An Effective Matrix Technique for the Numerical Solution of Second Order Differential Equations

Nathaniel Mahwash Kamoh; Bwebum Cleofas Dang; Comfort Soomiyol Mrumun

doi:doi:10.11648/j.acm.20241304.15

Research Article |

| Peer-Reviewed

An Effective Matrix Technique for the Numerical Solution of Second Order Differential Equations

Nathaniel Mahwash Kamoh^*

, Bwebum Cleofas Dang, Comfort Soomiyol Mrumun

Published in Applied and Computational Mathematics (Volume 13, Issue 4)

Received: 9 July 2024 Accepted: 29 July 2024 Published: 15 August 2024

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Abstract

In this paper, an effective technique for solving differential equations with initial conditions is presented. The method is based on the use of the Legendre matrix of derivatives defined on the close interval [-1,1]. Properties of the polynomial are outlined and further used to obtain the matrix of derivative which was used in transforming the differential equation into systems of linear and nonlinear algebraic equations. The systems of these algebraic equations were then solved using Gaussian elimination method to determine the unknown parameters required for approximating the solution of the differential equation. The advantage of this technique over other methods is that, it has less computational manipulations and complexities and also its availability for application on both linear and nonlinear second-order initial value problems is impressive. Other advantage of the algorithm is that high accurate approximate solutions are achieved by using a greater number of terms of the Legendre polynomial and once the operational matrix is obtained, it can be used to solve differential equations of higher order by introducing just a little manipulation on the operational matrix. Some existing sample problems from literature were solved and the results were compared to show the validity, simplicity and applicability of the proposed method. The results obtained validate the simplicity and applicability of the method and it also reveals that the method perform better than most existing methods.

Published in	Applied and Computational Mathematics (Volume 13, Issue 4)
DOI	10.11648/j.acm.20241304.15
Page(s)	111-117
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Legendre Polynomials, Matrix Calculus, Differential Equations

1. Introduction

The primary use of differential equations in general is to model motion, which is commonly called growth in economics. Specifically, a differential equation expresses the rate of change of the current state as a function of the current state.

In economics, differential calculus is used to compute marginal cost, marginal revenue, maxima and minima elasticity, partial elasticity and also enabling economists to predict maximum profit (or) minimum loss in a specific condition; one can also think of a change in general price level with respect to time as inflation. Second-order derivative with respect to time shows the rate of change of inflation, how inflation changes over time. Similarly, differentiating capital with respect to time shows investment.

Most ordinary differential equations arising in real-life applications cannot be solved exactly. These ordinary differential equations can be analyzed qualitatively. However, qualitative analysis may not be able to give accurate answers. A numerical method can be used to get an accurate approximate solution to a differential equation.

Motivated by these advantages, we will use Legendre operational matrix of derivatives through collocation method to approximate the solution of general second order differential equations with initial conditions.

The general second order differential equation is given as follows:

\frac{d^{2} y}{d x^{2}} = a (x) \frac{dy}{dx} + b (x) y + f (x)

(1)

where

a (x), b (x)

and

f (x)

are functions of

x

. Conventionally, (1) can be solved using different methods such as the method of (educated) guess, the method of variation of parameters and it can also be solved by reducing it to a system of first order differential equations, and then any method of solving first order differential equations can then be applied to solve it. The setbacks of this technique were reported in (

[2, 5, 12]

). The method of collocation and interpolation of the power series and other polynomial basis functions were used to generate approximate solution and these techniques were reported by many scholars among them are (

[1, 4, 6, 9, 11, 14]

) to mention a few. Their approaches and techniques generated implicit continuous linear multistep methods which require separate predictors for implementation; this method is called the predictor-corrector method. There are major setbacks of these methods, numerical techniques such the block linear multistep methods lately introduced by researchers such as (

[2-5, 7, 8, 10 13]

) have shown allot of advantages over the predictor-corrector method. However, the advantages are compensated by tedious computational work and the use of more advance software to enable it handle the work.

In this paper, a collocation technique based on the Legendre operational matrix of derivatives for second order differential equations is proposed. The advantages of this technique over other methods is that it has less computational manipulations and complexities because it only involves operational matrix of derivatives and its transpose and thus, reduces the time involve in the derivation of the schemes, analysis and implementation as is in the case with linear multistep methods.

