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Research article Β· 2022 Β· Full text

International Journal of Statistics and Applied Mathematics

A family of r-points 1-block implicit methods with optimized region of stability for stiff initial value problems in ordinary differential equations

I. M. EsuabanaSE EkoroFA AdieU. A. Abasiekwere

Volume 7(4) Β· Pages 22–35 Β· DOI: 10.22271/maths.2022.v7.i4a.842 β†—

Research areasNumerical methods for differential equationsMatrix Theory and AlgorithmsDifferential Equations and Numerical Methods

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Abstract

Author-written

In this paper, we proposed a family of r-points 1-block implicit methods with optimized region of stability. This family of methods is derived with Mathematical 10.4 software and the stability is investigated using boundary locus techniques. The block methods are consistence, zero stable, and A-stable and satisfy other stability requirements which finds them suitable for stiff problems in ODEs. Numerical experiments are presented and results are compared with other block methods and exact solutions of some stiff ordinary differential equations. The methods have been found to show competitiveness with other numerical methods.

Summary

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Plain-language reading of the full paper, prepared with AI and checked against the text. Each point links to the authors' own words.

  1. Why it matters

    • Differential equations are essential in science and technology for modeling deterministic relations between varying quantities and their rates of change. View evidence β†—

      Why it matters

      Differential equations are essential in science and technology for modeling deterministic relations between varying quantities and their rates of change.

      In the paper's own words

      β€œDifferential equations arise in many areas of science and technology, specifically whenever a deterministic relation involving some continuously varying quantities (modeled by functions) and the rate of change in space or time (expressed as derivatives) is known.”

      Page 1

      Found in: Introduction

      Read it in the full textOpen PDF at page 1 β†—
    • Because most complex differential equations lack analytical solutions, numerical methods are required to provide approximate solutions. View evidence β†—

      Why it matters

      Because most complex differential equations lack analytical solutions, numerical methods are required to provide approximate solutions.

      In the paper's own words

      β€œMost complex differential equations cannot be solved analytically, therefore approximation to the solution becomes imperative. This can be made possible by the application of suitable numerical methods.”

      Page 1

      Found in: Introduction

      Read it in the full textOpen PDF at page 1 β†—
    • Numerical block methods are advantageous because they are self-starting, minimize errors, and reduce computation time by solving simultaneously at all grid points. View evidence β†—

      Why it matters

      Numerical block methods are advantageous because they are self-starting, minimize errors, and reduce computation time by solving simultaneously at all grid points.

      In the paper's own words

      β€œNumerical block methods preserve the advantage of being self-starting methods and have minimal errors, it also has lesser time of computation since it solve simultaneously at all the grid points unlike some other numerical methods”

      Page 1

      Found in: Introduction

      Read it in the full textOpen PDF at page 1 β†—

  2. The question

    • The researchers aim to develop a family of block methods with an optimized stability region for both stiff and non-stiff initial value problems in ordinary differential equations. View evidence β†—

      The question

      The researchers aim to develop a family of block methods with an optimized stability region for both stiff and non-stiff initial value problems in ordinary differential equations.

      In the paper's own words

      β€œOur interest in this work is to formulate a family of block methods with optimized region of stability suitable for stiff and non-stiff Initial Value problems in Ordinary Differential Equations”

      Page 1

      Found in: Introduction

      Read it in the full textOpen PDF at page 1 β†—

  3. What they did

    • The authors used Mathematical 10.4 software for derivation and boundary locus techniques to investigate stability. View evidence β†—

      What they did

      The authors used Mathematical 10.4 software for derivation and boundary locus techniques to investigate stability.

      In the paper's own words

      β€œThis family of methods is derived with Mathematical 10.4 software and the stability is investigated using boundary locus techniques.”

      Page 1

      Found in: Conclusion

      Read it in the full textOpen PDF at page 1 β†—
    • The proposed methods were tested through numerical experiments and compared against exact solutions and other existing block methods. View evidence β†—

      What they did

      The proposed methods were tested through numerical experiments and compared against exact solutions and other existing block methods.

      In the paper's own words

      β€œNumerical experiments are presented and results are compared with other block methods and exact solutions of some stiff ordinary differential equations.”

      Page 1

      Found in: Introduction

      Read it in the full textOpen PDF at page 1 β†—

  4. What they found

    • The developed block methods are consistent, zero stable, and A-stable, making them appropriate for stiff ordinary differential equation problems. View evidence β†—

      What they found

      The developed block methods are consistent, zero stable, and A-stable, making them appropriate for stiff ordinary differential equation problems.

