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Global Research journal of Natural Science

& Technology (GRJNST)

Volume: 04 - Issue 3 (2026), 2089

ISSN P: 2790-7643 ISSN E: 2790-7651

www.grjnst.net

https://doi.org/10.53762/grjnst.04.03.10

Efficient Explicit Improved Scheme for Numerical Solution of Cauchy

Problems

Received: 29 March 2026. Accepted: 21 April 2026. Published: 18 May 2026

Muhammad Daud Kandhro (Corresponding Author)

Institute of Mathematics and Computer Science,

University of Sindh, Jamshoro, Sindh, Pakistan

Email: daud.kandhro@gmail.com

ORCHID ID: https://orcid.org/0009-0001-7160-2047

Khushbu Rajput

Department of Basic Engineering,

Sindh Agriculture University, Tandojam, Sindh, Pakistan

Email: rajputkhushi971@gmail.com

ORCHID ID: https://orcid.org/0000-0003-4850-6038

Zohaib Ali

Institute of Mathematics and Computer Science,

University of Sindh, Jamshoro, Sindh, Pakistan

Email: zohaib.ali@usindh.edu.pk

ORCHID ID: https://orcid.org/0009-0005-8077-2004

GRJNST, Volume: 04 - Issue 3 (2026) / ISSN P: 2790-7643

Article ID: 2089

https://doi.org/10.53762/grjnst.04.03.10

Commons Attribution 4.0 International (CC BY 4.0) license, which permits unrestricted use and distribution

G. 2089

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Abstract:

This study focuses on designing explicit splendid converging method to

estimate numerical results of initial-value-problems (IVP’s) for differential

equation in nature ordinary. The stability criteria of the improved scheme are

investigated, and corresponding stability region is depicted. Error analysis

carried out also affirms accuracy having third order. Error (LTE) is derived by

utilizing Taylor’s series expansion to skip higher order terms after matching

corresponding coefficients. Adding a partial derivative inside function

evaluation raises convergence and reduces both errors (maximum and last).

Based on results for improved scheme to illustrate the accuracy and efficiency.

The comparisons of the improved and existing schemes having same order of

local accuracy are discussed. Looking at numerical results, the improved scheme

gives the better results in the comparison of existing few same order methods.

Stability is proved to examine behavior of method and its region is visualized.

Consistency is investigated and shows how errors shrink to zero. The numerical

result is validated and illustrated graphically via utilizing MATLAB 2023a

software.

Keywords: Numerical Methods, Third order, Local Truncation Error,

Convergence

1. INTRODUCTION

Ordinary differential equations (ODEs) play a crucial role in modelling problems arising

in science and engineering. In particular, Cauchy problems (or initial value problems -

IVPs) are encountered in population dynamics, fluid dynamics and control theory.

These problems may not always have analytical solutions, and there is a need for

efficient and accurate numerical techniques. Many numerical methods have been

proposed to find approximate solutions with different levels of accuracy. But there is a

need for fast and accurate schemes. In this work, our goal is to address this issue by

designing a new numerical integration technique. The goal is to improve the accuracy of

the method without compromising its simplicity.

We have developed an efficient explicit third-order improved method for solving

Cauchy problems for ordinary differential equations. The modification of the new

scheme is based on the adjustment of classical numerical schemes and introducing

correction terms. They are done to improve the order of convergence of the scheme,

GRJNST, Volume: 04 - Issue 3 (2026) / ISSN P: 2790-7643

Article ID: 2089

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while keeping it efficient. This scheme is explicit, easy to implement and efficient. It

does not require the solution of nonlinear equations, as is required by implicit schemes.

The scheme is based on the method which has third-order accuracy. Hence, it can be

used to solve many problems.

The proposed scheme is derived using a Taylor series expansion, a standard approach in

numerical analysis to develop higher-order methods. The series expansion of the exact

solution is used to compare the coefficients with the numerical solution, thus revealing

the local truncation error. Terms of higher order are discarded to obtain the desired

accuracy. The presence of partial derivative terms in the evaluation of the function

increases the accuracy. This facilitates the tracing of solution curve in each step. This

leads to better convergence of the proposed scheme. The scheme is theoretically proven

to be third-order accurate.

