Available via license: CC BY 4.0
Content may be subject to copyright.
New computational methods using seventh derivative
type for the solution of first order initial value problems
1,∗V. O. Atabo, 2S. O. Adee, 3P. O. Olatunji, 4D. J. Yahaya
1,4Department of Mathematics and Statistics, Confluence University of Science and Technology, Osara, Kogi
State, Nigeria
2Department of Mathematics, Modibbo Adama University, Yola, Nigeria
3Department of Mathematical Sciences, Adekunle Ajasin University, Akungba-Akoko, Ondo, Ondo State,
Nigeria
Abstract
In this research, a class of implicit block methods of a seventh derivative type are examined
through interpolation and collocation techniques using finite power series as the basis function.
The discrete schemes, which are implicit two-point block methods, are obtained by carefully
and unevenly choose collocation points that ensure better methods’ stability via test. However,
these schemes require seventh derivative functions unlike other existing numerical formulae.
The new methods are found, investigated and proven to be convergent and A-stable. The
implementation of methods is achieved by using Newton Raphson’s method. Experiments show
the efficiency and accuracy of the developed formulae on different class of first-order initial value
problems, including SIR, growth models and Prothero-Robinson oscillatory problem and with
comparison to such existing methods. In addition, it is observed that uneven and positioning
of collocation points greatly influence the efficiency and accuracy of numerical methods.
Keywords: Seventh derivative functions, implicit block methods, Algorithm, numerical
stability, interpolation and collocation.
1. Introduction
Stiff differential equations have been studied over the years with a view to developing robust
numerical methods that will not only be robust but adequate. It is worthy to note that
[1], first examined the best approach in terms of numerical methods to solving stiff ODEs.
Several scholars have different definitions to this resounding area of research. Therefore, it
can be defined as equations that are ill-conditioned.To unveil the nature of stiffness of the
ill-conditioning and to motivate the need to formulate efficient numerical methods for stiff
∗Corresponding author tel. no: +234 9033472648
Email addresses: ataboov@custech.edu.ng (1,∗V. O. Atabo), solomonadee@mau.edu.ng (2S. O. Adee),
peter.olatunji@aaua.edu.ng (3P. O. Olatunji), yahayajd@custech.edu.ng (4D. J. Yahaya)
1
https://doi.org/10.32388/C5IA9C
differential equations, consider the first order initial value problems of the form:
y′=f(x, y), a ≤x≤b, y(0) = y0,(1)
where, xn=x0+nh,his the step size. Also, a stiff system of equations is one for which
|λmax|is enormous, so that either the stability or the error bound or both can only be assured
by unreasonable restrictions on h(i.e., an excessively small hrequiring too many steps to
solve the initial value problem). Enormous here means, enormous relative to a scale which
is 1
b. Thus, an equation with |λmax|small may also be viewed as stiff if we must solve it for
great values of time, where f: [xn, xN]×Rm→Rmin (1) is continuous and differentiable;
so that, fis assumed to satisfy the existence and uniqueness theorem within the interval of
[a, b]; while stability is clearly necessary, it is not sufficient to obtain accurate solutions to
stiff systems of ordinary differential equations. A phenomenon that is commonly observed
is that when applied to stiff problems, many implicit methods do not seem to achieve the
order of accuracy that is expected for the method. This phenomenon is called order reduction.
Certainly, order reduction occurs with Runge-Kutta methods, but not backward differentiation
formula methods. In addition, explicit methods fail on solving stiff ODEs as a result of step-size
being restricted to maintain the potential accuracy of the methods. This problem is overcome
by using appropriate implicit methods (see [2]). However, some of the famous numerical
methods, among others are the Euler method by [3], linear multistep methods in [4] and
Runge-Kutta methods in [5]. In addition, the methods mentioned above cannot solve difficult
problems with stiff nature that arise in many fields of science and engineering. Hence, the
need to develop more viable methods for approximation. Also, [6] formulated a diagonally
implicit block backward differentiation formula for stiff IVPs. In [8,11,12,14,16,17,19],
implicit linear block multistep methods for first-order stiff and non-stiff IVPs have been derived
and implemented respectively. Interestingly, [21] also developed and implemented an implicit
four-point hybrid block integrator on stiff models relating to some real-life situations with
method near optimal as with other existing methods. Another implicit block methods have
been considered for solving stiff IVPs using Chebyshev polynomial in [22,23]. However, their
methods depend on the perturbed collocation approximation with shifted Legendre polynomials
as perturbation term.
More recently, are the applications of multi-derivatives block methods to first-order stiff initial
value problems [24]. However, higher derivative methods have a general disadvantage of having
to provide and evaluate derivatives of function thereby resulting to more functions evaluations.
Therefore, this drawback could result to round-off errors in the global iterations if numerical
methods are not sufficiently stable, that is, the numerical errors are not under check by the
zero stability and consistency properties.
Consequently, [25] derived and implemented fourth derivative k-point block formula on first-
order stiff IVPs through interpolation and collocation techniques. Similarly, [26] proposed a
third derivative trigonometrically fitted block method of a low order 2 for solving Equation (1).
2
Others like [27], considered a family of third derivative multi-step methods for solving (1) and
a class of continuous third derivative block methods of order (k+3) for direct approximation of
(1) has also been derived through interpolation and collocation techniques by [28]. For second
derivative methods, [29,30,31] solved Equation (1) respectively.
Summarily, in this research, a class of seventh derivative implicit block methods are derived.
They are a collection of discrete schemes of a first order function with seventh derivative
type. The objectives are to derive higher-order derivative implicit block formulae which solves
(1) directly with increased stability and reduced computational time using interpolation and
collocation approach. The proposed methods require seven derivative functions unlike other
numerical methods. This technique makes the methods unique, though have the burden of
having to provide the aforementioned derivative functions, but the efficiency and accuracy of
the proposed methods prove their significance. Test on numerical examples indicate that our
derived formulae are viable on stiff IVPs.
Therefore, this research is organized as follows: section two gives the derivation of the proposed
methods, section three shows the analysis of the numerical properties, section four presents the
implementation strategy, section five shows the numerical experiment and section six displays
the real-life application of methods, section seven presents conclusion and future research.
2. Derivation of the seventh derivative methods
We consider the power series polynomial of the form:
y(x) =
k+8
X
j=0
ajxj(2)
with its derivatives given as:
y′(x) =
k+8
X
j=0
jajxj−1=f(x, y)(3)
y′′(x) =
k+8
X
j=0
j(j−1)ajxj−2=g(x, y)(4)
y′′′(x) =
k+8
X
j=0
j(j−1)(j−2)ajxj−3=u(x, y)(5)
y′′′′(x) =
k+8
X
j=0
j(j−1)(j−2)(j−3)ajxj−4=v(x, y)(6)
3
.
