Block Solver for Multidimensional Systems of Ordinary Differential
Equations
JIMEVWO GODWIN OGHONYON
Department of Mathematics, Covenant University,
Km 10. Idiroko, Canaan Land,
Ota, Ogun State, NIGERIA
https://covenantuniversity.edu.ng/
SOLOMON ADEWALE OKUNUGA
Department of Mathematics, University of Lagos
Lagos State, NIGERIA
https://unilag.edu.ng
PETER OLUWATOMI OGUNNIYI
Department of Mathematics, Covenant University
Km 10. Idiroko, Canaan Land
Ota, Ogun State, NIGERIA
https://covenantuniversity.edu.ng/
Abstract- This research study aimed at developing block solver for multidimensional systems (BSMS) of ordinary
differential equations. This method will be formulated via interpolation and collocation techniques with
multinomial as the basis function approximate. The block solver has the capacity to utilize each principal local
truncation errors to generate the convergence criteria that will ensure convergence. Some theoretical properties will
be stated. The process for executing the block solver will be done via the idea of the convergence criteria
introduced. Step by step implementation algorithm will be specified. Some selected model applications will be
worked out and a suitable step size will be determined to satisfy the convergence criteria in order to enhance the
accuracy and efficiency of the method. The implementation of BSMS is coded in Mathematica and executed under
the platform of Mathematica Kernel 9.
Key-words- Block solver; interpolation and collocation; multidimensional systems; variable step size; model
applications; convergence criteria, implementation algorithm; Mathematica Kernel
Received: May 23, 2021. Revised: March 21, 2022. Accepted: April 22, 2022. Published: May 24, 2022.
1 Introduction
A large number of physical occurrences are framed
with more than one equation and take more than one
subordinate variable. For instance, whenever we intend
to find out the population of two acting population like
foxes and rabbit, we will accept two subordinate
variables which constitute the two populations where
these populations rely on one autonomous variable
which constitutes time. Occurrences such as this give
rise to systems of differential equations. See [1] for
details.
Consider the initial value problem for a system of
first-order differential equations which possess the
general class of
(1)
Seeking for the solution functions on
some time-interval defined at .
The general class of (1) is cumbersome to figure
out, and it is not easy to determine the system when
writing a computer programming. The class of (1)
constitute the analytical solution and the differential
WSEAS TRANSACTIONS on FLUID MECHANICS
DOI: 10.37394/232013.2022.17.9
Jimevwo Godwin Oghonyon,
Solomon Adewale Okunuga,
Peter Oluwatomi Ogunniyi
E-ISSN: 2224-347X
89
Volume 17, 2022
equations by employing column vectors. This will be
announce as
(2)
with .
Then (1) will be seen as
(3)
The notational system of (2) is define as
. [1, 5]
Definition: A system of ordinary differential equations
is a concurrent set of equations that takes two or more
subordinate variables that rely on one autonomous
variable. A solution of the system is a set of functions
that meets each equation on some time interval I [1].
Theorem 1: Assume that each of the functions
,
and the partial derivatives
are continuous in a region
establishing the point . Then, the
initial-value problem
(4)
has a unique analytical solution
(5)
on the interval establishing [1-2, 15-16].
The system of initial value problems come into
existence by nature from any multidimensional system
or any system that possess more than one variable
quantity associated in a single model equation. Each
one of these variable quantities can be constituted by a
mathematical function of a single independent variable
quantity (usually time) [14].
Theorem 2: Let be a continuous function
and for each there exist a multinomial function
such that for all . Very
importantly, for whatever such , there exist a
succession of multinomial such that
uniformly on . [6]
Authors have suggested possible solutions to
handle (1). Some of these methods include the
continuous block backward differentiation formula for
solving stiff ODEs designed by [2]. This CBBDF
requires no starting values and implement implicit
block method for solving stiff ODEs. Block method
implemented by CBBDF has been well emphasized as
an advantage over other methods for ensuring better
efficiency and accuracy. Nevertheless, the step size
variation and tolerance level were not implemented.
The extended continuous block backward
differentiation formula for stiff systems is carried out
by [3]. This method successfully avoids the use of
starting values and thereby executed block method
approach with fixed step size. [3] utilizes the fixed step
approach to determine the numerical result. This
approach of fixed step size is not comparable with
variable step size change and tolerance level approach
which guarantees convergence of every iteration. Block
hybrid k-step backward differentiation formulas for
large stiff systems has be executed by [13]. This
method possesses the properties of block backward
differentiation formula and has the vantage of avoiding
the use of predictor method for initializing the process.