A detailed error analysis for the scheme is conducted. The local truncation error (LTE)

is computed by Taylor series expansion. The analysis confirms the order of the leading

error term as four and thus the convergence of the scheme as third order. Besides LTE,

the global error is also investigated by numerical experiments. These results indicate the

maximum error as well as the final error is significantly reduced. Hence, the new scheme

is superior to conventional approaches. This shows the advantage of adding more

correction procedures.

Another important aspect of the method is stability. We investigate the stability of the

method to assess its performance under different step sizes. We use a suitable test

equation to get the stability function. Using this, we compute and draw the stability

region. The plot gives an idea of how the method will perform in various cases. We

observe that the new scheme has a good stability region. As a result, the scheme can be

applied to solve a wide variety of problems without stability problems.

Theoretical analysis also indicates that the new scheme is consistent. A numerical scheme

is said to be consistent if the local truncation error (LTE) approaches zero as the step

size shrinks to zero. The LTE formula derived is indicative of this. This guarantees the

numerical solution will converge with the analytic solution as the step size goes to zero.

This, together with stability, ensures convergence as per the theory of numerical analysis.

Hence, the scheme meets the necessary conditions for a good numerical scheme. This

enhances the proposed method's reliability.

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To assess the effectiveness of the new approach, a number of examples are presented.

These are chosen to demonstrate the accuracy and efficiency of the scheme. These are

compared with the solutions of existing third-order schemes. Results are presented in

tables and figures to clearly show the differences. The enhanced method has better

accuracy with smaller error measures. This proves to be better than traditional schemes

of the same level. The results demonstrate the efficiency and accuracy of the proposed

scheme.

All numerical simulations and graphical validations are performed in MATLAB 2023a.

The proposed method is easy to apply as it is explicit. Plots of the exact and

approximate solutions visually verify the accuracy of the scheme. The results

demonstrate that the new scheme is close to the exact solution. This confirms the

theoretical results obtained in the paper. In conclusion, the proposed method is accurate,

stable and efficient. This method can thus be seen as an important addition to the

numerical solution of ordinary differential equations.

2. DERIVATION

We initiate by considering general form of ODE’s

dy

 f x_n, y_n



, yx₀



 y₀



1



dx

It is assumed, eqn. (1) on a specific integration interval confesses a unique solution.

Exact and numerical result is signified by

adopts new form

and

respectively. Integrate eqn. (1), it

x₀h

y_n1

f

x_n, y_n



dx

^{ y }

n

x₀

In discrete manner

If

s

^

y_n1 y_n h

_iw

i

i1

y_n1 y_n h₁w  ₂w₂ ₃w₃



2



1

Where w ,w₂and w₃are the slopes, determined by

1

w₁ f x_n, y_n



w₂ f



x_n₂h, y_n h₂₁w₁ h²



₂w₁f_y



1

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w₃ f



x_n₃h, y_n h₃₁w  ₃₂w₂ h²



₃₁w₁f_y



1

Taylor Series (TS) expansion for

Y

x_n, y_n



is

1

2

1

2

1

1

3

1

2

1

2

1

6













G

x_n, y_n  y

x



 hf  h²

f_x

f_yf  h³

f_xx

f 

xy

f_yyf 

f_y²f 

f_xf_y















6









1

8

1







f_xxx



3 f_xxy 3 f_xyyf  5 f_yf_xyf 

f_xf_xy

f_yyyf ³











⁵

24

1

24

h⁴

 O

3

 

^h







1

3 f_yf ²f_yy 3 f_xff_yy f_y³f 



f_xf_y² f_yf_xx













24

24

Slopes

and w₃are expended thru using Taylor’s Series then surrogate

,

w₂

w

w₂

1

andw₃into



2

. Finally, coefficients of are equating with



3



in power of h up to h³for

attaining system.