.
.
With the seventh derivative given as:
y(7)(x) =
k+8
X
j=0
j(j−1)(j−2)(j−3)(j−4)(j−5)(j−6)ajxj−7=q(x, y)(7)
Here, we define:
y′
n+i=fn+i, i = 0(1)k, y′′
n+i=gn+i, i = 1,...k,y′′′
n+i=un+i, i =k, y(4)
n+i=vn+i, i =k, y(5)
n+i=
wn+i, i =k, y(6)
n+i=mn+i, i =k, y(7)
n+i=qn+i, i =k. Where aj′s∈Rin Equations (2)–(7) are
found using Gaussian elimination method. Therefore, Equation (2) and Equations (3)–(7) are
then interpolated and collocated at xnand xn+l, l = 0(1)k(where kis the step number and
k= 2) to give the following system of equation using Maple 18 soft environment:
P X =Q(8)
Where,
P=
1xnx2
nx3
nx4
nx5
nx6
nx7
nx8
nx9
nx10
n
0 1 2 xn+l3x2
n+l4x3
n+l5x4
n+l6x5
n+l7x6
n+l8x7
n+l9x8
n+l10x9
n+l
.
.
.
.
.
.
.
.
.
.
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
.
0 0 2 6xn+k12x2
n+k20x3
n+k30x4
n+k42x5
n+k56x6
n+k72x7
n+k90x8
n+k
0 0 0 6 24xn+k60x2
n+k120x3
n+k210x4
n+k336x5
n+k504x6
n+k720x7
n+k
0 0 0 0 24 120xn+k360x2
n+k840x3
n+k1680x4
n+k3024x5
n+k5040x6
n+k
0 0 0 0 0 120 720xn+k2520x2
n+k6720x3
n+k15120x4
n+k30240x5
n+k
0 0 0 0 0 0 720 5040xn+k20160x2
n+k60480x3
n+k151200x4
n+k
0 0 0 0 0 0 0 5040 40320 xn+1 181440x2
n+1 604800x3
n+1
0 0 0 0 0 0 0 5040 40320 xn+k181440x2
n+k604800x3
n+k
X=
a0
a1
.
.
.
ak+8
, Q =yn, fn+l, f ′
n+k, f (2)
n+k, f (3)
n+k, f (4)
n+k, f (5)
n+k, f (6)
n+1, f (6)
n+k⊤
Where, l= 0(1)k.
Equation (8) is then solved for aj′s∈R, j = 0(1)kand substitution made into Equation (2) gives the
4
Linear Multi-step Method (LMM) of the form:
y(xn+ξ) = α0(ξ)yn+h
k
X
j=0
βj(ξ)fn+j+h2X
j=3
βj(ξ)gn+k+h3X
j=4
βj(ξ)un+k
+h4X
j=5
βj(ξ)vn+k+h5X
j=6
βj(ξ)wn+k+h6X
j=7
βj(ξ)mn+k+h7
9
X
j=8
βj(ξ)
k
X
i=1
qn+i,(9)
Therefore, the parameters α0(ξ) and βj(ξ) are obtained with ξ=x−xnas:
α0= 1 (10)
β0=ξ−3
2
ξ2
h+2
3
ξ3
h2+7ξ4
8h3−63 ξ5
40 h4+7
6
ξ6
h5−1
2
ξ7
h6+33 ξ8
256 h7−43 ξ9
2304 h8+3ξ10
2560 h9(11)
β1=128 ξ3
3h2−112 ξ4
h3+672 ξ5
5h4−280 ξ6
3h5+ 40 ξ7
h6−21
2
ξ8
h7+14 ξ9
9h8−1
10
ξ10
h9(12)
β2=3
2
ξ2
h−130 ξ3
3h2+889 ξ4
8h3−5313 ξ5
40 h4+553 ξ6
6h5−79 ξ7
2h6+2655 ξ8
256 h7−3541 ξ9
2304 h8+253 ξ10
2560 h9(13)
β3=−5
2ξ2+ 44 ξ3
h−441 ξ4
4h2+525 ξ5
4h3−91 ξ6
h4+ 39 ξ7
h5−1311 ξ8
128 h6+583 ξ9
384 h7−25 ξ10
256 h8(14)
β4= 2 ξ2h−45 ξ3
2+217 ξ4
4h−1281 ξ5
20 h2+133 ξ6
3h3−19 ξ7
h4+639 ξ8
128 h5−853 ξ9
1152 h6+61 ξ10
1280 h7(15)
β5=−ξ2h2+23 ξ3h
3−419 ξ4
24 +203 ξ5
10 h−14 ξ6
h2+ 6 ξ7
h3−101 ξ8
64 h4+15 ξ9
64 h5−29 ξ10
1920 h6(16)
β6=1
3ξ2h3−17 ξ3h2
9+ 4 ξ4h−109ξ5
24 +28 ξ6
9h−4
3
ξ7
h2+45 ξ8
128 h3−181 ξ9
3456 h4+13 ξ10
3840 h5(17)
β7=−1
15 ξ2h4+14 ξ3h3
45 −37 ξ4h2
60 +41 ξ5h
60 −67 ξ6
144 +1
5
ξ7
h−17 ξ8
320 h2+23 ξ9
2880 h3−ξ10
1920 h4(18)
β8=−2h5ξ2
315 +2h4ξ3
105 −1
36 h3ξ4+11 h2ξ5
450 −hξ6
72 +13 ξ7
2520 −7ξ8
5760 h+ξ9
6048 h2−ξ10
100800 h3(19)
β9=2h5ξ2
315 −5h4ξ3
189 +1
20 h3ξ4−49 h2ξ5
900 +1
27 hξ6−9ξ7
560 +5ξ8
1152 h−121 ξ9
181440 h2+ξ10
22400 h3(20)
We then evaluate Equations (11) –(20) at ξ= 1 and ξ= 2, substitute into Equation (9) to give
the new formulated seventh derivative implicit block methods, acronym as “7D2PIB1 and 7D2PIB2”
respectively.
yn+1 =yn+5639
23040hfn+121
45 hfn+1 −44551
23040hfn+2 +1289
768 h2gn+2 −7687
11520h3un+2
+287
1920h4vn+2 −583
34560h5wn+2 +1
5760h6mn+2 −257
604800h7qn+1 +121
907200h7qn+2 (21)
5
yn+2 =yn+11
45hfn+128
45 hfn+1 −49
45hfn+2 +4
3h2gn+2 −26
45h3un+2 +2
15h4vn+2
−2
135h5wn+2 −2
4725h7qn+1 +2
14175h7qn+2 (22)
Similarly, we derived the second formula 7D2PIB2 and it is presented as:
yn+1 =yn+1663
11520hfn+121
45 hfn+1 −21119
11520hfn+2 +2837
1920h2gn+2 −2687
5760h3un+2
−257
5040h4vn+1 +1343
20160h4vn+2 −113
120960h5wn+2 −37
33600h6mn+2 +121
907200h7qn+2 (23)
yn+2 =yn+13
90hfn+128
45 hfn+1 −89
90hfn+2 +17
15h2gn+2 −17
45h3un+2 −16
315h4vn+1
+16
315h4vn+2 +h5wn+2
945 −2
1575h6mn+2 +2
14175h7qn+2 (24)
3. The stability analysis of the methods
This section presents the numerical properties and theorems (without proof) in relation to the
proposed numerical methods.