[13] solved both linear stiff and non stiff systems
utilizing fixed step size without implementing step size
change and tolerance level to guarantee better
efficiency and accuracy. [17] implemented the
numerical solution of first order stiff ODEs using fifth
order block backward differentiation formulas. The
idea is basically for stiff ODEs which solved large
systems of ODEs simultaneously with fixed step size as
against finding a suitable step size and including
tolerance level. The parallel implementation of the
parallel block backward differentiation formulas
displays important benefits above the successive
implementation. The idea of step size change and
tolerance level was implemented but the technique of
finding a suitable step size for each tolerance level was
not examined by [23]. The derivation of block solver
for multidimensional systems is the main aim of this
research study. The introduction of block solver for
multidimensional systems (BSMS) of ODEs originates
as a result of the bounded stability attributes of the
block solver which contributes to the unfitness of the
system to show better efficiency and accuracy. Again,
BSMS is proposed by [15-16] to outwit the Dahlquist
WSEAS TRANSACTIONS on FLUID MECHANICS
DOI: 10.37394/232013.2022.17.9
Jimevwo Godwin Oghonyon,
Solomon Adewale Okunuga,
Peter Oluwatomi Ogunniyi
E-ISSN: 2224-347X
90
Volume 17, 2022
roadblocks. Again, [15-16] suggested the introduction
of variable step variable and variable step size as an
option to better accuracy and efficiency. BSMS will
ensure better efficiency and accuracy by the
introduction of variable step variable order and finding
a suitable variable step size to ensure the satisfaction of
the convergence criteria. BSMS possesses the idea of
block method and the implementation process is done
by a specially designed formula to achieve the desired
result thereby ensuring error control or monitoring.
The contribution of this research study is the
derivation and implementation of a block solver for
multidimensional systems via variable step variable
order and finding a suitable step size. Block solver as a
result of the multidimensional systems will be
implemented like other block backward differentiation
formulas adopting block method approach.
2 Developing the Block Solver of
Multidimensional Systems
The process of developing the block solver for
multidimensional systems of ODEs (BSMS) will be
done via interpolation and collocation together with
multinomial as the basis function approximate. This
propose block solver is constructed using
of block predictor mode of order while the block
corrector mode uses of order . This
combination is referred to as the block solver defined in
form of variable step and variable order.
The block predictor mode utilizes as
the interpolation point and as the
collocation points. On the other hand, the block
corrector mode takes on for interpolation
and as the points of collocation. The
derivation of block solver is subjected to a special
multinomial basis function approximate
, (6)
where constitute the unknown
physical quantities required to be examine specially.
Whenever (6) is utilized to approximate (1), theorem 2
is satisfied. Hence, the analytical solution of the points
of interpolation
will produce
, (7)
and points of collocation, , , ,
to bring forth
,, ,
. (8)
Equations (7) and (8) will be combined together to
produce the system of equations in the form of
. The solution loop will converge whenever the
absolute values of the pre-eminent diagonal
components of the constant coefficient square matrix A
of the system are larger than the total of
absolute values of the other constant coefficient of the
row [11]. Calculating and after then,
substituting the results into (6) as well as evaluating at
some selected interval of , will give
rise to block predictor mode and block corrector mode
of
(9)
Equation (9) is called block solver for
multidimensional systems of ODEs [18-22]. Block
solver developed by (9) has variable step and variable
order. The block predictor mode has 4-step of order 4
while the block corrector mode has 3-step of order 3.
This combination is a special design of a higher block
predictor mode with a lower block corrector mode.
2.1 Theoretical Properties of the Method
Theorem 3: Whenever the block solver (9) converges
to a certain
pth
order of equations then the order of (9)
is at leastwise
p
[9].
Theorem 4: The order of (9) for first order equations
must be greater than or equal to one whenever it is
convergent [9].See [9] for proof.
2.2 Executing the Convergence Criteria of
Block Solver
The usage of block solver for estimating the principal
local truncation error call for the block predictor mode-
block corrector mode to own ilk order. To realized this
we allow the block predictor to be Adams
Bashforth method and block corrector to be
Adams-Moulton method, both then own .
