The document proposes a new 4-point block method for direct integration of first-order ordinary differential equations. It derives the new method using interpolation and collocation techniques. The approximate solution is a combination of power series and exponential functions. The properties of the new integrator are investigated and it is found to be zero-stable, consistent, and convergent. The method is tested on some numerical examples and is found to perform better than some existing methods.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
Catalan Tau Collocation for Numerical Solution of 2-Dimentional Nonlinear Par...IJERA Editor
Tau method which is an economized polynomial technique for solving ordinary and partial differential
equations with smooth solutions is modified in this paper for easy computation, accuracy and speed. The
modification is based on the systematic use of „Catalan polynomial‟ in collocation tau method and the
linearizing the nonlinear part by the use of Adomian‟s polynomial to approximate the solution of 2-dimentional
Nonlinear Partial differential equation. The method involves the direct use of Catalan Polynomial in the solution
of linearizedPartial differential Equation without first rewriting them in terms of other known functions as
commonly practiced. The linearization process was done through adopting the Adomian Polynomial technique.
The results obtained are quite comparable with the standard collocation tau methods for nonlinear partial
differential equations.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
Catalan Tau Collocation for Numerical Solution of 2-Dimentional Nonlinear Par...IJERA Editor
Tau method which is an economized polynomial technique for solving ordinary and partial differential
equations with smooth solutions is modified in this paper for easy computation, accuracy and speed. The
modification is based on the systematic use of „Catalan polynomial‟ in collocation tau method and the
linearizing the nonlinear part by the use of Adomian‟s polynomial to approximate the solution of 2-dimentional
Nonlinear Partial differential equation. The method involves the direct use of Catalan Polynomial in the solution
of linearizedPartial differential Equation without first rewriting them in terms of other known functions as
commonly practiced. The linearization process was done through adopting the Adomian Polynomial technique.
The results obtained are quite comparable with the standard collocation tau methods for nonlinear partial
differential equations.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
On New Root Finding Algorithms for Solving Nonlinear Transcendental EquationsAI Publications
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In this paper we introduce a new operation on alpha cut for a symmetric hexagonal fuzzy numbers. We considered a transportation problem where the fuzzy demand and supply are in symmetric hexagonal fuzzy numbers and the minimum optimal cost is arrived .Transportation problems have various purposes in logistics and supply process for reducing the transportation cost’s The advantages of the proposed alpha
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The Analytical Nature of the Greens Function in the Vicinity of a Simple Poleijtsrd
It is known that the Green function of a boundary value problem is a meromorphic function of a spectral parameter. When the boundary conditions contain integro differential terms, then the meromorphism of the Greens function of such a problem can also be proved. In this case, it is possible to write out the structure of the residue at the singular points of the Greens function of the boundary value problem with integro differential perturbations. An analysis of the structure of the residue allows us to state that the corresponding functions of the original operator are sufficiently smooth functions. Surprisingly, the adjoint operator can have non smooth eigenfunctions. The degree of non smoothness of the eigenfunction of the adjoint operator to an operator with integro differential boundary conditions is clarified. It is indicated that even those conjugations to multipoint boundary value problems have non smooth eigenfunctions. Ghulam Hazrat Aimal Rasa "The Analytical Nature of the Green's Function in the Vicinity of a Simple Pole" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-4 | Issue-6 , October 2020, URL: https://www.ijtsrd.com/papers/ijtsrd33696.pdf Paper Url: https://www.ijtsrd.com/mathemetics/applied-mathamatics/33696/the-analytical-nature-of-the-greens-function-in-the-vicinity-of-a-simple-pole/ghulam-hazrat-aimal-rasa
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method is illustrated by solving second and third order FIVPs. The results show that the proposed method suits
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In this paper, we present new iterative algorithms to find a root of the given nonlinear transcendental equations. In the proposed algorithms, we use nonlinear Taylor’s polynomial interpolation and a modified error correction term with a fixed-point concept. We also investigated for possible extension of the higher order iterative algorithms in single variable to higher dimension. Several numerical examples are presented to illustrate the proposed algorithms.