Theorem 3.1. Convergence [5]: The necessary and sufficient conditions for the linear multistep
method (LMM) of Equations (21)–(24) to be convergent are that it must be consistent and zero
stable.
Theorem 3.2. The necessary and sufficient condition for the method given by Equations (21)–(24)
to be zero stable is that it satisfies the root condition (See [5]).
Definition 3.1. Zero stability [10]
The numerical methods in Equations (21)–(24) are said to be zero stable if no root of te first
characteristic polynomial has a modulus greater than one and that every root with modulus one is
simple.
Definition 3.2. A-stability : A numerical method is said to be A-stable if the whole of the left-half
plane z:ℜ(z)≤0 is contained in the region z:ℜ(z)≤1. Where ℜ(z) is the stability polynomial of
the proposed method. (See [5]).
Definition 3.3. A(α)-stability : A numerical algorithm is said to be A(α) stable for some α∈[0,π
2]
if the wedge Sα={z:|Arg(−z)|< α, z = 0}is contained in its region of absolute stability. (See, [7]).
3.1. The Order of the 7D2PIB1 and 7D2PIB2
To establish the order of the derived methods, Equations (21) –(24) are rewritten in block form to
give the linear operator:
L(y(x); h) = A(1)Ym−A(0) Ym−1−B(0)Fm−1−B(1)Fm−h2C1Gm−h3C2Um
−h4C3Vm−h5C4Wm−h6C5Mm−h7C6Qm(25)
6
Where,
A(1) =
1 0
0 1
, A(0) =
0 1
0 1
, B(0) =
05639
23040
011
45
, B(1) =
121
45
128
45
−44551
23040 −49
45
,
C(1) =
01289
768
04
3
, C(2) =
0−7687
11520
0−26
45
, C(3) =
0287
1920
02
15
, C(4) =
0−583
34560
0−2
135
,
C(5) =
0−1
5760
0−2
135
C(6) =
−257
604800
121
907200
−2
4725
2
14175
, Ym=
yn+1
yn+2
, Ym−1=
yn−(k−1)
yn
,
F(Ym) =
fn+1
fn+k
, F (Ym−1) =
fn−(k−1)
fn
, Gm=
f′
n+1
f′
n+k
, Um=
f(2)
n+1
f(2)
n+k
, Vm=
f(3)
n+1
f(3)
n+k
,
Wm=
f(4)
n+1
f(4)
n+k
, Mm=
f(5)
n+1
f(5)
n+k
, Qm=
f(6)
n+1
f(6)
n+2
.
Note that fn+l, l = 0(1)kare the first-order derivative functions in x, y.
Equation (25) is expanded using Taylor series expansion, comparing their coefficients of powers of h
to give:
L(y(x); h) = q0y(x) + q1hy′(x) + q2h2y′′ (x) + . . . +qphpyp(x) + . . .
+qp+1hp+1yp+1(x) + . . .
(26)
Therefore, the linear operator L(y(x); h) in Equation (25) and the associated continuous linear
multistep methods in Equations (21)–(24) are said to be of order p if q0=q1=q2=. . . =qp= 0 and
qp+1 = 0. qp+1 is the error constant and the local truncation error is given by:
tn+k=qp+1h(p+1)y(p+1)(xn) + 0(hp+2)(27)
Therefore, using MAPLE 18, the order and error constants for “7D2PIB1 and 7D2PIB2” are
investigated as:
7
Table 1: Order and error constants
Method Order Error constant (qp+1)
7D2PIB1 10 −5881
7185024000 D(11) (y) (x)h11 +O(h12)
10 −23
28066500 D(11) (y) (x)h11 +O(h12)
7D2PIB2 10 −3931
12573792000 D(11) (y) (x)h11 +O(h12)
10 −31
98232750 D(11) (y) (x)h11 +O(h12)
3.2. Zero Stability
The zero stability polynomial of the formulated block methods in Equations (21)–(24) can be expressed
by evaluating:
R(t) =
(A(0)t−A(1))
(28)
Therefore, Equation (28) is then equated to zero and solved for tto give the characteristic roots each
for 7D2PIB1 and 7D2PIB2 as:
t= 0,1.Therefore, by Definition 3.1, it follows that the methods in Equations (21)–(24) are zero
stable.
3.3. Consistency
The necessary and sufficient condition that a numerical method be consistent is that its order, p≥1
. (See [4]).
Thus, the new methods whose order is 10 each, are certainly consistent.
3.4. Convergency
Inline with Theorem 3.1, the new derived block methods are convergent since they are both zero
stable and consistent. Let yiand y(xi) be the approximate and exact solution of (1) respectively,
then the absolute error is evaluated by using the formula:
AbsErr =|(yi)t−(y(xi))t|, 1 ≤t≤NS,
Where, NS is the total number of steps.
3.5. Linear stability
The absolute stability polynomials are presented below in the light of [13], using the test equations:
y′=λh, y(i)=λihi, i = 2(1)k+ 5,
8
So that substituting the above into Equations (21)–(24) yields:
M(w, z) = −A1w+A0+zB0+zB1w+z2B2w+z3B3w+z4B4w+z5B5w
+z6B6w+z7B7w(29)
Where z=λh and wis the difference equation shift operator. From which we have the following
expression as the stability polynomial:
πi(w, z) = |M(w, z)|, i = 1,2.(30)
The absolute stability regions are obtained by evaluating Equation (30) to give the following stability
functions for 7D2PIB1 and 7D2PIB2 respectively:
π1(w, z) = −w2
285768000z14 +w2
13608000z13 −23 w2
27216000z12 +w2
151200z11 −67 w2
1814400z10
+29 w2
201600z9+−1291 w2
3628800 −w
3628800z8+−127 w2
604800 −w
604800z7+11 w2
1350 z6
−7w2
135 z5+19 w2
90 z4−11 w2
18 z3+223 w2
180 −7w
180z2+−8
5w2−2
5wz+w2−w
π2(w, z) = −w2
2381400z11 +w2
113400z10 −23 w2
226800z9+w2
1260z8−367 w2
75600 z7+1817 w2
75600 z6
+−2923 w2
30240 −w
30240z5+523 w2
1680 −w
5040z4−7w2
9z3+64 w2
45 −w
45z2
+−17 w2
10 −3
10wz+w2−w
From which π1and π2are then coded in a MATLAB software environment and the region of absolute
stability for each derived method is as shown in Figures 1and 2below. Figures 1and 2indicate
Re(z)
-1012345
Im(z)
-4
-3
-2
-1
0
1
2
3
4
7D2PIB1
Figure 1: Absolute Stability Region of 7D2PIB1
9
Re(z)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Im(z)
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
7D2PIB2
Figure 2: Absolute Stability Region of 7D2PIB2
Re(z)
-1012345
Im(z)
-4
-3
-2
-1
0
1
2
3
4
7D2PIB1
7D2PIB2
Figure 3: Compared absolute stability region of methods
the region of absolute stability of the methods. The first method, 7D2PIB1, whose unstable region
is the closed region, is larger than the second method, 7D2PIB2, as precisely shown in Figure 3;
implying that 7D2PIB2 has an open region of larger stability region than 7D2PIB1. However, both
methods have regions of absolute stability that are left symmetric. Hence, both developed formulae
are A-stable inline with Definition 3.2.