1.1. Legendre Polynomials

The Legendre polynomials exhibit simple and convenient form for calculation, compared with other orthogonal polynomials (Chebyshev polynomials, shifted Legendre polynomials...). They are well known family of orthogonal polynomials on the interval

[- 1, 1]

. They are solutions to the popular Legendre differential equation given as follows;

(1 - x^{2}) \frac{d^{2} y}{d x^{2}} - 2 x \frac{dy}{dx} + n (n + 1) y = 0

(2)

They are widely used because of their smooth properties in the approximation of functions

[15]

. Equation (2) can be solved by series solution method (See

[14]

). The first few Legendre polynomials using the Rodriquez formula are:

l_{0} (x) = 1

l_{1} (x) = x

l_{2} (x) = \frac{1}{2} (3 x^{2} - 1)

l_{3} (x) = \frac{1}{2} (5 x^{3} - 3 x)

l_{4} (x) = \frac{1}{8} (35 x^{4} - 30 x^{2} + 3) l_{5} (x) = \frac{1}{8} (63 x^{5} - 70 x^{3} + 15 x)

l_{6} (x) = \frac{1}{16} (231 x^{6} - 315 x^{4} + 105 x^{2} - 5)

,...

The recurrence relation for Legendre polynomial is given by

l_{0} (x) = 1

l_{1} (x) = x

l_{k + 1} (x) = \frac{(2 k + 1)}{(k + 1)} x l_{k} (x) - \frac{k}{(k + 1)} l_{k - 1} (x), k = 1, 2, 3, \dots

The Rodrigues formula for the Legendre polynomial is

l_{k} (x) = \frac{1}{2^{k} k!} \times \frac{d^{k}}{d x^{k}} {(x^{2} - 1)}^{k}

k =, 0, 1, 2, . . .

Properties of Legendre polynomials

The following properties hold for Legendre polynomials are

l_{k} (- x) = {(- 1)}^{n} l_{k} (x)

l_{k} (1) = 1

ii.

l_{k} (- 1) = {(- 1)}^{n}

iii.

l_{k} (0) = 0, k

odd

iv.

{l^{'}}_{k} (0) = 0, k

even

Thus, the condition for orthogonality is:

\int_{0}^{1} l_{k} (x) l_{i} (x) dx = \{\begin{matrix} \frac{1}{2 k + 1}, if k = i \\ 0, if k \neq i \end{matrix}

This implies that any function

y (x) \in [- 1, 1]

can be approximated by Legendre polynomials as follows:

y (x) ≅ \sum_{k = 0}^{\infty} c_{k} l_{k} (x)

(3)

where

c_{k} = < y (x), l_{k} (x) > = (2 k + 1) \int_{0}^{1} y (x), l_{k} (x) dx

1.2. Preliminaries

We introduce the Legendre vector

L (x)

in the form

L (x) = [l_{0} (x), l_{1} (x), \dots, l_{n} (x)]

, then the derivative of the vector

L (x)

, can be expressed in matrix form by

{({L (x)}^{'})}^{T} = M {(L (x))}^{T} = {L (x) (M)}^{T}

(4)

Where

{({L (x)}^{'})}^{T} = (\begin{matrix} {l_{0}}^{'} (x) \\ {l_{1}}^{'} (x) \\ \begin{matrix} {l_{2}}^{'} (x) \\ . \\ \begin{matrix} . \\ . \\ l^{'} (x) \end{matrix} \end{matrix} \end{matrix}), M = (\begin{matrix} \begin{matrix} 0 \\ 1 \\ \begin{matrix} 0 \\ 1 \\ \begin{matrix} . \\ . \\ 0 \end{matrix} \end{matrix} \end{matrix} & \begin{matrix} 0 \\ 0 \\ \begin{matrix} 3 \\ 0 \\ \begin{matrix} . \\ . \\ 6 \end{matrix} \end{matrix} \end{matrix} & \begin{matrix} \begin{matrix} 0 \\ 0 \\ \begin{matrix} 0 \\ 5 \\ \begin{matrix} . \\ . \\ 0 \end{matrix} \end{matrix} \end{matrix} & \begin{matrix} \dots \\ \dots \\ \begin{matrix} \dots \\ \dots \\ \begin{matrix} \dots \\ \dots \\ \dots \end{matrix} \end{matrix} \end{matrix} & \begin{matrix} 0 \\ 0 \\ \begin{matrix} 0 \\ 0 \\ \begin{matrix} . \\ . \\ 0 \end{matrix} \end{matrix} \end{matrix} \end{matrix} \begin{matrix} 0 \\ 0 \\ \begin{matrix} 0 \\ 0 \\ \begin{matrix} . \\ . \\ (2 n - 1) \end{matrix} \end{matrix} \end{matrix} \end{matrix} \begin{matrix} 0 \\ 0 \\ \begin{matrix} 0 \\ 0 \\ \begin{matrix} . \\ . \\ 0 \end{matrix} \end{matrix} \end{matrix})

