http://www.openmathscience.com SOME MULTI-STEP ITERATIVE METHODS FOR SOLVING NONLINEAR EQUATIONS MUHAMMAD SAQIB 1, MUHAMMAD IQBAL Abstract.
More informationSecond Derivative Generalized Backward Differentiation Formulae for Solving Stiff Problems
IAENG International Journal of Applied Mathematics, 48:, IJAM_48 Second Derivative Generalized Bacward Differentiation Formulae for Solving Stiff Problems G C Nwachuwu,TOor Abstract Second derivative generalized
More informationHybrid Methods for Solving Differential Equations
Hybrid Methods for Solving Differential Equations Nicola Oprea, Petru Maior University of Tg.Mureş, Romania Abstract This paper presents and discusses a framework for introducing hybrid methods share with
More informationSolution of Differential Equations of Lane-Emden Type by Combining Integral Transform and Variational Iteration Method
Nonlinear Analysis and Differential Equations, Vol. 4, 2016, no. 3, 143-150 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/nade.2016.613 Solution of Differential Equations of Lane-Emden Type by
More informationChapter 5 Exercises. (a) Determine the best possible Lipschitz constant for this function over 2 u <. u (t) = log(u(t)), u(0) = 2.
Chapter 5 Exercises From: Finite Difference Methods for Ordinary and Partial Differential Equations by R. J. LeVeque, SIAM, 2007. http://www.amath.washington.edu/ rjl/fdmbook Exercise 5. (Uniqueness for
More informationSome New Three Step Iterative Methods for Solving Nonlinear Equation Using Steffensen s and Halley Method
British Journal of Mathematics & Computer Science 19(2): 1-9, 2016; Article no.bjmcs.2922 ISSN: 221-081 SCIENCEDOMAIN international www.sciencedomain.org Some New Three Step Iterative Methods for Solving
More information