10
4. Implementation of the methods
The simultaneous approximation of yn+lin the new methods was done using Newton Raphson’s
techniques on MATLAB software environment. Therefore,
y(i+1)
n+l=y(i)
n+l−f(y(i)
n+l)
f′(y(i)
n+l), l = 1(1)k. (31)
So that, Equation (31) can be rewritten as:
y(i+1)
n+l−y(i)
n+l= [f(y(i)
n+l)][f′(y(i)
n+l)]−1(32)
From which we get:
ej+1
n+1 = [f(y(i)
n+l)][f′(y(i)
n+l)]−1(33)
where,
f(y(i)
n+l) =
yn+1 −yn−1663
11520 hfn−121
45 hfn+1 +21119
11520 hfn+2 −2837
1920 h2gn+2
+2687
5760 h3un+2 +257
5040 h4vn+1 −1343
20160 h4vn+2 +113
120960 h5wn+2
+37
33600 h6mn+2 −121
907200 h7qn+2
yn+2 −yn−29
180 hfn+832
45 hfn+1 −3659
180 hfn+2 +64
15 h2gn+1
+159
10 h2gn+2 −539
90 h3un+2 +7
5h4vn+2 −59
270 h5wn+2 +1
45 h6mn+2
−17
14175 h7qn+2
and ej+1
n+1 =y(i+1)
n+l−y(i)
n+l; while f(y(i)
n+l) is a system of equations and f′(y(i)
n+l) is a (2 ×2) Jacobian
matrix, g, u, v, w, m and qare second, third, fourth, fifth, sixth and seventh derivatives respectively.
Since the new block is self-starting, it does not require starting formula to incorporate all the initial
values for the stiff IVPs. Therefore, approximate solutions yn+lare simultaneously generated.
11
Algorithm 1 Proposed Methods Algorithm
Input: Define initial guess: f(x), df (x), N, h, [a, b],where f(x) is the problem to be solved
and df(x) is the derivative function, eis the tolerance, Ntotal number of iterations and h
is the step-size and [a, b] is the iterations interval.
Output: ynew =y(i+1)
n+l
1: Define yold =y(i)
n+l,[a, b], h =(b−a)
N
2: for j= 1 : N−1, do
3: x(j) = x0+jh
4: while |(yold −ynew)|> tol,do
5: ynew =yold −f′(x,y)
f(x,y)
6: Print ynew =yold
7: for j=j+ 1, do
8: Goto 3
9: end for
10: if j≥N,then
11: Goto 4
12: end if
13: end while
14: Goto 6
15: end for
16: Stop
5. Numerical Experiment
The following first order stiff initial value problems are used to test the performance of the new
method and comparison, where possible, are made with some selected existing methods of close or
higher orders. The test problems considered here are either mild or highly stiff first order IVPs.
Problem 1. Consider the first order system of stiff initial value problem:
y′
1=−8y1+ 7y2, y1(0) = 1, h = 0.1,
y′
2= 42y1−43y2, y2(0) = 8,
Exact Solution:
y1(t)=2e−x−e−50x
y2(t)=2e−x+ 6e−50x
Source: Skwame et al. [9]
Table 2shows the results from solving Problem 1with their efficiency curves shown in Figures 4and
5. The figures show that at many grid and approximate points of iterations, proposed methods show
small scale error with better accuracy in 7D2PIB2 than 7D2PIB1 and method in [9]. It is evident that
with h= 0.1, the proposed formulae show sufficient efficiency and improved accuracy. This efficiency
of the derived methods indicate that with smaller step-sizes, the methods could have smaller scale
errors and therefore approximate solutions could tend to their true solutions.
12
Table 2: Comparison of Absolute Error for Problem 1with h= 0.1
xError in [9], p= 10 7D2PIB1, p= 10 7D2PIB2, p= 10
y1y2y1y2y1y2
0.1 1.32e-06 8.10e-02 4.36e-03 2.62e-02 6.31e-04 3.79e-02
0.2 1.90e-08 5.50e-04 7.78e-05 4.67e-04 8.11e-06 4.870e-05
0.3 4.00e-09 3.70e-06 1.06e-06 6.37e-06 7.82e-08 4.69e-07
0.4 4.00e-09 2.10e-08 1.31e-08 7.88e-08 6.71e-10 4.02e-09
0.5 2.00e-09 3.00e-09 1.55e-10 9.28e-10 5.39e-12 3.24e-11
0.6 3.00e-09 2.00e-09 1.73e-12 1.07e-11 3.78e-14 2.55e-13
0.7 4.50e-09 2.90e-09 2.37e-14 1.68e-13 4.33e-15 6.77e-15
0.8 4.10e-09 3.70e-09 3.88e-14 4.44e-14 4.00e-15 4.11e-15
0.9 4.60e-09 4.00e-09 3.49e-14 3.81e-14 3.55e-15 3.78e-15
1.0 4.80e-09 4.60e-09 3.24e-14 3.59e-14 2.55e-15 2.44e-15
Problem 2. Consider the first order system of stiff initial value problem:
y′
1=−9y1+ 95y2, y1(0) = 1, h = 0.1,
y′
2=−y1−97y2, y2(0) = 1,
Exact Solution:
y1(t) = 95
47 e−2x−48
47 e−96x
y2(t) = 48
47 e−96x−1
47 e−2x
Problem 2was solved in [9] by using an implicit block method of a uniform order 10. The same
is solved using the proposed formulae. The results are presented in Table 3with their efficiency
curves shown in Figures 6and 7. It is clear that the proposed methods show better accuracy with
7D2PIB2 outperforms 7D2PIB1 and such method in [9]. The figures show that at many grid points,
the proposed methods have smaller scale absolute errors, which indicate consistency in terms of
numerical properties, as errors decrease as iterations proceed.
