Solutions of Stiff Systems of Ordinary Differential Equations Using Residual Power Series Method

Qayyum, Mubashir; Fatima, Qursam

doi:https://doi.org/10.1155/2022/7887136

Journal of Mathematics

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Abstract Introduction Conclusion Data Availability Conflicts of Interest References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 7887136 | https://doi.org/10.1155/2022/7887136

Solutions of Stiff Systems of Ordinary Differential Equations Using Residual Power Series Method

Mubashir Qayyum¹and Qursam Fatima¹

Academic Editor: Clemente Cesarano

Received30 Oct 2021

Accepted04 Feb 2022

Published04 Mar 2022

Abstract

The stiff differential equations occur in almost every field of science. These systems encounter in mathematical biology, chemical reactions and diffusion process, electrical circuits, meteorology, mechanics, and vibrations. Analyzing and predicting such systems with conventional numerical techniques require more time and memory; still accurate solution is completely uneconomical and uncertain. Most of the numerical techniques have stability issues while dealing with stiff systems. To overcome these limitations, residual power series method (RPSM) is proposed for stiff systems of differential equations (DEs). RPSM is applied to various linear and nonlinear stiff systems, and closed-form solutions are achieved. This indicates the effectiveness of proposed scheme for stiff family of DEs. Since this method leads to better results with less computational cost, it can be extended for more complex systems which arise in different areas of engineering and sciences.

1. Introduction

Stiff system of ordinary differential equations (ODEs) arises in modeling of various real-world physical problems. Solutions of such systems are of great interest for the scientific community. Several numerical, analytical, and seminumerical schemes have been proposed for the solution of such systems. Spijker provided comprehensive review of different stiffness in [1]. He also shows theoretical analysis of numerical methods in stiff situation. Darvishi et al. performed comparative analysis for stiff systems using variation iteration method (VIM) and Adomian decomposition method (ADM) [2]. Abelman and Patidar attempted combustion equation which falls in stiff family of differential equations using finite difference methods [3].Classical numerical techniques, such as finite difference and finite element methods, are computationally expensive and usually affected by round-off errors, which may lead to inaccurate results. Analytical methods are widely used for solving functional equations which are very restrictive for stiff systems and can be used only in special cases [4, 5]. In 2013, Arqub et al. proposed a method based on Taylor series for the solution of functional equations and systems [6]. This method overcomes the limitation of pure numerical schemes to some extent as it does not require linearization, perturbation, or discretization. Also, it has less computational cost and does not effected by round-off errors. In this method, the solution is considered as the sum of an infinite series, which converges rapidly to the exact solution. This technique is used by many authors for solving linear and nonlinear DEs and systems. Arqub used RPSM to Lane Emden equation (LEE) in [7]. Al-Smadi used this method to different linear and nonlinear IVPs [8]. Mahmood and Yousaf apply the RPSM to Boussinesq–Burgers equations [9]. Kumar and Kumar applied this method to fractional burger type equations in [10]. Kumar et al. also extended this method to fractonal diffusion equation (16). Alquran analyzed fractional form of drainange equation using RPSM [11]. Arqub also altered RPSM to fuzzy DEs [12]. Yao and Kumar applied RPSM to fractional Fokker–Planck equations (21). Kumar and Baleanu apply this scheme to fractional coupled Boussinesq–Burgers equations [13]. El-Ajou et al. used RPSM to nonlinear fractional KdV-Burgers equation [14]. Arqub et al. also apply this method to solve the higher order IVPs [6]. Komashynska et al. used this method to solve the system of multi-panto-graph differential equations in [15]. Koashynska et al. also analyzed Fredhlom integral equation with the same method [16].Analysis of stiff systems with traditional numerical techniques requires more time and effort; still accurate solution is uncertain in most of the cases due to stability issues. To overcome these issues, RPSM is proposed for stiff systems in this manuscript. PRSM is tested against various linear and nonlinear stiff systems to check the efficiency and effectiveness of proposed scheme. In the rest of the manuscript, the proposed approach is presented in Section 2, which is then applied to various linear and nonlinear stiff systems in Section 3, while the conclusion is in Section 4.