AN ALPHA -CUT OPERATION IN A TRANSPORTATION PROBLEM USING SYMMETRIC HEXAGONAL...ijfls
In this paper we introduce a new operation on alpha cut for a symmetric hexagonal fuzzy numbers. We considered a transportation problem where the fuzzy demand and supply are in symmetric hexagonal fuzzy numbers and the minimum optimal cost is arrived .Transportation problems have various purposes in logistics and supply process for reducing the transportation cost’s The advantages of the proposed alpha
cut operations over existing methods is simpler and computationally more efficient in day to day applications.
The Analytical Nature of the Greens Function in the Vicinity of a Simple Poleijtsrd
It is known that the Green function of a boundary value problem is a meromorphic function of a spectral parameter. When the boundary conditions contain integro differential terms, then the meromorphism of the Greens function of such a problem can also be proved. In this case, it is possible to write out the structure of the residue at the singular points of the Greens function of the boundary value problem with integro differential perturbations. An analysis of the structure of the residue allows us to state that the corresponding functions of the original operator are sufficiently smooth functions. Surprisingly, the adjoint operator can have non smooth eigenfunctions. The degree of non smoothness of the eigenfunction of the adjoint operator to an operator with integro differential boundary conditions is clarified. It is indicated that even those conjugations to multipoint boundary value problems have non smooth eigenfunctions. Ghulam Hazrat Aimal Rasa "The Analytical Nature of the Green's Function in the Vicinity of a Simple Pole" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-4 | Issue-6 , October 2020, URL: https://www.ijtsrd.com/papers/ijtsrd33696.pdf Paper Url: https://www.ijtsrd.com/mathemetics/applied-mathamatics/33696/the-analytical-nature-of-the-greens-function-in-the-vicinity-of-a-simple-pole/ghulam-hazrat-aimal-rasa
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Seikkala derivative of fuzzy process is studied. The fourth order Runge-Kutta method based on Centroidal Mean
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method is illustrated by solving second and third order FIVPs. The results show that the proposed method suits
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We seek to explore the effects of three basic types of Collocation points namely points at zeros of Legendre polynomials, equally-spaced points with boundary points inclusive and equally-spaced points with boundary point non-inclusive. Established in literature is the fact that type of collocation point influences to a large extent the results produced via collocation method (using orthogonal polynomials as basis function). We
analyse the effect of these points on the accuracy of collocation method of solving second order BVP. For equally-spaced points we further consider the effect of including the boundary points as collocation points. Numerical results are presented to depict the effect of these points and the nature of problem that is best handled by each.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
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problems and establishing a process tending towards a higher ordered by alloying the already proved
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problems and establishing a process tending towards a higher ordered by alloying the already proved
conventional methods like Newton-Raphson method (N-R), Regula Falsi method (R-F) & Bisection method
(BIS). The present method is good to solve those nonlinear and transcendental equations that cannot be
solved by the basic algebra. Comparative analysis are also made with the other racing formulas of this
group and the result shows that present method is faster than all such methods of the class.
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Generally a range of equation solvers for estimating the solution of an equation contain the derivative of
first or higher order. Such solvers are difficult to apply in the instances of complicated functional
relationship. The equation solver proposed in this paper meant to solve many of the involved complicated
problems and establishing a process tending towards a higher ordered by alloying the already proved
conventional methods like Newton-Raphson method (N-R), Regula Falsi method (R-F) & Bisection method
(BIS). The present method is good to solve those nonlinear and transcendental equations that cannot be
solved by the basic algebra. Comparative analysis are also made with the other racing formulas of this
group and the result shows that present method is faster than all such methods of the class.
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability.