The order ABM pair is therefore
WSEAS TRANSACTIONS on FLUID MECHANICS
DOI: 10.37394/232013.2022.17.9
Jimevwo Godwin Oghonyon,
Solomon Adewale Okunuga,
Peter Oluwatomi Ogunniyi
E-ISSN: 2224-347X
91
Volume 17, 2022
(10)
Whenever we presume (10) to be utilized in
mode, then, in the second of (10),
will be replaced by
, and the value
displayed on the right side by
, the remaining
values of being replaced by
. We can surmount this task by specifying
as
with the one value
replaced as
. That is,
(11)
We may rewrite (30) in the class
(12)
and represent (10) as the notational system.
Right away, the pair of (10) is utilized in
mode, and employs the mode of Adams methods to
construct a type of ABM method that is very tedious
and difficult to handle in terms of the computation. The
type is defined as follows:
(13)
(14)
whenever .
To utilize the block solver, we demand the calculation
of
. Deducting (13) from (14) with
we have
.
Since
and , the block solver
estimate will be seen as
. (15)
Whenever the principal local truncation error is at
, the is achieved as
.
Whenever
, wherefrom
. See [4, 7, 15-16, 18-22] for
more info.
2.3 Step by Step Implementation Algorithm of
Block Solver
Step 1: choose a step size
h
and vary the step size until
a suitable variable step size
h
is found
Step 2: use Taylor’s series of order four to prime the
block solver.
Step 3: write the code of block solver using
Mathematica
Step 4: run equation (13) with step 1 under the platform
of Mathematica Kernel 9.
Step 5: if step 4 fails repeat the process again as
prescribed by step 1.
Step 6: if step 5 is successful after determining the
suitable variable step size
h
then proceed to step 7.
Step 7: print the maximum errors of the block solver.
3 Results and Discussion
Three model applications will be examine to show case
better efficiency and accuracy of the block solver. The
computational results of BSMS be compared with the
analytical result and verified using some selected
convergence criteria of
. The BSMS
is applied under a proficient mode in the manner of
to examine the convergence criteria,
efficiency, accuracy and maximum error. The block
solver of (9) will be devised and carried out under the
platform of Mathematica to solve the model
applications of the multidimensional systems of ODEs.
3.1 Numerical Examples
We utilized the idea of compartment analysis to
transforms the diagram into a system of linear
differential equations. The concept has been utilized to
formulate real life modes in various topics such as
environment science, chemical science, heating,
cooling, kinetics, mechanics and electrical energy.
A compartment diagram consists of the following
elements.
Variable Names : Each compartment is marked with a
variable X.
WSEAS TRANSACTIONS on FLUID MECHANICS
DOI: 10.37394/232013.2022.17.9
Jimevwo Godwin Oghonyon,
Solomon Adewale Okunuga,
Peter Oluwatomi Ogunniyi
E-ISSN: 2224-347X
92
Volume 17, 2022
Arrows: Each arrow is labeled with flow rat R.
Input Rate: An arrowhead directing at compartment X
documents input rate R.
Output Rate: An arrowhead directing away from
compartment X documents output rate R.
0
Fig. 1: A diagram of compartment analysis. The
diagram represents the classical brine tank problem of
figure 2.
Gathering the single linear differential equation for
a diagram compartment X is carried out by composing
for the left hand side of the differential equation.
Again, in algebraic manner we add the input and output
rates to get the right hand side of the differential
equation, allowing to the equilibrium law.
Conventionally, a compartment that has no coming
arrowhead possess input zero, and a compartment that
has no coming out possess output zero.
These model applications of the systems of ODEs
are as follows
Recycled brine tank cascade [10]
Biomass Transfer [10].
Population problems [1].
Model Application 1: Three brink tanks in cascade with
recycling
Let brine tanks A, B, C be given volumes of
, respectively, as in figure 2
.
Fig. 2: Three brine tanks in cascade with recycling.
Suppose that fluid drains from tank A to B at rate r,
drains from tank B to C at rate r, then drains from tank
C to A at rate r. The tank volumes remain constant due
to constant recycling of fluid. For the purpose of
illustration, let . Uniform stirring of each tank is
assumed, which implies uniform salt concentration
throughout each tank.
Let denote the amount of salt at
time t in each tank. No salt is lost from the system, due
to recycling. Using compartment analysis, the recycled
cascade is modeled by the non-triangular system
,
,
.
The analytical solution is given by
,
,
.
At infinity,
. This implies that the
total amount of salt is uniformly distributed in the
tanks, in the ratio [10].