13
Table 3: Comparison of Absolute Error for Problem 2with h= 0.1
xError in [9], p= 10 7D2PIB1, p= 10 7D2PIB2, p= 10
y1y2y1y2y1y2
0.1 1.74×10−41.74 ×10−43.70×10−43.70×10−49.22×10−49.22×10−4
0.2 5.40×10−85.30×10−81.84×10−71.84×10−77.07×10−77.07×10−7
0.3 1.00×10−94.00×10−11 8.01×10−11 8.07×10−11 5.94×10−10 5.94×10−10
0.4 2.30×10−93.50×10−11 4.27×10−13 2.99×10−14 4.46×10−13 4.95×10−13
0.5 2.20×10−93.10×10−11 3.69×10−13 3.88×10−15 3.71×10−14 7.95×10−16
0.6 1.80×10−92.70×10−11 2.92×10−13 3.08×10−15 2.81×10−14 2.93×10−16
0.7 1.60×10−92.20×10−11 2.34×10−13 2.47×10−15 1.93×10−14 2.04×10−16
0.8 1.40×10−92.00×10−11 1.85×10−13 1.95×10−15 1.44×10−14 1.49×10−16
0.9 1.20×10−91.60×10−11 1.45×10−13 1.53×10−15 9.44×10−15 9.98×10−17
1.0 9.10×10−10 1.40×10−11 1.16×10−13 1.22×10−15 6.50×10−15 6.68×10−17
Problem 3. Consider the first order stiff initial value problem:
y′=x−y, 0≤x≤1, h = 0.1,
Exact Solution:
y(x) = x+e−x−1
The above Problem has been considered in [32] with a uniform block order of 13. Their method was
directly employed without starting values. The results of the derived formulae are presented in Table 4
with the efficiency curves shown in Figure 8. A clear comparison of our derived methods indicates that
7D2PIB2 outperformed 7D2PIB1 of the same order 10, though with minimal comparable performance
in accuracy while outperformed such a method of order 13 in [32]. Figure 7shows the competitive
performance of 7D2PIB1, 7D2PIB2 and with such existing methods in [32]. It is clear that 7D2PIB1
and 7D2PIB2 show convergence at the last two grid points of the iterations.
14
Table 4: Comparison of Absolute Error for Problem 3with h= 0.1
xError in [32], p= 13 7D2PIB1, p= 10 7D2PIB2, p= 10
0.1 1.9595 ×10−11 3.8165×10−17 3.29598×10−17
0.2 3.54623×10−11 4.85723×10−17 3.81639×10−17
0.3 4.81315×10−11 4.85723×10−17 6.93889×10−17
0.4 5.80680×10−11 1.66534×10−16 1.38778×10−16
0.5 6.56779×10−11 2.77556×10−17 2.77556×10−17
0.6 7.13132×10−11 1.66534×10−16 1.66534×10−16
0.7 7.52814×10−11 1.11022×10−16 1.11022×10−16
0.8 7.78485×10−11 5.55112×10−17 5.55112×10−17
0.9 7.92403×10−11 0.00000×10−00 0.00000×10−00
1.0 7.96712×10−11 0.00000×10−00 0.00000×10−00
Problem 4. Consider the first order stiff initial value problem:
y′=−y, 0≤x≤1, h = 0.1,
Exact Solution:
y(x) = e−x
Problem 4has been solved in [18] with one-step and two-step hybrid block methods of uniform order
10 and 18 respectively. The proposed methods are applied using a step-size, h= 0.1. Results from
application, as shown in Table 5, indicate the competitive performance of our first formula, 7D2PIB1
with those in [18] particularly, 2 SHBM of order 18. Figure 9depicts the convergence at some grid
points of the iterations in 7D2PIB1 while outperforms, particularly 2 SHBM of a uniform order 18.
15
Table 5: Comparison of Absolute Error for Problem 4with h= 0.1
x1 SHBM [18], p= 10 2 SHBM [18], p= 18 7D2PIB1, p= 10
0.1 0.00e+00 4.10e-20 0.00e+00
0.2 1.10e-20 6.10e-20 0.00e+00
0.3 2.10e-20 8.10e-20 1.11e-16
0.4 1.10e-20 1.11e-20 0.00e+00
0.5 1.10e-20 1.21e-19 2.22e-16
0.6 2.10e-20 1.31e-19 2.22e-16
0.7 1.10e-20 1.41e-19 1.67e-16
0.8 2.10e-20 1.41e-19 0.00e+00
0.9 2.10e-20 1.51e-19 5.55e-17
1.0 3.10e-20 1.41e-20 0.00e+00
Problem 5. Consider the first order non-linear stiff initial value problem of the form:
y′=xy, 0≤x≤1, h = 0.1,
Exact Solution:
y(x) = ex2
2
Problem 5has been solved in [32] with a uniform block method of order 13. Proposed methods
are directly employed without starting values to solve the same problem. The results of the derived
formulae are presented in Table 7with the efficiency curves shown in Figure 11. Results show
that proposed methods of uniform order 10 give improved accuracy with 7D2PIB2 giving potential
advantage over 7D2PIB1 of the same order. A method of a uniform order 13 has been compared and
our derived methods evidently show adequate accuracy over it. Observe from Figure 10 that, for this
nonlinear Problem 5, as the log(x) increases from left to right, the log of absolute error also increases.
This numerical results are usually common with all non-linear stiff IVPs.
16
Table 6: Comparison of Absolute Error for Problem 5with h= 0.1
xError in [32], p= 13 7D2PIB1, p= 10 7D2PIB2, p= 10
0.1 2.6067 ×10−11 1.2213×10−14 4.8850×10−15
0.2 8.4790×10−11 3.5083×10−14 1.3545×10−14
0.3 1.8684×10−10 7.2831×10−14 2.8422×10−14
0.4 3.5701×10−10 1.3123×10−13 5.1070×10−14
0.5 6.1054×10−09 2.2116×10−13 8.5931×10−14
0.6 1.0157×10−09 3.5727×10−13 1.3922×10−13
0.7 1.6445 ×10−09 5.6355×10−13 2.1960×10−13
0.8 2.6158×10−09 8.7708×10−13 3.4195×10−13
0.9 4.1110×10−09 1.3551×10−12 5.2913×10−13
1.0 6.4070×10−09 2.0863×10−12 8.1535×10−13
6. Application problems
Problem 6.