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Derivation and Application of Multistep Methods to a Class of First-order Ord...AI Publications
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A Fast Numerical Method For Solving Calculus Of Variation Problems
Gm2511821187
1. M.R. Odekunle, A.O. Adesanya, J. Sunday / International Journal of Engineering Research and
Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue 5, September-October 2012, pp.1182-1187
4-Point Block Method for Direct Integration of First-Order
Ordinary Differential Equations
M.R. Odekunle*, A.O. Adesanya*, J. Sunday**
*(Department of Mathematics, Modibbo Adama University of Technology, Yola, Nigeria)
**(Department of Mathematical Sciences, Adamawa State University, Mubi, Nigeria)
ABSTRACT
This research paper examines the development and application of block methods.
derivation and implementation of a new 4-point More recently, authors like [9], [10], [11], [12],
block method for direct integration of first-order [13], [14], [15], [16], have all proposed LMMs to
ordinary differential equations using generate numerical solution to (1). These authors
interpolation and collocation techniques. The proposed methods in which the approximate
approximate solution is a combination of power solution ranges from power series, Chebychev’s,
series and exponential function. The paper Lagrange’s and Laguerre’s polynomials. The
further investigates the properties of the new advantages of LMMs over single step methods have
integrator and found it to be zero-stable, been extensively discussed by [17].
consistent and convergent. The new integrator In this paper, we propose a new
was tested on some numerical examples and Continuous Linear Multistep Method (CLMM), in
found to perform better than some existing ones. which the approximate solution is the combination
of power series and exponential function. This work
Keywords: Approximate Solution, Block Method, is an improvement on [4].
Exponential Function, Order, Power Series
AMS Subject Classification: 65L05, 65L06, 65D30 2. METHODOLOGY: CONSTRUCTION
OF THE NEW BLOCK METHOD
1. INTRODUCTION Interpolation and collocation procedures
Nowadays, the integration of Ordinary are used by choosing interpolation point s at a grid
Differential Equations (ODEs) is carried out using point and collocation points r at all points giving
some kinds of block methods. Therefore, in this rise to s r 1 system of equations whose
paper, we propose a new 4-point block method for
coefficients are determined by using appropriate
the solution of first-order ODEs of the form:
procedures. The approximate solution to (1) is taken
y ' f ( x, y), y(a) a x b (1) to be a combination of power series and exponential
where f is continuous within the interval of function given by:
4 5
jxj
y( x) a j x j a5
integration [a, b] . We assume that f satisfies
(3)
Lipchitz condition which guarantees the existence j 0 j 0 j!
and uniqueness of solution of (1). The problem (1)
with the first derivative given by:
occurs mainly in the study of dynamical systems
4 5
j x j 1
and electrical networks. According to [1] and [2],
y '( x) ja j x j 1 a5 (4)
equation (1) is used in simulating the growth of
j 0 ( j 1)!
j 1
populations, trajectory of a particle, simple
where a j , for j 0(1)5 and y ( x) is
j
harmonic motion, deflection of a beam etc.
Development of Linear Multistep Methods (LMMs) continuously differentiable. Let the solution of (1)
for solving ODEs can be generated using methods
such as Taylor’s series, numerical integration and be sought on the partition N : a x0 < x1 < x 2 < .
collocation method, which are restricted by an . . < x n < xn 1 < . . .< x N = b , of the integration
assumed order of convergence, [3]. In this work, we
will follow suite from the previous paper of [4] by
interval a, b , with a constant step-size h , given
deriving a new 4-point block method in a multistep by, h xn 1 xn , n 0,1,..., N .
collocation technique introduced by [5].
Block methods for solving ODEs have Then, substituting (4) in (1) gives:
initially been proposed by [6], who used them as 4 5
j x j 1
starting values for predictor-corrector algorithm, [7] f ( x, y) ja j x j 1 a5 (5)
developed Milne’s method in form of implicit j 0 j 1 ( j 1)!