      In the paper's own words

      β€œThe block methods are consistence, zero stable, and A-stable and satisfy other stability requirements which finds them suitable for stiff problems in ODEs.”

      Page 1

      Found in: Conclusion

      Read it in the full textOpen PDF at page 1 β†—

  5. What it means

    • The authors conclude that the proposed block method is suitable and recommend it for systems of first-order initial value problems in ordinary differential equations. View evidence β†—

      What it means

      The authors conclude that the proposed block method is suitable and recommend it for systems of first-order initial value problems in ordinary differential equations.

      In the paper's own words

      β€œTherefore proposed block method is suitable and also recommended for system of first order IVPs in ODEs.”

      Page 13

      Found in: Conclusion

      Read it in the full textOpen PDF at page 13 β†—

Visual summary

3 visuals

Illustrations explain the paper; charts are replotted from the paper's own table values.

AI-generated illustration
AI-generated conceptual illustration (a complex plane coordinate system). It explains the idea and does not show study data.β†— Enlarge
Stability Region Concept
AI-generated conceptual illustration (a complex plane coordinate system). It explains the idea and does not show study data.

AI-generated conceptual illustration (a complex plane coordinate system). It explains the idea and does not show study data.

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Stability Region Concept

AI-generated conceptual illustration (a complex plane coordinate system). It explains the idea and does not show study data.

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Replotted from source data
Replotted from Table 1 (page 12) of the original paper. Every bar is the paper's own table value.β†— Enlarge
0.1: Error Between Exact... vs Error Between Exact...
Replotted from Table 1 (page 12) of the original paper. Every bar is the paper's own table value.

Replotted from Table 1 (page 12) of the original paper. Every bar is the paper's own table value.

Open full-size image β†—

0.1: Error Between Exact... vs Error Between Exact...

Replotted from Table 1 (page 12) of the original paper. Every bar is the paper's own table value.

View source evidence β†—
Replotted from source data
Replotted from Table 1 (page 12) of the original paper. Every bar is the paper's own table value.β†— Enlarge
0.2: Error Between Exact... vs Error Between Exact...
Replotted from Table 1 (page 12) of the original paper. Every bar is the paper's own table value.

Replotted from Table 1 (page 12) of the original paper. Every bar is the paper's own table value.

Open full-size image β†—

0.2: Error Between Exact... vs Error Between Exact...

Replotted from Table 1 (page 12) of the original paper. Every bar is the paper's own table value.

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Figures & tables

From the original paper
A-stable method for r =2, d= -1 and s= -1.Figure 1Page 10A-stable method for r =3 at d = -1 and s = -6.5Figure 2Page 10A-stable method for r = 4, d = -0.35 and s = -6.8Figure 3Page 11A-stable method for r =5, d = -0.16 and s = -10.2Figure 4Page 11Numerical solution for problem 1Table 1Page 12Numerical solution for problem 2Table 2Page 12Numerical solution for problem 3Table 3Page 13for problem 4Table 4Page 13
FULL TEXT Original article Β· reflowed for reading

Introduction

Differential equations arise in many areas of science and technology, specifically whenever a deterministic relation involving some continuously varying quantities (modeled by functions) and the rate of change in space or time (expressed as derivatives) is known. Such areas include Hamilton's equations in classical mechanics, Newton's second law in dynamics; Radioactive decay in nuclear complex analysis; the heat equation in thermodynamics; Verhulst equation in biological population growth, Lotka volterra equations in biological population dynamics; exogenous growth model etc. The history of differential equations is traced from calculus, which was independently invented by English physicist Isaac Newton in 1676 and a German mathematician Gottfried Leibniz in 1693. Other mathematicians like Jacob Bernoulli and Leonard Euler also made remarkable contributions to the solutions of differential equations which also arise as a result of some physical phenomena in every day's life. Most complex differential equations cannot be solved analytically, therefore approximation to the solution becomes imperative. This can be made possible by the application of suitable numerical methods. Researchers like [6,3,1,13,10,11,14,20] , etc have proposed different numerical methods for solving IVPs in ODEs but some of these methods are not self-starting methods and required single step methods to generate other starting values which is computationally cumbersome and prone to enormous error since the starting method and the method are not of the same order. Our interest in this work is to formulate a family of block methods with optimized region of stability suitable for stiff and non-stiff Initial Value problems in Ordinary Differential Equations of the form 𝑦 β€² = 𝑓(π‘₯), 𝑦(π‘₯ 0 ) = 𝑦 0, π‘₯ ∈ [π‘Ž, 𝑏]

The proposed methods can also perform well for system of first order ODEs. Numerical block methods preserve the advantage of being self-starting methods and have minimal errors, it also has lesser time of computation since it solve simultaneously at all the grid points unlike some other numerical methods [11,5,15,1,7,18,19] etc. have worked on some numerical block methods for solution of (1) and have presented results that are competitive.