1

6

1

₁ ₂ ₃1

₂₃₃₂

₂₂₃₃

2

1

6

1

2

1



₂²₂₃²₃





₂₂₁ ₃₃₁ ₃₃₂

₂₂₂₁₃₃₃₁₃₃₃₂



4



2

3

1

6

1

₂₂₁ ₃₃₁ ₃₂₁₃₂

₂₂²₁ ₃₃²₁ ₃₃²₂ ₃₃₁₃₂

2

6

Above system is nonlinear which has equations and 10 variable quantities. Now we are

investigating this system for solution. So, it has multiple solutions because of free

variables. One of the accurate and reliably solution we have

w  f x_n, y_n

1



2h

2

3







w₂ f  x_n

, y  hw

 hf 

₁

_y

n





3











2h

1

3

4





w  f x 

, y_n

hw  hw₂ h²w f



5





3

n

1



^y

3

15

5

h

y_n1 y_n

3w  4w₂ 5w₃

1

12

New proposed method has driven now further investigation is mandatory to check

accuracy, convergence and compute error.

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3. LOCAL TRUNCATION ERROR

The enhanced extended method having local truncation error is expressed as T_n1, where

T_n1 Gx  h y_n1

Where

is gained by TS and y_n1is used as an approximate solution. Taylor

G

x  h

series is utilized to expand these around

and similar terms collect in . Now we

expand proposed accelerated explicit Method present in equation (2) upto , we get

1

49

1

49









(

f_yyyf 

f_xyy) f ²

f_xy( ff_y f_x) 

f_yf_xx

f f

xxy

216

19

648

5

72

1

648

1

4

T_n1



h  Oh⁵



6







1

5



f_y³f  f_yy( f_yf ²

f_xf ) 

f_xyyf 

f_xf_y²

f_xxx









24

6

72

81

24

216

4. CONSISTENCY ANALYSIS

Definition 4.1 NM together IVP with an increment function

be consistent, if



x_n, y_n;h is accepted to

lim

h0



x_n, y_n;h  f x_n, y_n

From proposed method

1



x_n, y_n;h



3w  4w₂ 5w₃

1

12

Proceeds lim on both side

h0

1

lim x_n, y_n;h



lim 3w  4w₂ 5w₃

1

h0

12









2h

2

3









3 f x_n, y_n 4 f  x_n

, y  hw

 hf  

₁

_y

n





3

1













lim

^h0





12

2h

1

3

4





5 f x 

, y_n

hw  hw₂ h²w f





n

1

^y



3

15

5



 f x_n, y_n

Hence proved the consistent with at least third order accuracy for improved method.

5. LINEAR STABILITY ANALYSIS

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Stability is shown by taking Dahlquist’s test problem which is in form

dy

 q y

x



;

y



0



 y₀, qC

dx

we have attained polynomial function; we called it stability polynomial by employing (5)

on this test problem. In figure 1 unshaded region displays linear stability region.

2









w  q y_n; w₂ q y_n1 hq h²q ²; w₃ q y_n1 hq 6h²q ² 3h³q ³

1









3









Substituting above all values in (5), the stability function is originated

z²

z

z²









R



z



 1 z 

1



z  hq

where

2

3

2





6. NUMERICAL INVESTIGATION

To investigate the behaviour of methods either they have better converges or not, we

examined and tested few problems for proving better result. We have chosen three

methods, two methods are taken from open literature, and one is proposed. Before

taking method, one thing is kept in mind all methods have same order. So here with the

help of MATLAB2023a, error such absolute last and maximum is evaluated by using

these methods, graph of each problem is displayed separately. In graph three curvy lines

are drawn. Numerically and graphically result give the clear evidence on which proposed

method is better in all aspects. RK3M and RK3HM both less converges than developed

one.

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Table 1. Shows Numerically Illustration of Problem 1

Problem 1.

dy

2

 xy³ y,

y



0

1

Exact 

RK3M

2  4x  2e^2x

dx

Step-size

/method

0.1s

RK3HM

Proposed

2.2406e-004

2.1363e-004

0e0

2.1008e-005

1.5944e-005

0e0

5.8246e-006

2.2251e-006

0e0

0.05

0.025

0.0125

5.1698e-005

4.9489e-005

0e0

1.2497e-005

1.1988e-005

0e0

3.0773e-006

2.9553e-006

0e0

2.5084e-006

1.8970e-006

0e0

3.0722e-007

2.3198e-007

0e0

3.8027e-008

2.8697e-008

0e0

5.7261e-007

6.9545e-008

0e0

6.2151e-008

3.1638e-009

0e0

7.1587e-009

1.1110e-009

0e0

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Table 2. Shows Numerically Illustration of Problem 2