and

{(L (x))}^{T} = (\begin{matrix} l_{0} (x) \\ l_{1} (x) \\ \begin{matrix} l_{2} (x) \\ . \\ \begin{matrix} . \\ . \\ l_{n} (x) \end{matrix} \end{matrix} \end{matrix})

Where the matrix

M

is an

(N + 1) \times (N + 1)

matrix calculus, similarly, the

k^{th}

derivative of

L (x)

can be obtained from the following relation;

\begin{matrix} {({L (x)}^{'})}^{T} = M {(L (x))}^{T} \to {L (x)}^{1} = L (x) M^{T} \\ {({L (x)}^{2})}^{T} = {L (x)}^{1} M^{T} = L (x) {M^{T} M}^{T} = L (x) {(M^{T})}^{2} \\ {({L (x)}^{3})}^{T} = {L (x)}^{2} M^{T} = L (x) {(M^{T})}^{2} M^{T} = L (x) {(M^{T})}^{3} \\ . \\ . \\ . \\ {({L (x)}^{k})}^{T} = L (x) {(M^{T})}^{k} \end{matrix}\}

(5)

In this paper, we shall use the collocation method based on Legendre matrix calculus to solve numerically the general second order differential equation.

2. Derivation of the Method

We now derive the algorithm for solving (1.1), that is

y^{''} = f (x, y, y')

Let us suppose the solution of (1) is to be approximated by the first

(N + 1)

terms of the Legendre polynomial; thus, we can write (3) as

y_{N} (x) ≅ \sum_{j = 0}^{N} c_{j} (x) l_{j} (x) = L (x) C^{T}

(6)

where the Legendre coefficients

C

vector and the Legendre vector

L (x)

are given by

\begin{matrix} C = [c_{0}, c_{1}, c_{2}, \dots, c_{N}] \\ L (x) = [l_{0} (x), l_{1} (x), l_{2} (x), \dots, l_{N} (x)] \end{matrix}\}

(7)

The second derivative of

(6)

can be expressed as follows

y_{N}^{(2)} (x) = \sum_{j = 0}^{N} c_{j} (x) l_{j}^{(2)} (x) = L^{(2)} (x) C^{T} = L (x) {(M^{T})}^{(2)} C^{T}

(8)

where

M

is the matrix calculus defined in (4) above. Now substituting

(6)

and

(8)

into (1), we have

L (x) {(M^{T})}^{(2)} C^{T} = a (x) L (x) {(M^{T})}^{(1)} C^{T} + b (x) L (x) C^{T} + f (x)

(9)

Finally, to find the approximate solution, we collocate the transformed equation (9) at different collocation points

x_{j} = ({\frac{j}{N - k} h}_{j}), j = 0, 1, 2, \dots, N - k

, to obtain

N - 2

nonlinear algebraic equations using

L (x_{j}) {(M^{T})}^{(2)} C^{T} = a (x_{j}) L (x_{j}) {(M^{T})}^{(1)} C^{T} + b (x_{j}) L (x_{j}) C^{T} + f (x_{j}), j = 0, 1, ., N - 2

(10)

These equations together with the initial conditions give

(N + 1) \times (N + 1)

nonlinear systems of algebraic equations which can be solved using Newton’s iterative method for the unknown constants. Finally,

y_{N} (x)

given in

(6)

can be calculated.

3. Numerical Illustrations

The following numerical experiments are performed with the aid of MAPLE 18 and Scientific Workplace software packages in order to further affirm the applicability, simplicity and accuracy of the proposed method.