Model Application 2: Biomass Transfer
Consider a European forest having one or two
varieties of trees. We select some of the oldest trees,
those expected to die off in the next few years, and then
follow the cycle of living trees into dead trees. The
dead trees eventually decay and fall from seasonal and
biological events. Finally, the fallen trees become
humus. Let variables be
.
A typical biological model is
,
,
A
B
C
WSEAS TRANSACTIONS on FLUID MECHANICS
DOI: 10.37394/232013.2022.17.9
Jimevwo Godwin Oghonyon,
Solomon Adewale Okunuga,
Peter Oluwatomi Ogunniyi
E-ISSN: 2224-347X
93
Volume 17, 2022
Suppose there are no dead trees and no humus at
, with initially units of living tree biomass.
These assumptions imply initial conditions
. The analytical solution is seen as
,
,
.
The live tree biomass decreases
according to a Malthusian decay lay from its initial size
. It decays to of its original biomass in one year
[10].
Model Application 3: Population problem
The rate at which population changes is
,
,
where the initial population
while ,
and are
given.
The exact solutions are
,
,
[1].
TERMINOLOGY
BSMS: block solver for multidimensional
systems of ODEs.
: tolerance level of the convergence
criteria.
: maximal error(s).
Table 1. Results of Model Application 1
Method Used
MAXE
Convergence
Criteria
BSMS
BSMS
BSMS
BSMS
BSMS
BSMS
BSMS
BSMS
BSMS
Table 2. Results of Model Application 2
Method Used
MAXE
Convergence
Criteria
BSMS
BSMS
BSMS
BSMS
BSMS
BSMS
BSMS
BSMS
BSMS
Table 3. Results of Model Application 3
Method Used
MAXE
Convergence
Criteria
BSMS
BSMS
BSMS
BSMS
BSMS
BSMS
BSMS
BSMS
BSMS
4 Conclusion
The block solver for multidimensional systems
(BSMS) of ODEs has been suggested. The block solver
is product of block predictor mode and block corrector
mode which is formulated utilizing variable step and
variable order. Three model applications with
exponentially and trigonometrically solutions in nature
were examined. The block solver adopted the idea of
variable step-variable order and variable step size to
implement the procedure. The convergence criteria
apply (15) together with (9) to ensure the
implementation The convergence criteria of
were utilized
to decide the MAXE results. The mathematical
WSEAS TRANSACTIONS on FLUID MECHANICS
DOI: 10.37394/232013.2022.17.9
Jimevwo Godwin Oghonyon,
Solomon Adewale Okunuga,
Peter Oluwatomi Ogunniyi
E-ISSN: 2224-347X
94
Volume 17, 2022
expression of (15) is used to decide whether to accept
or reject the results. The high level of efficiency and
accuracy achieved were made possible with the
determination of a suitable step size and block solver
designed via variable step and variable order. Block
solver derived has the capacity to proffer solution to
multidimensional systems of ODEs with oscillating and
vibration behavior via the efficient utilization of
variable step size-variable order and suitable variable
step size. A step by step approach for realizing the
result is specified. The execution of block solver is
implemented under the Mathematica Kernel 9. Thus,
makes it easier to achieve faster computation and
precise results. Furtherwork is required to build a
block solver to handle stiff oscillating and vibration
solutions.
Acknowledgements:
The authors would like to appreciate Covenant
University, Ota for their continuous sponsorship
throughout this research project.
References:
[1] M. L. Abell, J. P. Braselton, Differential
equations with Mathematica, Elsevier
Academic Press, USA, 2004.
[2] O. A. Akinfenwa, S. N. Jator, N. M. Yao,
Continuous block backward differentiation
formula for solving stiff ordinary differential
equations, Computers and Mathematics with
applications, Vol. 65, No. 7, 2013, pp. 996-
1005.
[3] O. A. Akinfenwa, S. N. Jator, Extended
continuous block backward differentiation
formula for stiff systems, Fasciculi
Mathematici, Vol. 2015, 2015, pp. 1-15.
[4] U. M. Ascher, L. R. Petzoid, Computer
methods for ordinary differential equations
and differential-algebraic equations, SIAM,
USA, 1998.
[5] K. Atkinson, W. Han, D. Stewart, Numerical
solution of ordinary differential equations,
John Wiley & Sons, Inc, New Jersey, 2009.
[6] M. Bond, Convolutions and the Weierstrass
approximation theorem, Department of
Mathematics, Michigan State University, USA,
2009.
[7] J. R. Dormand, Numerical methods for
differential equations, CRC Press, New York,
1996.