Derivation of the Methods

Our interest in this section is to construct the proposed Linear Multistep Methods that generate the block method. We consider polynomial interpolant of the form:

which also approximates the solution of (1) as

Collocating and interpolating at the different grid points generates the block in the form of

, B B A

are rXr matrices, A 0 a unit diagonal matrix, h is the fixed step size, d and s are stability parameters of the methods and r = 2, 3,....

Expand (3) we obtain https://www.mathsjournal.com

Obtaining the unknown parameters in (4) with Mathematica (10.4) we obtain , 24

B

Obtaining the unknown parameters in (7) with Mathematica (10.4) we have Consistency of the Method: The LMM method can be proved to be consistent if and only if the following conditions are satisfied [9] .

Definition (1.1): Block method is consistent if it has order at least one [4] .

Zero Stability: Block method is said to be zero stable if the roots r of the characteristics polynomial ρ(r) defined by ρ(r) =det[ rA 0 -A 1 ] satisfies |r| ≀ 1 and every root with |r 0 | = 1 has multiplicity not exceeding one in the limit as h β†’ 0

Therefore, the block method is zero stable.

Theorem (1.1):

The necessary conditions for LMM to converge are; it must be consistent and zero stable [8] . Since the method satisfies these conditions, therefore it is convergent.

Stability of the method: Consider the block method below (2) above

Applying the test equation ( 13) on (2) we have

Where (𝐴 0 -𝑧 𝑑 𝐡 Is defined as a unique integer p such that 𝐢 𝑝 = 0, π‘ž = 0(1)𝑝 π‘Žπ‘›π‘‘ 𝐢 𝑝+1 β‰  0

Where the 𝐢 π‘ž are constant defined as

The associated linear operator ( 2) is

The methods ( 2) is said to be of order p if 𝑐 0 = 𝑐 1 = β‹― = 𝑐 𝑝 = 0 π‘Žπ‘›π‘‘ 𝑐 𝑝+1 β‰  0 where 𝑐 𝑝+1 is the error constant of the block method. Therefore the order p and the associated error constant for each r-point is given as follows.

for r =2 is of order p = 4 with error constant

3):

A numerical method is said to be A-Stable if the stability region contain the entire left hand side of the complex plane [4] . One of the basic requirements for every numerical method to solve stiff IVPs in ODEs is that the method must be A-stable. Applying the boundary locus techniques on the amplification matrix of (2) we have Exact Solution 𝑦(π‘₯) = 𝑒 π‘₯ [1] .

Problem 4: Consider the system of equations

π‘Œ(𝑑) = 𝑒 -1000𝑑 -𝑒 -𝑑 . [2] . [17] Proposed Method Order P=6

Error Between Exact Solution and 3PBEBDF [17] Error Between Exact Solution and Proposed Method of order p=6 0.0 [17] Proposed Method Order P=6

Absolute Error Between Exact Solution and 3PBEBDF [17] Error Between Exact Solution and Proposed Method of order p =6

. [1] Proposed Methods at Order p=4

Error Between Exact solution and Ajileye et al. [1] Error Between Exact solution and

The proposed methods at p=4

Conclusion

A new family of block methods have been developed using a polynomial interpolant as a basis function. The family of the block methods has small error constant at high order which is a good property of every efficient numerical integrator. The boundary Locus shows that the method is A-stable for all values of r = 2, 3, 4, and 5with varying stability parameters s and d. Numerical Integrator for stiff IVPs in ODEs must have a wider region of absolute stability and which also our proposed block method has and therefore, it is suitable for stiff IVPs in ODEs it also satisfies other stability requirements like consistency and zero stability. The method was demonstrated in three stiff problems and results are displayed in table 1, 2, 3,.. This result shows high competitiveness with the exact solutions and some existing numerical methods. Therefore proposed block method is suitable and also recommended for system of first order IVPs in ODEs.