Example 1

Let us first consider the second order pantograph equation solved by

[9]

given by

y^{''} (x) = \frac{3}{4} y (x) + y (\frac{x}{2}) - x^{2} + 2, x \in [0, 1], h = 0.01

Subject to the initial conditions

y (0) = y^{'} (0) = 0

, the exact solution to this problem is known to be

y (x) = x^{2}

Applying our technique with

N = 3

, we have the following expression;

L (x) {(M^{T})}^{(2)} C^{T} = \frac{3}{4} L (x) C^{T} + L (\frac{x}{2}) C^{T} - x^{2} + 2 (ϑ_{1})

Using the initial conditions, we have respectively,

L (0) C^{T} = 0, L (0) {(M^{T})}^{(1)} C^{T} = 0 (ϑ_{2})

Collocating (

ϑ_{1}

) at

x = 0, \frac{1}{2}

and evaluating

(ϑ_{2})

x = 0

, we have the following algebraic systems of equations

(\begin{matrix} 3 c ₂ = \frac{7}{4} c ₀ - \frac{7}{8} c ₂ + 2 \\ 3 c ₂ + \frac{3}{40} c ₃ = \frac{7}{4} c ₀ + \frac{1}{160} c ₁ - \frac{69997}{80000} c ₂ - \frac{239993}{25600000} c ₃ + \frac{79999}{40000} \\ c ₁ - \frac{3}{2} c ₃ = 0 \\ c ₀ - \frac{1}{2} c ₂ = 0 \end{matrix})

Solving for the unknown coefficients

[c_{0}, c_{1}, c_{2}, c_{3}]

, we have

[c ₀ = \frac{1}{3}, c ₁ = 0, c ₂ = \frac{2}{3}, c ₃ = 0]

Substituting these approximate values into

(6)

, we get the approximate solution to the problem as

y_{N} (x) = x^{2}

The approximate solution is the same as the exact solution showing the accuracy of the method.

Example 2

Consider the second order differential equation solved by

[5]

given by

y^{''} (x) = y (x) + x e^{3 x}, x \in [0, 1], h = 0.0025

Subject to the initial conditions

y (0) = - \frac{3}{32}, y^{'} (0) = - \frac{5}{32}

, the exact solution to this problem is known to be

y (x) = \frac{4 x - 3}{32 e^{- 3 x}}

Applying our technique with

N = 5,

we have the following expression:

L (x) {(M^{T})}^{(2)} C^{T} = L (x) C^{T} + x e^{3 x} (ϑ_{1})

Using the initial conditions, we have respectively,

L (0) C^{T} = - \frac{3}{32}, L (0) {(M^{T})}^{(1)} C^{T} = - \frac{5}{32} (ϑ_{2})

Collocating (

ϑ_{1}

) at

x = 0, \frac{1}{3}, \frac{2}{3}, \frac{1}{2}

and evaluating

(ϑ_{2})

x = 0

, we have the following Values of the unknown coefficients

[c ₀ = - 6 . 015660797 02 \times 10 ⁻², c ₁ = 2 . 794908665 50 \times 10 ⁻², c ₂ = 0.109373977 228,

c ₃ = 0.159771274 744, c ₄ = 5 . 624959089 12 \times 10 ⁻², c ₅ = 2 . 957750691 22 \times 10 ⁻²]

Substituting these approximate values into

(6)

, we get the approximate solution to the problem as

0.232922866 934 x ⁵ + 0.246091960 149 x ⁴ + 0.140625001 378 x ³ - 0.046875 x ² - 0.156250000 001 x - 0.09375

Table 1. Showing the numerical comparison of example 2 for

N = 5

$x$

Exact

Approximate solution by LOM

Absolute error by LOM

Absolute error by

[

5 ]