Problem 2.

x²

y

1

dy

dx

Exact 



,

y



0



1

3 9  6x³

Step-size

/method

0.1

RK3HM

RK3M

Proposed

1.5241e-003

0e0

2.2290e-005

0e0

2.6463e-006

2.0957e-006

0e0

0.05

0.025

0.0125

3.8551e-004

0e0

9.7000e-005

0e0

2.4332e-005

0e0

2.8590e-006

0e0

3.6139e-007

0e0

4.5408e-008

0e0

4.1078e-007

0e0

6.1315e-008

0e0

8.3075e-009

0e0

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Table 3. Shows Numerically Illustration of Problem 3

Problem 3.

dy

sin x

Exact  3 2cos x



, y(0) 1

dx

y

Step-size

/method

0.1

RK3HM

RK3Ms

Proposed

2.2846e-003

1.9600e-003

0e0

1.4261e-005

1.3062e-005

0e0

5.1507e-006

3.8481e-007

0e0

0.05

0.025

0.0125

6.5681e-004

5.5494e-004

0e0

1.8632e-004

1.5560e-004

0e0

5.2173e-005

4.3185e-005

0e0

1.8175e-006

1.6664e-006

0e0

2.2894e-007

2.0996e-007

0e0

2.8709e-008

2.6336e-008

0e0

6.2727e-007

3.3522e-008

0e0

7.7070e-008

9.3205e-009

0e0

9.5465e-009

1.4863e-009

0e0

s

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Table 4. Shows Numerically Illustration of Problem 4

Problem 4.

x³

dy

 x²y,

y



0

1

3

^{Exact }e

dx

Step-size

/method

0.1

RK3HM

RK3M

Proposed

3.1171e-003

0e0

6.3568e-005

0e0

1.4165e-005

1.3359e-005

0e0

0.05

0.025

0.0125

8.2411e-004

0e0

2.1194e-004

0e0

5.3744e-005

0e0

8.4079e-006

0e0

1.0805e-006

0e0

1.3694e-007

0e0

1.6414e-006

1.2474e-006

0e0

1.9677e-007

1.2857e-007

0e0

2.4068e-008

1.4347e-008

0e0

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7. RESULTS AND DISCUSSIONS

The proposed method is constructed to find more suitable and accurate solution of

IVP’s for ODE’s. Proposed method has accuracy order three, so we compare it with

order third methods. AT different step size such as 0.1, 0.05, 0.025 and 0.0125 error is

computed including absolute last and maximum and CPU time is observed in seconds.

From the data and graphs listed above clearly observed that the extended proposed

scheme produced a smaller error then existing method, having the same order of

accuracy and CPU time. The results have achieved numerically by proposed method

which comes quickly closer to exact point of the solution in comparison of RK3M and

RK3HM. Conclusively, the amended proposed scheme is most effective to solve Cauchy

differential equations and converges faster to the required solution.

7. CONCLUSION

This study had achieved these objectives, firstly it focuses on designing explicit splendid

converging methods to estimate numerical result of IVP’s for differential equation in

nature ordinary. Secondly, the stability criteria of the improved scheme are investigated,

and corresponding stability region is depicted. Thirdly, Error analysis carried out also

affirms accuracy having third order. Error (LTE) is derived by utilizing Taylor’s series

expansion to skip higher order terms after matching corresponding coefficients. Adding

a partial derivative inside function evaluation raises convergence and reduces both errors

(maximum and last). Based on results for improved scheme to illustrate the accuracy and

efficiency. The comparisons of the improved and existing schemes having same order of

local accuracy are discussed. Looking at numerical results, the improved scheme gives the

better results in comparison with the existing few same order methods. Stability is

proved to examine behavior of method and its region is visualized. Finally, Consistency

is investigated and shows how errors shrink to zero. The numerical result is validated

and illustrated graphically via utilizing MATLAB 2023a software.

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[1]

Abdul Majid Wazwak ‘’ A Modified Third Order Runge Kutta Method”Al

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Page 13

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Mukaddes Okten Turacı1, 2·Turgut ¨Ozis¸1 ‘’Derivation of Three-Derivative

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