As discussed in [15], the SIR model is an epidemiological model that computes the theoretical number
of people infected with a contagious illness in a closed population over time. The name of this class of
models is derived from the fact that they involve coupled equations relating the number of susceptible
people S(t), number of people infected I(t) and the number of people who have recovered R(t). This
is a good and simple model for many infectious diseases Including measles, mumps and rubella. It is
given by the following three coupled equations:
dS
dt =µ(1 −S)−γI S
dI
dt =−µI −γI +βIS
dR
dt =−µR +γI
(34)
Where µ,γand βare positive parameters to be determined. Therefore, let ybe given by:
y=S+I+R(35)
By taking the derivative of Equation (35) and summing Equations (34) and (35) to give the SIR
17
model of the form:
y′=µ(1 −y),0≤x≤1, h = 0.01,(36)
Whose exact solution is:
y(x)=1 −0.5e−0.5x.
Problem 6in Equation 36 is solved using the proposed methods. The results as presented in Table 8
depict the absolute errors and time taken (seconds) at each point of iterations. The efficiency curves
are also plotted using the logarithm of absolute errors against the log of time and are shown in Figure
11. Because of the stiff nature present in the modeled problem, it is clear from Figure 11 that, the
scale absolute errors, particularly in 7D2PIB2 inter-nodes. Table 8also clearly indicates the near
convergence of 7D2PBI2 unlike 7D2PBI1. However, effective time cost is observed in 7D2PBI1 in
comparison with 7D2PBI2. Reasonably, for this particular Problem 6in Equation (36), 7D2PBI1
presents efficient time cost over 7D2PBI2 but improved accuracy is seen in 7D2PBI2. While time of
iterations is important, accuracy of numerical methods is most significant as it shows the consistency
and zero stability of methods. Finally, the proposed formulae present improved efficiency and accuracy
over the compared method in [15], as shown in Table 8.
Table 7: Comparison of Absolute Error for Problem 6in Equation (36) (SIR Model) with h= 0.01
xError in [15], p= 8 Time 7D2PIB1, p= 10 Time 7D2PIB2, p= 10 Time
0.010 1.2165824e-12 0.043527 4.4408921e-16 0.008644 0.0000000e-00 0.009575
0.020 7.0361494e-12 0.048093 1.1102230e-15 0.011411 1.1102230e-16 0.012471
0.030 1.6891821e-11 0.053913 1.5543122e-15 0.013618 1.1102230e-16 0.015296
0.040 3.0793479e-11 0.059570 1.9984014e-15 0.016258 2.2204461e-16 0.018339
0.050 5.0472182e-11 0.063933 2.5535130e-15 0.018438 1.1102230e-16 0.021397
0.060 7.1624151e-11 0.080116 2.9976022e-15 0.020523 1.1102230e-16 0.025175
0.070 1.0171974e-10 0.085281 3.5527137e-15 0.022605 2.2204461e-16 0.027427
0.080 1.2969015e-10 0.093241 4.1078252e-15 0.024689 3.3306691e-16 0.030011
0.090 1.6615576e-10 0.097912 4.5519144e-15 0.027011 3.3306691e-16 0.036050
0.100 2.0496926e-10 0.104638 5.2180482e-15 0.029718 5.5511151e-16 0.038381
Problem 7. Consider the growth model as solved in [15]:
A bacteria culture is known to grow at a rate proportional to the amount present. After one hour,
1000 strands of the bacteria are observed in the culture; and after four hours, bacteria are observed
18
in the culture to be 3000 strands. Find the number of strands of the bacteria present in the culture
at time t, where, 0 ≤t≤1.
Let N(t) denote the number of bacteria strands in the culture at time t, the initial value problem
modeling this problem is given by:
dN
dt = 0.366N, N (0) = 694,(37)
The exact solution is given by:
N(t) = 694e0.366t.
Problem 7in Equation (37), which is a population growth model, has been solved with h= 0.01 in
[15]. The new methods are also applied for the approximations and the absolute errors are as shown
in Table 9with the efficiency curves in Figure 12. Results indicate improved accuracy with reduced
computational time in 7D2PBI1 than with 7D2PBI2. Method, 7D2PIB1 performed excellently over
7D2PIB2 in terms of efficiency and accuracy as shown in Table 9. Comparison with a method in [15]
showed clear performance in terms of accuracy and time of iterations in the proposed formulae.
Table 8: Comparison of Absolute Error for Problem 7in Equation (37) (Growth Model) with h= 0.01
xError in [15], p= 8 Time 7D2PIB1, p= 10 Time 7D2PIB2, p= 10 Time
0.010 6.7871042e-11 0.022520 0.0000000e-00 0.008482 0.0000000e-00 0.008264
0.020 2.9922376e-10 0.050186 0.0000000e-00 0.010774 2.2737368e-13 0.010404
0.030 6.8837380e-10 0.070106 0.0000000e-00 0.013659 2.2737368e-13 0.029010
0.040 1.2363444e-09 0.090180 1.1368684e-13 0.015907 4.5474735e-13 0.031195
0.050 1.9656454e-09 0.110484 1.1368684e-13 0.018652 4.5474735e-13 0.033432
0.060 2.8278464e-09 0.133450 1.1368684e-13 0.021897 4.5474735e-13 0.035572
0.070 3.9101451e-09 0.152786 2.2737368e-13 0.024202 3.4106051e-13 0.037686
0.080 5.0885092e-09 0.175301 3.4106051e-13 0.026631 2.2737368e-13 0.039777
0.090 6.4850383e-09 0.210087 4.5474735e-13 0.029239 1.1368684e-13 0.041873
0.100 8.0320888e-09 0.232457 4.5474735e-13 0.031873 3.4106051e-13 0.043992
Problem 8. Consider the Prothero-Robinson oscillatory problem:
y′=L(y−sinx) + cosx, y(0) = 0, L =−1, h = 0.1(38)
19
The exact solution is given by:
y(x) = sinx
Problem 8in Equation (38), which is a Prothero-Robinson oscillatory problem has been solved in
[15] and the proposed formulae is applied also. Results are presented in Table 9and efficiency curves
clearly shown in Figure 13. It is evident that the proposed methods show reduced computational
time and improved accuracy in terms of absolute errors. However, for this Problem 8, 7D2PIB2
show improved efficiency but certainly not accuracy. Accuracy has clearly been lost to 7D2PIB1 at
all points in the iterations. Therefore, each method show uniqueness in itself and 7D2PIB1 showed
overall improved accuracy in terms of absolute errors, even with comparison with a method in [15].