methods, and [8] also contributed greatly to the
1182 | P a g e
2. M.R. Odekunle, A.O. Adesanya, J. Sunday / International Journal of Engineering Research and
Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue 5, September-October 2012, pp.1182-1187
Now, interpolating (3) at point xn s , s 0 and A [a0 , a1 , a2 , a3 , a4 , a5 ]T
collocating (5) at points xn r , r 0(1)4 , leads to U [ yn , f n , f n 1 , f n 2 , f n 3 , f n 4 ]T
the following system of equations: and
AX U (6)
where
2 xn 3 xn 4 xn 5 xn
2 3 4 5
1 xn 2
xn 3
xn 4
xn 1 xn
2 6 24 120
3 xn 4 xn 5 xn
2 3 4
0 1 2 xn 2
3xn 3
4 xn xn
2
2 6 24
0 3 xn 1 4 xn 1 5 xn 1
2 3 4
xn 1
2 3 2
1 2 xn 1 3 xn 1 4 xn 1
2 6 24
X
3 xn 2 4 xn 2 5 xn 2
2 3 4
0 xn 2
2 3 2
1 2 xn 2 3 xn 2 4 xn 2
2 6 24
0 3 xn 3 4 xn 3 5 xn 3
2 3 4
2 xn 3
2 3
1 2 xn 3 3 xn 3 4 xn 3
2 6 24
3 xn 4 4 xn 4 5 xn 4
2 3 4
xn 4
2 3 2
0 1 2 xn 4 3 xn 4 4 xn 4
2 6 24
Solving (6), for a j ' s, j 0(1)5 and substituting Ym yn1 , yn 2 , yn3 , yn 4
T
back into (3) gives a continuous linear multistep
y n yn 3 , yn 2 , yn 1 , yn
T
method of the form:
F(Ym ) f n 1 , f n 2 , f n 3 , f n 4
4
y ( x ) 0 ( x ) yn h j ( x ) f n j
T
(7)
j 0
f (y n ) f n3 , f n2 , f n1 , f n
T
where 0 1 and the coefficients of
f n j gives:
1 0 0 0 0 0 0 1
0
1 0 0 0 0 0 1
1 A(0) , E ,
0 (6t 5 75t 4 350t 3 750t 2 720t ) 0 0 1 0 0 0 0 1
720
0 0 0 1 0 0 0 1
(12t 135t 520t 720t )
1
1 5 4 3 2
360 251
0 0 0
1
720
2 (3t 5 30t 4 95t 3 90t 2 )
60 0 0 0
29
1 90
3 (12t 5 105t 4 280t 3 240t 2 ) d
360 0 27
0 0
1 80
4 (6t 45t 110t 90t )
5 4 3 2
720
14
0 0 0
(8) 45
where t ( x xn ) h . Evaluating (7) at
t 1(1)4 gives a block scheme of the form:
A(0) Ym Ey n hdf (y n ) hbF (Ym ) (9)
where
1183 | P a g e
3. M.R. Odekunle, A.O. Adesanya, J. Sunday / International Journal of Engineering Research and
Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue 5, September-October 2012, pp.1182-1187
323 11 53 19 Expanding (10) using Taylor series and comparing
360 30 360 720 the coefficients of h gives:
L y ( x); h c0 y ( x) c1hy '( x) c2 h 2 y ''( x)
62 4 2 1
45 ... c p h p y p ( x) c p 1h p 1 y p 1 ( x) ...
15 45 90
b
51 9 21 3 (11)
40 10 40 80 3.1.1 Definition
64 8 64 14 The linear operator L and the associated
continuous linear multistep method (7) are said to
45 15 45 45 be of order p if
3. ANALYSIS OF BASIC
c0 c1 c2 ... c p and c p 1 0. c p 1 is
PROPERTIES OF THE NEW BLOCK
METHOD called the error constant and the local truncation
3.1 Order of the New Block Method error is given by:
Let the linear operator L y( x); h tnk cp1h( p1) y( p1) ( xn ) O(h p2 ) (12)
associated with the block (9) be defined as: For our method:
L y( x); h A(0)Ym Eyn hdf ( yn ) hbF (Ym )
(10)
251 323 11 53 19
720 360 30 360 720 f n
1 0 yn 1 1
0 0
29 62 4 2 1 f
0
1 0 0 yn 2 1 90 90
n 1
L y ( x); h yn h
45 15 45
f n2 0
0 0 1 0 yn 3 1 27 51 9 21 3
80 80 f n 3
0 0 0 1 yn 4 1 40 10 40
14 64 8 64 14 f n 4
45 45 15 45 45
(13)
Expanding (13) in Taylor series gives:
( h) j j 251h ' h j 1 j 1 323 j 11 53 19
j ! yn yn yn yn (1) (2) j (3) j (4) j
j 0 720 j 0 j ! 360 30 360 720
(2h) j
j
29h '
h j 1
62 4 2 1 0
yn yn yn ynj 1 (1) j (2) j (3) j (4) j
j 0 j ! 90 j 0 j ! 45 15 45 90 0
(3h) j 27 h ' h j 1 j 1 51 j 9 0
21 j 3 j
yn yn
j
yn yn (1) (2) (3) (4)
j
j 0 j ! 80 j 0 j! 40 10 40 80 0
j 1
(4h) y j y 14h y ' h y j 1 64 (1) j 8 (2) j 64 (3) j 14 (4) j
j
j ! n
j 0
n
45
n j ! n 45
j 0 15 45 45
(14)
c1 c2 c3 c4 c5 0, c6 1.88(02), 1.11(02), 1.88(02), 8.47(03) .