$0.0025$

$- 0.094140915761800$

$- 0.0941409157619$

$2.5000 \times 10 ⁻¹⁵$

$7.020 \times 10^{- 14}$

$0.0050$

$- 0.094532404142338$

$- 0.09 45324041423$

$5.6388 \times 10 ⁻¹⁵$

$1.217 \times 10^{- 13}$

$0.0075$

$- 0.094924451608388$

$- 0.0949244516084$

$1.9695 \times 10 ⁻¹⁴$

$3.396 \times 10^{- 12}$

$0.0100$

$- 0.095317044390700$

$- 0.0953170443908$

$9.6425 \times 10 ⁻¹⁴$

$8.122 \times 10^{- 12}$

$0.0125$

$- 0.095710168480980$

$- 0.0957101684814$

$3.8902 \times 10 ⁻¹³$

$1.453 \times 10^{- 11}$

$0.0150$

$- 0.096103809629100$

$- 0.0961038096304$

$1.2407 \times 10 ⁻¹²$

$2.233 \times 10^{- 11}$

$0.0175$

$- 0.096497953340300$

$- 0.0964979533436$

$3.3021 \times 10 ⁻¹²$

$3.156 \times 10^{- 11}$

$0.0200$

$- 0.096892584872264$

$- 0.0968925848799$

$7.6781 \times 10 ⁻¹²$

$4.220 \times 10^{- 11}$

$0.0225$

$- 0.097289689232184$

$- 0.0972876892483$

$1.6104 \times 10 ⁻¹¹$

$5.421 \times 10^{- 11}$

$0.0250$

$- 0.097683251173919$

$- 0.0976832512051$

$3.1149 \times 10 ⁻¹¹$

$6.754 \times 10^{- 11}$

xample 3

Consider the following nonlinear second order boundary value problem solved in

[16]

given as:

y^{''} (x) = {(y^{'} (x))}^{2} - y (x) - 16 x^{6} + 2, x \in [- 1, 1], h = 0.0025

subject to the initial conditions

y (- 1.0) = 0, y^{'} (1.0) = - 2

, the exact solution to this problem is known to be

y (x) = x^{2} - x^{4}

Applying our technique with

N = 4

we have the following expression;

L (x) {(M^{T})}^{(2)} C^{T} = {(L (x) {(M^{T})}^{(1)} C^{T})}^{2} - L (x) C^{T} - 16 x^{6} + 2 (ϑ_{3})

Using the initial conditions, we have respectively,

L (0) C^{T} = 0, L (0) {(M^{T})}^{(1)} C^{T} = 0 (ϑ_{4})

Collocating (

ϑ_{3}

) at

x = 0, \frac{1}{2}, \frac{1}{4}

and evaluating

(ϑ_{4})

x = 0

, we have the following Values of the unknown coefficients

[c ₀ = 0.133333333 333, c ₁ = 0.0, c ₂ = 9 . 523809523 81 \times 10 ⁻², c ₃ = 0.0, c ₄ = - 0.228571428 571]

Substituting these approximate values into

(6)

, we get the approximate solution to the problem as

- 1 . 000000000 00 x ⁴ + 1 . 000000000 00 x ² - 1 . 750000000 00 \times 10 ⁻¹³

Table 2. Showing the numerical comparison of example

3

for

N = 4

$x$	Exact	Approximate solution by LOM	Absolute error LOM $\|y_{N} (x) - y (x)\|$
$0.0025$	$6.24996093750 \times 10 ⁻⁶$	$6.24996076250 \times 10 ⁻⁶$	$1 . 75 \times 10 ⁻¹³$
$0.0050$	$2.4999 3750000 \times 10 ⁻⁵$	$2.49993748250 \times 10 ⁻⁵$	$1 . 75 \times 10 ⁻¹³$
$0.0075$	$5.6246 8359375 \times 10 ⁻⁵$	$5.62468357625 \times 10 ⁻⁵$	$1 . 75 \times 10 ⁻¹³$
$0.0100$	$9.999000000000 \times 10 ⁻⁵$	$9.99899998250 \times 10 ⁻⁵$	$1 . 75 \times 10 ⁻¹³$
$0.0125$	$1.562255859375 \times 10 ⁻⁴$	$1.56225585763 \times 10 ⁻⁴$	$1 . 75 \times 10 ⁻¹³$
$0.0150$	$2.249493750000 \times 10 ⁻⁴$	$2.24949374825 \times 10 ⁻⁴$	$1 . 75 \times 10 ⁻¹³$
$0.0175$	$3.061562109375 \times 10 ⁻⁴$	$3.06156210763 \times 10 ⁻⁴$	$1 . 75 \times 10 ⁻¹³$
$0.0200$	$3.998400000000 \times 10 ⁻⁴$	$3.99839999825 \times 10 ⁻⁴$	$1 . 75 \times 10 ⁻¹³$
$0.0225$	$5.059937109375 \times 10 ⁻⁴$	$5.05993710763 \times 10 ⁻⁴$	$1 . 75 \times 10 ⁻¹³$
$0.0250$	$6.24609375000 \times 10 ⁻⁴$	$6.24609374825 \times 10 ⁻⁴$	$1 . 75 \times 10 ⁻¹³$

Example 4

Consider the second order differential equation solved by

[5]

given by

y^{''} (x) = {x (y^{'} (x))}^{2}, x \in [- 1, 1], h = 0.0025

Subject to the initial conditions

y (0) = 1, y^{'} (0) = \frac{1}{2}

, the exact solution to this problem is known to be

y (x) = 1 + \frac{1}{2} \ln (\frac{2 + x}{2 - x})