2. Residual Power Series Method for Stiff Systems

In this section, RPSM is explained for the -order system of IVPs. Consider the following system:with initial conditionswhere and are nonlinear continuous functions of and are unknown functions to be determined. and are real finite constants with . It is assumed that and are analytical functions on the given interval.

Let us assume solution of the formwhere are terms of approximations and given as = . When , satisfy the initial conditions (2), where are the initial guesses of , we have , .

If we choose as initial guesses approximations of , then we can calculate for and approximate the solutions of system of IVPs (1) by the truncated series

Rewrite the system in the following form

Using (4) in (5), the residual function isand the residual function is

This show that is infinitely many times differentiable at .which is iterative process of obtaining higher order solution.

3. Application of RPSM to Stiff Systems

3.1. Test Problem 1

Consider the following linear stiff system [17]:with initial conditions

Exact solutions of the problem is

Let we suppose . Also, the approximate solution for the given problem is

Using initial condition with in (13) and (14) gives .

Rewrite (9) as

Now, plugging the truncated series ((13)–(14)) in (15) and (16), we obtainand

To get approximate solution, use (17) and (18) along with (8) with . This leads to , . Proceeding in a similar pattern, RPS solution for the given problem iswhich is closed-form solution.

3.2. Test Problem 2

Consider the following nonlinear stiff system [17]:with initial conditions

Exact solutions of the given system are

Let us suppose . Also, suppose that approximate solution for the given system is

The values of and can be computed from (26) and (27) by taking along with initial conditions.

Rewrite (22) and (23) as follows:

Plugging the truncated series (26), (27) in (28) and (29), we get

To obtain approximate solution put in (30) and (31), then use these residual functions in (8) will give and . Proceeding in this way, the higher order coefficients can be obtained, and the RPS solution iswhich is closed-form (exact) solution.

3.3. Test Problem 3

Consider the following linear stiff system [18]:with initial conditions

Exact solutions of the problem are

Approximate solution for the given problem is

The values of and can be computed from (39) and (40) by taking along with initial conditions.

Rewriting the given (35) and (36) in the following form:and plugging the truncated series (39) and (40) in (41) and (42) to get the following residual functionsand

To obtain approximate solution put in (43) and (44), then use these residual functions in (8) will give and . Proceeding in this way, we can obtain higher order coefficients, and RPS solution iswhich is closed-form solution.

3.4. Test Problem 4

Consider the following nonlinear stiff system [18]:with initial conditions

Exact solutions of the given problem is

Approximate solution for the system is

The values of and can be computed from (52) and (53) by taking along with initial conditions.

Rewriting the given system in the following form:and plugging the truncated series (52) and (53) in (54) and (55) to obtain the following residual functionsand

To obtain approximate solution put in (56) and (57), we use these residual functions in (8) leading to and . Proceeding in this way, higher order coefficients can easily be found, and the RPS solution iswhich is closed-form solution.

3.5. Test Problem 5

Consider the following system [19]:with initial conditions

Exact solutions of the problem are

Approximate solution for the given problem is

The values of , , and can be computed from (66) by taking along with initial conditions.

Rewriting the given IVP in the following form:

Plugging the truncated series (66) in (67), (68), and (69) to obtain the following residual functionsand

For approximate solution, put in (70)–(72), we use these functions in (8) to get the following coefficients:

Repeat the recursive process to obtain higher order coefficients. Hence, RPS solution iswhich is closed-form solution.

3.6. Test Problem 6

Consider the following nonlinear system [19]:with initial conditions

Exact solutions of the given problem is

Approximate solution for the given problem is

The values of , , and can be computed from (83) by taking along with initial conditions.

Rewriting the given system in the following form:

Plugging the truncated series (83) in (84), (85), and (86) to obtain the following residual functions:and

For approximate solution, put in (87)–(89), we use these functions in (8) to get the following coefficients:

Repeat the process to get higher order coefficients of the series solution. Hence, RPS solution iswhich is closed-form solution.

4. Conclusion

In this manuscript, the residual power series method is applied to various linear and nonlinear stiff systems of ODEs and closed-form solutions are obtained. These solutions shows that RPSM is simple and easy to use technique for stiff systems and can effectively be used for wide variety of problems in science and engineering.

Data Availability

All the relevant data are available within the manuscript.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright

Copyright © 2022 Mubashir Qayyum and Qursam Fatima. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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