T
Hence, c0
Therefore, the new block method is of order five.
3.2 Zero Stability
3.2.1 Definition every root satisfying zs 1 have multiplicity not
The block method (9) is said to be zero-
exceeding the order of the differential equation.
stable, if the roots zs , s 1, 2,..., k of the first
Moreover, as h 0, ( z ) z ( z 1) where
r
characteristic polynomial ( z) defined by
is the order of the differential equation, r is the
( z ) det( zA (0) E) satisfies zs 1 and
1184 | P a g e
4. M.R. Odekunle, A.O. Adesanya, J. Sunday / International Journal of Engineering Research and
Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue 5, September-October 2012, pp.1182-1187
( z , h) ( z ) h ( z ) 0 satisfies
(0)
order of the matrices A and E (see [11] for
details). zs 1, s 1, 2,..., n where
For our new integrator,
f
1 0 0 0 0 0 0 1 h h and .
0 1 0 0 0 0 0 1 y
( z) z 0 (15) We shall adopt the boundary locus method for the
0 0 1 0 0 0 0 1 region of absolute stability of the block method.
Substituting the test equation y ' y into the
0 0 0 1 0 0 0 1
block formula gives,
( z ) z 3 ( z 1) 0, z1 z2 z3 0, z4 1 A (0) Ym (r ) Ey n (r ) h Dyn (r ) h BYm (r )
. Hence, the new block method is zero-stable.
(16)
3.3 Convergence
Thus:
The new block method is convergent by
consequence of Dahlquist theorem below. A(0)Ym (r ) Eyn (r )
h( r , h) (17)
3.3.1 Theorem [18] Dyn (r ) BYm (r )
The necessary and sufficient conditions Writing (17) in trigonometric ratios gives:
that a continuous LMM be convergent are that it be A(0)Ym ( ) Eyn ( )
consistent and zero-stable. h( , h) (18)
Dyn ( ) BYm ( )
3.4 Region of Absolute Stability where r ei . Equation (18) is our
3.4.1 Definition
characteristic/stability polynomial. Applying (18) to
The method (9) is said to be absolutely
our method, we have:
stable if for a given h , all the roots z s of the
characteristic polynomial
(cos 2 )(cos 3 )(cos ) (cos 2 )(cos 3 )(cos 4 )(cos )
h( , h) (19)
1 1
(cos 2 )(cos 3 )(cos ) (cos 2 )(cos 3 )(cos 4 )(cos )
5 5
which gives the stability region shown in fig. 1 below.
5
y
4
3
2
1
-5 -4 -3 -2 -1 1 2 3 4 5
-1 x
-2
-3
-4
-5
Fig. 1: Region of Absolute Stability of the 4-Point Block Method
4. NUMERICAL IMPLEMENTATIONS EOAS- Error in [4]
We shall use the following notations in the tables EBM- Error in [13]
below; EMY- Error in [12]
ERR- |Exact Solution-Computed Result|
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5. M.R. Odekunle, A.O. Adesanya, J. Sunday / International Journal of Engineering Research and
Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue 5, September-October 2012, pp.1182-1187
Problem 1 This problem was solved by [12] and [4].