Applying our technique with

N = 4

, we have the following expression;

L (x) {(M^{T})}^{(2)} C^{T} = x {(L (x) {(M^{T})}^{(1)} C^{T})}^{2} (ϑ_{5})

Using the initial conditions, we have respectively,

L (0) C^{T} = 1, L (0) {(M^{T})}^{(1)} C^{T} = \frac{1}{2} (ϑ_{6})

Collocating (

ϑ_{5}

) at

x = 0, \frac{1}{2}, \frac{1}{4}

and evaluating

(ϑ_{6})

x = 0

, we have the following values of the unknown coefficients

[c ₀ = 1 . 000003906 25, c ₁ = 0.524999990 234, c ₂ = 1 . 116072282 34 \times 10 ⁻⁵,

c ₃ = 1 . 666666015 62 \times 10 ⁻², c ₄ = 4 . 464289129 35 \times 10 ⁻⁶]

Substituting these approximate values into

(6)

, we get the approximate solution to the problem as

1 . 953126494 09 \times 10 ⁻⁵ x ⁴ + 4 . 166665039 05 \times 10 ⁻² x ³ + 3 . 75 \times 10 ⁻¹⁷ x ² + 0.500000000 000 x + 1 . 000000000 00

Table 3. Numerical comparison of example 4 for

N = 4 .

$x$

Exact

Approximate solution by LOM

Absolute error LOM $|y_{N} (x) - y (x)|$

Absolute error by

[

5 ]

|y_{N} (x) - y (x)|

$0.0025$

$1 . 001250000 65$

$1.001250000 60$

$1.017 \times 10 ⁻¹⁶$

$1.339 \times 10 ⁻¹⁴$

$0.0050$

$1 . 002500005 21$

$1.002500004 82$

$9.359 \times 10 ⁻¹⁵$

$7.321 \times 10 ⁻¹⁴$

$0.0075$

$1 . 003750017 58$

$1.003750016 29$

$9.339 \times 10 ⁻¹⁴$

$1.000 \times 10 ⁻¹³$

$0.0100$

$1 . 005000041 67$

$1.005000038 63$

$4.460 \times 10 ⁻¹³$

$1.250 \times 10 ⁻¹²$

$0.0125$

$1 . 006250081 38$

$1.006250075 49$

$1.462 \times 10 ⁻¹²$

$1.372 \times 10 ⁻¹²$

$0.0150$

$1 . 007500140 63$

$1.007500130 52$

$3.812 \times 10 ⁻¹²$

$4.824 \times 10 ⁻¹²$

$0.0175$

$1 . 008750223 32$

$1.008750207 37$

$8.514 \times 10 ⁻¹²$

$6.314 \times 10 ⁻¹²$

$0.0200$

$1 . 010000333 35$

$1.010000309 73$

$1.701 \times 10 ⁻¹¹$

$2.801 \times 10 ⁻¹²$

$0.0225$

$1 . 011250474 65$

$1.011250441 25$

$3.122 \times 10 ⁻¹¹$

$4.322 \times 10 ⁻¹²$

$0.0250$

$1 . 012500651 10$

$1.012500605 63$

$5.367 \times 10 ⁻¹¹$

$6.757 \times 10 ⁻¹¹$

4. Conclusion

In this work, a collocation technique based on the Legendre matrix calculus for solving general second order linear and nonlinear differential equations was presented. The derivation of this algorithm was essentially based on choosing a set of Legendre polynomials. The advantage of this technique over other methods is that it has less computational manipulations and complexities and also its availability for application on both linear and nonlinear second-order initial value problems. Other importance of the algorithm is that high accurate approximate solutions are achieved by using a few numbers of terms of the Legendre polynomial which result to simple matrix calculus and its transpose and thus, reduces the time involve in the derivation of the schemes to be used for implementation as compared to the case of linear multistep methods, more importantly the same matrix can be used to solve higher order differential equations by introducing just a little manipulation io it and also reduces the computational run time. The comparison of the results shows that the method is a very simple and efficient mathematical tool for solving initial value problems of differential equations.

Acknowledgments

The author expresses their sincere thanks to the referees for the careful and details reading of their earlier version of the paper and for the very helpful suggestions.