[8] S. O. Fatunla, Numerical methods for initial
value problems in ordinary differential
equations, Academic Press, Inc, New York,
1988.
[9] C. W. Gear, Numerical value problems in
ODEs, Prentice-Hall, Inc;, New Jersey, USA,
1971.
[10] ‘Systems of ODEs’, available at
http://www.math.utah.edu/~gustafso/2250syste
ms-de.pdf.
[11] S. R. K. Iyengar, R. K. Jain, Numerical
methods, New Age International (P) Ltd, New
Delhi, 2009.
[12] M. K. Jain, S. R. K. Iyengar, R. K. Jain,
Numerical methods for scientific and
Engineering computation, New Age
International (P) Limited, New Delhi, India,
2007.
[13] S. N. Jator, E. Agyingi, Block hybrid k-step
backward differentiation formulas for large
stiff systems, International Journal of
Computational Mathematics, Vol. 2014, 2014,
1-9.
[14] R. Khoury, H. Wilhelm, Numerical methods
and modelling for engineering, Springer
Nature, Switzerland, 2016.
[15] J. D. Lambert, Computational methods in
ordinary differential equations, John Wiley &
Sons, New York, 1973.
[16] J. D. Lambert, Numerical methods for ordinary
differential systems, John Wiley & Sons, New
York, 1991.
[17] A. A. M. N. Nor, B. I. Zarina, I. O. Khairil, S.
Mohamed, Numerical solution of first order
stiff ordinary differential equations using fifth
order block backward differentiation formulas,
Sains Malaysiana, Vol. 41, No. 4, 2012, pp.
489-492.
[18] J. G. Oghonyon, S. A. Okunuga, N. A.
Omoregbe, O. O. Agboola, A computational
approach in estimating the amount of pond and
determining the long time behavioural
representation of pond pollution, Global
Journal of Pure and Applied Mathematics,
Vol. 11, No 5, 2015, pp. 2773-2785.
[19] J. G. Oghonyon, J. Ehigie, S. K. Eke,
Investigating the convergence of some selected
properties on block predictor-corrector
WSEAS TRANSACTIONS on FLUID MECHANICS
DOI: 10.37394/232013.2022.17.9
Jimevwo Godwin Oghonyon,
Solomon Adewale Okunuga,
Peter Oluwatomi Ogunniyi
E-ISSN: 2224-347X
95
Volume 17, 2022
methods and it’s applications, Journal of
Engineering and Applied Sciences, Vol. 11, No
11, 2402-2408, 2016.
[20] J. G. Oghonyon, O. A. Adesanya, H. Akewe,
H. I. Okagbue, “Softcode of multi-processing
Milne’s device for estimating first-order
ordinary differential equations, Asian Journal
of Scientific Research, Vol. 11, No. 4, 2018,
pp. 553-559.
[21] J. G. Oghonyon, O. F. Imaga, P. O. Ogunniyi,
The reversed estimation of variable step size
implementation for solving nonstiff ordinary
differential equations, International Journal of
Civil Engineering and Technology, Vol. 9, No.
8, 2018, pp. 332-340.
[22] J. G. Oghonyon, S. A. Okunuga, H. I.
Okagbue, Expanded trigonometrically matched
block variable-step-size technics for computing
oscillating vibrations, Proceedings of the
International Multi-Conference of
Engineerings and Computer Scientists, Vol.
2019, 2019, pp. 552-557.
[23] B. Zarina, I. Khairil, O. Iskandar, Parallel
block backward differentiation formulas for
solving large systems of ordinary differential
equations, World Academic of Science,
Engineering and Technology, Vol. 4, No. 4,
2010, 1-4.
Conflicts of Interest
There is no conflict of interest among the authors.
Contribution of Individual Authors to the Creation
of a Scientific Article (Ghostwriting Policy)
-Jimevwo Godwin Oghonyon develop the idea, method
and write the code using Mathematica.
-Solomon Adewale Okunuga supervised the research
work.
-Peter Oluwatomi Ogunniyi provided the logistics and
technical support.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
This research study is funded by Covenant University,
Ota.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the Creative
Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en_U
S
WSEAS TRANSACTIONS on FLUID MECHANICS
DOI: 10.37394/232013.2022.17.9
Jimevwo Godwin Oghonyon,
Solomon Adewale Okunuga,
Peter Oluwatomi Ogunniyi
E-ISSN: 2224-347X
96
Volume 17, 2022