Consider a linear first order IVP: They adopted block methods of order four to solve
y ' y, y(0) 1, 0 x 2, h 0.1 (20) the problem (20). We compare the result of our
with the exact solution: new block method (9) (which is of order five) with
their results as shown in table 1 below.
y ( x) e x (21)
Table 1: Performance of the New Block Method on y ' y, y(0) 1, 0 x 1, h 0.1
_____________________________________________________________________________________
x Exact solution Computed Solution ERR EOAS EMY
---------------------------------------------------------------------------------------------------------------------------------------
0.1000 0.9048374180359595 0.9048374025462963 1.5490(-08) 2.3231(-07) 2.5292(-06)
0.2000 0.8187307530779818 0.8187307437037037 9.3743(-09) 1.0067(-07) 2.0937(-06)
0.3000 0.7408182206817178 0.7408182037500000 1.6932(-08) 3.2505(-07) 2.0079(-06)
0.4000 0.6703200460356393 0.6703200296296297 1.6406(-08) 4.6622(-07) 1.6198(-06)
0.5000 0.6065306597126334 0.6065306344848305 2.5228(-08) 3.4071(-07) 3.1608(-06)
0.6000 0.5488116360940264 0.5488116163781553 1.9716(-08) 4.8161(-07) 2.7294(-06)
0.7000 0.4965853037914095 0.4965852802878691 2.3504(-08) 5.6328(-07) 2.5457(-06)
0.8000 0.4493289641172216 0.4493289421226676 2.1995(-08) 4.4956(-07) 2.1713(-06)
0.9000 0.4065696597405991 0.4065696328791496 2.6862(-08) 5.3518(-07) 3.1008(-06)
1.0000 0.3678794411714423 0.3678794189516900 2.2220(-08) 5.7870(-07) 2.7182(-06)
---------------------------------------------------------------------------------------------------------------------------------------
Problem 2 This problem was solved by [13] and [4].
Consider a linear first order IVP: They adopted a self-starting block method of order
y ' xy, y(0) 1, 0 x 2, h 0.1 (22) six and four respectively to solve the problem (22).
with the exact solution: We compare the result of our new block method (9)
(which is of order five) with their results as shown
x2
in table 2.
y ( x) e 2 (23)
Table 2: Performance of the New Block Method on y ' xy, y(0) 1, 0 x 1, h 0.1
_____________________________________________________________________________________
x Exact solution Computed Solution ERR EOAS EBM
---------------------------------------------------------------------------------------------------------------------------------------
0.1000 1.0050125208594010 1.0050122069965277 3.1386(-07) 5.2398(-07) 5.29(-07)
0.2000 1.0202013400267558 1.0202011563888889 1.3364(-07) 1.6913(-07) 1.77(-07)
0.3000 1.0460278599087169 1.0460275417187499 3.1819(-07) 8.7243(-07) 8.99(-07)
0.4000 1.0832870676749586 1.0832870577777778 9.8972(-09) 3.0098(-06) 3.09(-06)
0.5000 1.1331484530668263 1.1331477578536935 6.9521(-07) 1.7466(-06) 1.91(-06)
0.6000 1.1972173631218102 1.1972169551828156 4.0794(-07) 4.1710(-06) 4.48(-06)
0.7000 1.2776213132048868 1.2776205400192113 7.7319(-07) 9.6465(-06) 1.02(-05)
0.8000 1.3771277643359572 1.3771270561253053 7.0821(-07) 6.7989(-06) 7.74(-05)
0.9000 1.4993025000567668 1.4992996318515335 2.8682(-06) 1.2913(-05) 1.44(-05)
1.0000 1.6487212707001282 1.6487192043043850 2.0664(-06) 2.6575(-05) 2.93(-05)
---------------------------------------------------------------------------------------------------------------------------------------
5. CONCLUSION natural laws of growth and decay. Pacific
In this paper, we have proposed a new 4- Journal of Science and Technology, 12(1),
point block numerical method for the solution of 237-243, 2011.
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integrator proposed was found to be zero-stable, Gunavathy, Numerical methods (S. Chand
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Sofoluwe, A generalized 2-step continuous
implicit linear multistep method of hybrid
type. Journal of Institute of Mathematics
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Vol. 2, Issue 5, September-October 2012, pp.1182-1187
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