Author Contributions

Nathaniel Mahwash Kamoh: Conceptualization, Methodology, Project administration, Software, Writing – original draft, Writing – review & editing

Bwebum Cleofas Dang: Conceptualization, Methodology, Writing – original draft

Comfort Soomiyol Mrumun: Methodology, Project administration, Writing – original draft, Writing – review & editing

Conflicts of Interest

The authors declare no conflicts of interest.

References

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Kamoh, N. M., Dang, B. C., Mrumun, C. S. (2024). An Effective Matrix Technique for the Numerical Solution of Second Order Differential Equations. Applied and Computational Mathematics, 13(4), 111-117. https://doi.org/10.11648/j.acm.20241304.15

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Kamoh, N. M.; Dang, B. C.; Mrumun, C. S. An Effective Matrix Technique for the Numerical Solution of Second Order Differential Equations. Appl. Comput. Math. 2024, 13(4), 111-117. doi: 10.11648/j.acm.20241304.15

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AMA Style

Kamoh NM, Dang BC, Mrumun CS. An Effective Matrix Technique for the Numerical Solution of Second Order Differential Equations. Appl Comput Math. 2024;13(4):111-117. doi: 10.11648/j.acm.20241304.15

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@article{10.11648/j.acm.20241304.15,
  author = {Nathaniel Mahwash Kamoh and Bwebum Cleofas Dang and Comfort Soomiyol Mrumun},
  title = {An Effective Matrix Technique for the Numerical Solution of Second Order Differential Equations
},
  journal = {Applied and Computational Mathematics},
  volume = {13},
  number = {4},
  pages = {111-117},
  doi = {10.11648/j.acm.20241304.15},
  url = {https://doi.org/10.11648/j.acm.20241304.15},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20241304.15},
  abstract = {In this paper, an effective technique for solving differential equations with initial conditions is presented. The method is based on the use of the Legendre matrix of derivatives defined on the close interval [-1,1]. Properties of the polynomial are outlined and further used to obtain the matrix of derivative which was used in transforming the differential equation into systems of linear and nonlinear algebraic equations. The systems of these algebraic equations were then solved using Gaussian elimination method to determine the unknown parameters required for approximating the solution of the differential equation. The advantage of this technique over other methods is that, it has less computational manipulations and complexities and also its availability for application on both linear and nonlinear second-order initial value problems is impressive. Other advantage of the algorithm is that high accurate approximate solutions are achieved by using a greater number of terms of the Legendre polynomial and once the operational matrix is obtained, it can be used to solve differential equations of higher order by introducing just a little manipulation on the operational matrix. Some existing sample problems from literature were solved and the results were compared to show the validity, simplicity and applicability of the proposed method. The results obtained validate the simplicity and applicability of the method and it also reveals that the method perform better than most existing methods.
},
 year = {2024}
}

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TY  - JOUR
T1  - An Effective Matrix Technique for the Numerical Solution of Second Order Differential Equations

AU  - Nathaniel Mahwash Kamoh
AU  - Bwebum Cleofas Dang
AU  - Comfort Soomiyol Mrumun
Y1  - 2024/08/15
PY  - 2024
N1  - https://doi.org/10.11648/j.acm.20241304.15
DO  - 10.11648/j.acm.20241304.15
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 111
EP  - 117
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20241304.15
AB  - In this paper, an effective technique for solving differential equations with initial conditions is presented. The method is based on the use of the Legendre matrix of derivatives defined on the close interval [-1,1]. Properties of the polynomial are outlined and further used to obtain the matrix of derivative which was used in transforming the differential equation into systems of linear and nonlinear algebraic equations. The systems of these algebraic equations were then solved using Gaussian elimination method to determine the unknown parameters required for approximating the solution of the differential equation. The advantage of this technique over other methods is that, it has less computational manipulations and complexities and also its availability for application on both linear and nonlinear second-order initial value problems is impressive. Other advantage of the algorithm is that high accurate approximate solutions are achieved by using a greater number of terms of the Legendre polynomial and once the operational matrix is obtained, it can be used to solve differential equations of higher order by introducing just a little manipulation on the operational matrix. Some existing sample problems from literature were solved and the results were compared to show the validity, simplicity and applicability of the proposed method. The results obtained validate the simplicity and applicability of the method and it also reveals that the method perform better than most existing methods.