Table 1. Numerical solution for problem 1
XTheoretical Solution3PBEBDF ORDERP=6 [17]Proposed Method Order P=6Error Between Exact Solution and 3PBEBDF [17]Error Between Exact Solution and Proposed Method of order p=6
0.01.83333331.83333331.833333300
0.10.85260200.85274500.85263500.00014300.0000420
0.20.86917120.86905730.86915730.00011390.0000128
0.30.88354740.88297670.88297670.00057070.0002505
0.40.89610600.89608590.89609590.00002010.0000101
0.50.90713590.90650190.90682390.00063400.0003120
0.60.91686470.91554970.91676170.00131500.0001030
0.70.92547600.92530580.92541580.00017020.0000602
0.80.93312030.93215620.93287830.00096410.0002420
0.90.93992270.93816800.93916800.00175470.0007547
1.00.94598840.94396500.94497620.00202340.0010122

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Table 1
Numerical solution for problem 1

Numerical solution for problem 1

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Figures and tables from the paper

Figure 1. A-stable method for r =2, d= -1 and s= -1.
A-stable method for r =2, d= -1 and s= -1.β†— Enlarge
Figure 1
A-stable method for r =2, d= -1 and s= -1.

A-stable method for r =2, d= -1 and s= -1.

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Original paper figureSource PDF Β· page 10 β†—
Figure 2. A-stable method for r =3 at d = -1 and s = -6.5
A-stable method for r =3 at d = -1 and s = -6.5β†— Enlarge
Figure 2
A-stable method for r =3 at d = -1 and s = -6.5

A-stable method for r =3 at d = -1 and s = -6.5

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Original paper figureSource PDF Β· page 10 β†—
Figure 3. A-stable method for r = 4, d = -0.35 and s = -6.8
A-stable method for r = 4, d = -0.35 and s = -6.8β†— Enlarge
Figure 3
A-stable method for r = 4, d = -0.35 and s = -6.8

A-stable method for r = 4, d = -0.35 and s = -6.8

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Original paper figureSource PDF Β· page 11 β†—
Figure 4. A-stable method for r =5, d = -0.16 and s = -10.2
A-stable method for r =5, d = -0.16 and s = -10.2β†— Enlarge
Figure 4
A-stable method for r =5, d = -0.16 and s = -10.2

A-stable method for r =5, d = -0.16 and s = -10.2

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Original paper figureSource PDF Β· page 11 β†—
Table 2. Numerical solution for problem 2
Numerical solution for problem 2β†— Enlarge
Table 2
Numerical solution for problem 2

Numerical solution for problem 2

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Table 3. Numerical solution for problem 3
XTheoretical SolutionAjileye et al. [1]Proposed Methods at Order p=4Error Between Exact solution and Ajileye et al. [1]Error Between Exact solution and The proposed methods at p=4
0.11.15017091871.15017091781.15017091860.00000000090.0000000001
0.21.22140275811.22140275761.22140275800.00000000050.0000000001
0.31.34985880751.34985880671.34985880730.00000000080.0000000002
0.41.49182469761.49182469641.4918246977-0.0000000012-0.0000000001
0.51.64872127071.64872122691.64872127040.00000004380.0000000003
0.61.82211880031.82211879821.82211880120.0000000021-0.0000000009
0.72.01375270742.01375270472.01375270720.00000000270.0000000002
0.82.22554092842.22554092502.22554092810.00000000340.0000000003
0.92.45960311112.45960310682.45960311010.00000000430.0000000001
1.02.71828182842.71828182412.71828182830.00000000420.0000000001

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Table 3
Numerical solution for problem 3

Numerical solution for problem 3

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Table 4. for problem 4
hβ€”Exact Solution X(t)Proposed Method of Order P=4Error Between Exact and The Proposed Method of Order p=4
0.0x(t)1.0000000001.0000000000.0000000000
y(t)0.0000000000.0000000000.0000000000
0.2x(t)1.6374615061.6374615060.0000000000
y(t)-0.8187307531-0.81873075070.0000000002
0.4x(t)1.3406400921.3406400920.0000000000
y(t)-0.6703200460-0.67032004640.0000000004
0.6x(t)1.0976232721.0976232720.0000000000
y(t)-0.5488116361-0.54881163860.0000000025
0.8x(t)0.8993530500.8993999290.000046879
y(t)-0.4493289641-0.44932896620.0000000021
1.0x(t)0.7364703890.7364580880.000012301
y(t)-0.3678794412-0.36787944250.0000000013

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for problem 4β†— Enlarge
Table 4
for problem 4

for problem 4

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References

20 in the original paper
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