Table 9: Comparison of Absolute Error for Problem 8in Equation (38) (oscillatory problem) with h= 0.1
xError in [15], p= 8 Time 7D2PIB1, p= 10 Time 7D2PIB2, p= 10 Time
0.10 1.2439794e-09 0.181433 2.2689489e-12 0.004828 3.8868035e-10 0.004438
0.20 4.8347478e-09 0.380374 2.1383478e-11 0.013938 9.4229044e-10 0.010371
0.30 1.0511839e-08 0.564763 6.8256345e-11 0.017435 1.6392342e-09 0.012579
0.40 1.8015317e-08 0.751144 1.3565132e-10 0.019693 2.4580119e-09 0.014814
0.50 2.7086332e-08 0.952488 2.2087732e-10 0.022883 3.3772897e-09 0.017027
0.60 3.7467873e-08 1.129044 3.2125658e-10 0.025142 4.3759819e-09 0.026412
0.70 4.8905666e-08 1.332819 4.3413462e-10 0.028643 5.43334100e-09 0.028844
0.80 6.1149209e-08 1.620844 5.5688854e-10 0.031100 6.5290587e-09 0.031626
0.90 7.3952914e-08 1.904968 6.8693884e-10 0.036798 7.6433704e-09 0.033859
1.00 8.7077323e-08 2.125000 8.2176266e-10 0.039081 8.7571647e-09 0.036455
20
log(x)
-2.5 -2 -1.5 -1 -0.5 0
log(AbsErr)
-40
-38
-36
-34
-32
-30
-28
-26
-24
-22
Error in [32]
Error in 7D2PIB1
Error in 7D2PIB2
Figure 8: Efficiency curves of Table 4
.
log(x)
-2.5 -2 -1.5 -1 -0.5 0
log(AbsErr)
-46
-44
-42
-40
-38
-36
Error in 1 SHBM [18]
Error in 2 SHBM [18]
Error in 7D2PIB1
Figure 9: Efficiency curves of Table 5
.
23
log(x)
-2.5 -2 -1.5 -1 -0.5 0
log(AbsErr)
-34
-32
-30
-28
-26
-24
-22
-20
-18
-16
Error in [32]
Error in 7D2PIB1
Error in 7D2PIB2
Figure 10: Efficiency curves of Table 6
.
log(Time)
-5 -4.5 -4 -3.5 -3 -2.5 -2
log(AbsErr)
-38
-36
-34
-32
-30
-28
-26
-24
-22
Error in [15]
Error in 7D2PIB1
Error in 7D2PIB2
Figure 11: Efficiency curves of Table 7(SIR Model)
.
24
log(Time)
-5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1
log(AbsErr)
-36
-34
-32
-30
-28
-26
-24
-22
-20
-18
Error in [15]
Error in 7D2PIB1
Error in 7D2PIB2
Figure 12: Efficiency curves of Table 8(GROWTH MODEL)
.
log(Time)
-6 -5 -4 -3 -2 -1 0 1
log(AbsErr)
-28
-26
-24
-22
-20
-18
-16
Error in [15]
Error in 7D2PIB1
Error in 7D2PIB2
Figure 13: Efficiency curves of Table 9
.
25
7. Conclusion and future research
A new family of computational methods, with seventh derivative type of implicit two-point block
for the direct approximation of first order stiff initial value problems of uniform order 10 each have
been developed. Formulae were derived through interpolation and collocation techniques. The new
methods considered uneven points of collocation. They require seventh derivative type, though of
a first-order function. It has been established that uneven points of collocation affect numerical
schemes efficiency in terms of computational time and accuracy in terms of absolute errors. The
new methods are found to be A-stable and convergent. The convergence were shown through test
problems on first order stiff IVPs, including real-life problems as SIR model, growth model and
oscillatory problem with comparison with some other existing methods. Results indicate that the
new methods showed different numerical behaviors on different problems considered, either in terms
of accuracy or efficiency while outperformed such existing methods in literature. Summarily, 7D2PIB2
displayed better accuracy and effective time cost than 7D2PIB1. This is not far-fetched as 7D2PIB2
has a larger open region of absolute stability than 7D2PIB1. In general, we have formulated numerical
methods with uneven collocation points that are computationally stable with effective time cost for
direct solution of (1). These methods outperformed such existing formulae in literature, as compared
in this research. Our next future research will focus on developing and implementing efficient and
robust numerical methods with uneven collocation points to real-life problems in chemical reaction
in chemical engineering, models on drug magnetic nano-particle transport, population growth model,
tumor immune interaction model, e.t.c and application to higher-order stiff IVPs may be considered
also.
Acknowledgments
The authors wish to thank the referees and editors for their comments and suggestions at making
this work a success.
Availability of data
The data used to support the results of the study are duly enclosed in the paper.
Conflict of interest
The authors declare that there is no conflict of interests.
Funding
No funding available for this research paper.
26
Author’s contribution
V. O. Atabo: Conceptualization, methodology, former analysis and software. S. O. Adee :
investigation, review and editing. P. O. Olatunji: validity and visualization. D. J. Yahaya:
confirmation and original draft preparation. All content of manuscript were written via author’s
contribution, read and agreed to publish the final manuscript.
References
[1] C. F. Curtiss & J. O. Hischfelder, Integration of stiff equations, Proc. Nat. Acad.Sci. U.S.A.,
38(1952) 235.
[2] L. M. Willard, Numerical methods for stiff equations and singular perturbation problems, D.
Reidel Publishing Company, Dordretch: Holland/Boston, (1981).
[3] K. A. Atkinson, An introduction to Numerical Analysis:, John Wiley & Sons, (1989), ISBN
978-0-471-50023-0.
[4] E. Hairer, S. P. Norsett & G. Wanner, Solving Ordinary Differential Equations 1- Non-Stiff
Problems. Springer Series in Computational Mathematics, (1993), ISSN: 0179-3632. https://
org.doi.10.1007/978-3-540-78862-1
[5] J. D. Lambert, Numerical Methods for Ordinary Differential Systems: The Initial Value Problem;
John Wiley & Sons, Inc.: Hoboken, NJ, USA (1991).
[6] J. S. Aksah, Z. B. Ibrahim & I. S. M. Zawawi, “Stability Analysis of Singly Diagonally Implicit
Block Backward Differentiation Formulas for Stiff Ordinary Differential Equations”, MDPI,
https://doi.org/10.3390/math7020211
[7] C. W. Gear, “The automatic integration of stiff ODEs”, pp. 187-193 in A.J.H. Morrell (ed).
Information processing 68: Proc. IFIP Congress, Edinurgh (1968), Nor-Holland, Amsterdam.
[8] Y. Skwame, J. Sabo & T. Y. Kyagya, “The Construction of Implicit One-step Block Hybrid
Methods with Multiple Off-grid Points for the Solution of Stiff Differential Equations”, Journal
of Scientific Research and Reports,16(2017) 1-7, ISSN: 2320-0227, http://doi.org/10.9734/
JSRR/2017/36187
[9] Y. Skwame, J. Sabo, P. Tumba and T. Y. Kyagya, “Order ten implicit one-step hybrid block
method for the solution of stiff second-order ordinary differential equations”, International
Journal of Engineering and Applied Sciences (IJEAS), ISSN: 2394-3661, Volume-4, Issue-12,
December, 2017. www.ijeas.org.