VL  - 13
IS  - 4
ER  -

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Author Information

Nathaniel Mahwash Kamoh

Department of Mathematics, University of Jos, Jos, Nigeria

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http://orcid.org/0000-0001-6333-1913
Bwebum Cleofas Dang

Department of Mathematics, University of Jos, Jos, Nigeria

Contact Email
Comfort Soomiyol Mrumun

Department of Mathematics, Benue State University, Makurdi, Nigeria

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Plain Text BibTeX RIS

APA Style

Kamoh, N. M., Dang, B. C., Mrumun, C. S. (2024). An Effective Matrix Technique for the Numerical Solution of Second Order Differential Equations. Applied and Computational Mathematics, 13(4), 111-117. https://doi.org/10.11648/j.acm.20241304.15

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ACS Style

Kamoh, N. M.; Dang, B. C.; Mrumun, C. S. An Effective Matrix Technique for the Numerical Solution of Second Order Differential Equations. Appl. Comput. Math. 2024, 13(4), 111-117. doi: 10.11648/j.acm.20241304.15

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AMA Style

Kamoh NM, Dang BC, Mrumun CS. An Effective Matrix Technique for the Numerical Solution of Second Order Differential Equations. Appl Comput Math. 2024;13(4):111-117. doi: 10.11648/j.acm.20241304.15

Copy | Download

@article{10.11648/j.acm.20241304.15,
  author = {Nathaniel Mahwash Kamoh and Bwebum Cleofas Dang and Comfort Soomiyol Mrumun},
  title = {An Effective Matrix Technique for the Numerical Solution of Second Order Differential Equations
},
  journal = {Applied and Computational Mathematics},
  volume = {13},
  number = {4},
  pages = {111-117},
  doi = {10.11648/j.acm.20241304.15},
  url = {https://doi.org/10.11648/j.acm.20241304.15},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20241304.15},
  abstract = {In this paper, an effective technique for solving differential equations with initial conditions is presented. The method is based on the use of the Legendre matrix of derivatives defined on the close interval [-1,1]. Properties of the polynomial are outlined and further used to obtain the matrix of derivative which was used in transforming the differential equation into systems of linear and nonlinear algebraic equations. The systems of these algebraic equations were then solved using Gaussian elimination method to determine the unknown parameters required for approximating the solution of the differential equation. The advantage of this technique over other methods is that, it has less computational manipulations and complexities and also its availability for application on both linear and nonlinear second-order initial value problems is impressive. Other advantage of the algorithm is that high accurate approximate solutions are achieved by using a greater number of terms of the Legendre polynomial and once the operational matrix is obtained, it can be used to solve differential equations of higher order by introducing just a little manipulation on the operational matrix. Some existing sample problems from literature were solved and the results were compared to show the validity, simplicity and applicability of the proposed method. The results obtained validate the simplicity and applicability of the method and it also reveals that the method perform better than most existing methods.
},
 year = {2024}
}

Copy | Download

TY  - JOUR
T1  - An Effective Matrix Technique for the Numerical Solution of Second Order Differential Equations

AU  - Nathaniel Mahwash Kamoh
AU  - Bwebum Cleofas Dang
AU  - Comfort Soomiyol Mrumun
Y1  - 2024/08/15
PY  - 2024
N1  - https://doi.org/10.11648/j.acm.20241304.15
DO  - 10.11648/j.acm.20241304.15
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 111
EP  - 117
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20241304.15
AB  - In this paper, an effective technique for solving differential equations with initial conditions is presented. The method is based on the use of the Legendre matrix of derivatives defined on the close interval [-1,1]. Properties of the polynomial are outlined and further used to obtain the matrix of derivative which was used in transforming the differential equation into systems of linear and nonlinear algebraic equations. The systems of these algebraic equations were then solved using Gaussian elimination method to determine the unknown parameters required for approximating the solution of the differential equation. The advantage of this technique over other methods is that, it has less computational manipulations and complexities and also its availability for application on both linear and nonlinear second-order initial value problems is impressive. Other advantage of the algorithm is that high accurate approximate solutions are achieved by using a greater number of terms of the Legendre polynomial and once the operational matrix is obtained, it can be used to solve differential equations of higher order by introducing just a little manipulation on the operational matrix. Some existing sample problems from literature were solved and the results were compared to show the validity, simplicity and applicability of the proposed method. The results obtained validate the simplicity and applicability of the method and it also reveals that the method perform better than most existing methods.

VL  - 13
IS  - 4
ER  -

Copy | Download