[10] C. T. Gerald & P. O. Wheatley, Applied numerical analysis, Addison-Wesley Publishing Company
Inc. (1994).
27
[11] Z. B. Ibrahim & A. A. Nasarudin, “A Class of Hybrid Multistep Block Methods with A-Stability
for the Numerical Solution of Stiff Ordinary Differential Equations”. MDPI,http://doi.org/
10.3390/math8060914.
[12] N. N. Mohamad, Z. B. Ibrahim & F. Ismail, “Numerical Solution for Stiff Initial Value Problems
Using 2-point Block Multistep Method”, 1132(2018), IOP Publishing Ltd.http://doi.org/10.
1088/1742-6596/1132/1/012017.
[13] J. C. Butcher, “Numerical Methods for ordinary differential equations,” John Wiley & Sons Ltd.,
The Atrium, Southern Gate, Chichester, Sussex PO19 8SQ, England, (2008). www.wiley.com.
[14] A. A. Nasarudin, Z. B. Ibrahim & H. Rosali, “On the Integration of Stiff ODEs Using Block
Backward Differentiation Formulas of Order Six”. MDP1, Symmetry Journal, (2020). https:
//doi.org/10.3390/sym12060952.
[15] E. O. Omole, O. A. Jeremiah and L.O. Adoghe, “A Class of Continuous Implicit Seventh-
eight method for solving y′=f(x, y) using power series”, International journal of Chemistry,
Mathematics and Physics (IJCMP), Vol-4, Issue-3, May-Jun, 2020. http://dx.doi.org/10.
22161/ijcmp.4.3.2
[16] O. O. Olanegan & O. I. Aladesote, “Effficient fifth-order class for the numerical solution of
first order ordinary differential equations”, FUDMA Journal of Sciences (FJS),4(2020) 207,
https://doi.org/10.33003/fis-2020-0403-171
[17] M. A. Rufai, M. K. Duromola & A. A. Ganiyu, “Derivation of One-Sixth Hybrid Block Method
for Solving General First Order Ordinary Differential Equations”, IOSR Journal of Mathematics
(IOSR-JM), p-ISSN:2319-765X. 12(2016) 20-27. http://doi.org/10.9790/5728-1205022027.
[18] C. B. Ononogbo, I. E. Airemen and U. J. Ezurike, ”Numerical algorithm for one and two-step
hybrid block methods for the solution of first order initial value problems in ordinary differential
equations”, Applied Engineering, 6(2022) 13-23. https://doi:10.11648/j.ae.20220601.13.
[19] Y. Skwamw, J. Z. Donald & J. M. Althemai, “Formation of Multiple Off-Grid Points for the
Treatment of Systems of Stiff Ordinary Differential Equations”, Academic Journal of Applied
Mathematical Sciences, Academic Research Publishing Group, ISSN(p): 2415-5225, 4(2018) 1.
http://arpgweb.com/?ic=journal&journal=17&info=aims.
[20] L. O. Adoghe, “A New L-Stable Third Derivative Hybrid Method for Solving First Order
Ordinary Differential Equations”, Asian Research Journal of Mathematics, 17(2021) 58-69,
Article no.ARJOM.71481. http://doi.org/10.9734/ARJOM/2021/v17i630310.
[21] J. Sunday, G. M. Kumleng, N. M. Kamoh, J. A. Kwanamu, Y. Skwame, O. Sarjiyus, “Implicit
Four-Point Hybrid Block Integrator for the Simulations of Stiff Models”, J. Nig. Soc. Phys. Sci.,
4(2022) 287–296. http://doi.org/10.46481/jnsps.2022.777.
[22] I. O. Isah, A. S. Salawu, K. S. Olayemi & L. O. Enesi, “An efficient 4-step block method for
solution ff first order initial value problems via shifted chebyshev polynomial”, Tropical Journal
of Science and Technology, 1(2), 25-36, 2020. . http://doi.org/10.47524/tjst.v1i2.5.
28
[23] S. Yimer, A. Shiferaw, S. Gebregiorgis, “Block Procedure for Solving Stiff First Order Initial
Value Problems Using Chebyshev Polynomials”, Ethiop. J. Educ. & Sc. Vol. 15 No. 2 March,
(2020) 34.
[24] D. G. Yakubu, M. Aminu & A. Aminu, “The Numerical Integration of Stiff Systems Using Stable
Multistep Multiderivative Methods”, Journal of Modern Methods in Numerical Mathematics,
8(2017) 99. Modern Science Publishers, http://dx.doi.org/10.20454/mmnm.2017.1319.
[25] L.A. Ukpebor and L.O. Adoghe, “Continuous fourth derivative block method for solving first
order stiff order ordinary differential equations”, Abacus (Mathematics Science Series), Vol. 44,
No 1, Aug. 2019.
[26] M. Kida, S. Adamu, O. O. Aduroja and T. P. Pantuvo, “Numerical solution of stiff and oscillatory
problems using third derivative trigonometrically fitted block method”, J. Nig. Soc. Phys. Sci.
4(2022) 34–48. http://doi.org/10.46481/jnsps.2022.271.
[27] AK. Ezzeddine and G. Hojjati, “Third derivative multistep methods for stiff systems”, Intl. J.
of Nonlinear Sci., 14(2012) 443–50.
[28] O. A. Akinfenwa, B. Akinnukawe & S. B. Mudasiru,“A family of Continuous Third Derivative
Block Methods for solving stiff systems of first order ordinary differential equations”, Journal of
the Nigerian Mathematical Society,34(2015) 160-168. http://dx.doi.org/10.1016/j.jnnms.
2015.06.002
[29] G. Ismail and I. Ibrahim, “New efficient second derivative multistep methods for stiff systems”.
Appl. Math. Model, 23(1999) 279–88.
[30] G. Hojjati, M. Rahimi, SM. Hosseini, “New second derivative multistep methods for stiff
systems”. Appl.Math. Model, 30(2006) 466–76.
[31] A. I. Bakari, B. Sunday, T. Pius and A. Danladi, “Seven-step hybrid block extended second
order derivative backward differentiation formula”, Dutse Journal of Pure and Applied Sciences
(DUJOPAS), 6(2020).
[32] A. A. James, A. O. Adesanya and J. Sunday,“Uniform Order Continuous Block Hybrid Method
for the Solution of First Order Ordinary Differential Equations”, IOSR Journal of Mathematics
(IOSR-JM), ISSN: 2278-5728.Volume 3, Issue 6 (Sep-Oct. 2012), PP 08-14. www.iosrjournal.
org.
29
