Comparative study of the improved Euler’s method and fadugba-falodun scheme for the solution of second order ordinary differential equation

1. Introduction

Many scientific and technological problems in the field of sciences and technology are mathematically modeled using differential equations. For instance, in physics, phenomena such as heat flow and wave propagation are well-defined by partial differential equations. Interestingly, the differential equations arising from the modeling of physical phenomena often lack exact solutions. Therefore, the utilization of numerical methods to obtain approximate solutions becomes necessary. Numerical solutions for ordinary differential equations hold great significance in scientific computation as they are widely used to model real-life problems.

In the pursuit of obtaining solutions for initial value problems in ordinary differential equations, extensive work and effort have been dedicated to modifying existing numerical methods or developing new ones with improved accuracy, stability, and consistency properties. Numerous authors have enhanced the well-known explicit Runge-Kutta method, which is widely used by numerical analysts, by reducing the number of slope evaluations required per integration step in their modified techniques/methods. These modifications require fewer slope evaluations than the original Runge-Kutta methods. Many numerical methods have been developed for solving initial value problems in ordinary differential equations in the literature, including references [1-9], to name just a few. The aim of this paper is to compare two numerical methods, namely the Improved Euler Method (IEM) and the Fadugba-Falodun Scheme (FFS), for solving the initial value problem of second-order ordinary differential equations in the context of the exact solution.

The rest of the paper is organized as follows: In Section 2, we present an overview of FFS and IEM. Section 3 provides the algorithms for these two methods used to solve the initial value problem of second-order ordinary differential equations. Section 4 includes an illustrative example, and Section 5 serves as the conclusion of the paper.

2. Overview of the Methods

The overview of the two methods under consideration is presented as follows:

2.1. Fadugba-Falodun Scheme

Consider the interpolating function given by \[\label{GrindEQ__1_} y\left(x\right)=\sum^1_{i=0}{{\propto }_ix^i{+\ \propto }_2e^{-x}}\,. \tag{1}\] Equation (1) consists of polynomial and exponential functions, where \(\varpropto^{'}_is\ \left(i=0\left(1\right)2\right)\ \mathrm{are}\ \)constants. The integration interval is defined as; \[\label{GrindEQ__2_} {a=x}_0\ <x_1\dots <\ x_{n+1}\dots <x_{\mathrm{N}}=b\,, \tag{2}\] with the step length \(h\) given by \[\label{GrindEQ__3_} h=\ x_{n+1}-x_n\ or\ \frac{b-a}{N}\,, \tag{3}\] Such that n = 0, 1, 2,…, N-1. The mesh points \[\label{4a} x_n=a+nh,\quad\quad n=1,\ 2,\ 3,\dots\,, \tag{4}\] or \[\label{4b} x_{n+1}=a+\left(n+1\right)h,\quad\quad n=0,\ 1,\ 2,\dots \tag{5}\] Evaluating (1) at the points \(\ {x=x}_n\ \mathrm{and\ }{x=x}_{n+1}\) yields, \[\label{GrindEQ__5_} {F(x}_n)=\ {\propto \ }_0+\ {\propto \ }_1{x\ }_n+{\propto \ }_2e^{{-x}_n}\,, \tag{6}\] and \[\label{GrindEQ__6_} {F(x}_{n+1})=\ {\propto \ }_0+\ {\propto \ }_1{x\ }_{n+1}+{\propto \ }_2e^{{-x}_n+1}\,, \tag{7}\] respectively. From Equation (6), the following derivatives were obtained, \[\begin{aligned} \label{GrindEQ__7_} F^{\mathrm{\textrm{'}}}(x_n)&=\ {\propto \ }_1-\ {\propto \ }_2e^{-x_n}=\ f_n\,,\\ \label{GrindEQ__8_} F^{\mathrm{\textrm{'}}\mathrm{\textrm{'}}}(x_n)&=\ {\propto \ }_2e^{-x_n}=\ {f_n}^{(1)}\,. \end{aligned} \tag{8,9}\] Solving (8) and (9) further yields \[\label{GrindEQ__9_} {\propto \ }_1=\ f_n+{f_n}^{(1)}\,, \tag{10}\] and \[\label{GrindEQ__10_} {\propto \ }_2=\ \frac{{f_n}^{(1)}}{e^{-(a+nh)}}\,. \tag{11}\] Subtracting (6) from (7) and using (10) and (11) with \(a = 0,\) one obtains \[\label{GrindEQ__11_} {f(x}_{n+1})-{f(x}_n)=hf_n+(h+(e^{-h}-1){f_n}^{(1)}\,. \tag{12}\] Using the fact that \[\label{GrindEQ__12_} y_{n+1}-y_n\equiv {f(x}_{n+1})-{f(x}_n)\,, \tag{13}\] the Equation (12) becomes \[\label{GrindEQ__13_} y_{n+1}-y_n={hf}_n+[h+\left(e^{-h}-1\right)]{f_n}^{(1)}={hf}_n+\left[h+e^{-h}-1\right]{f_n}^{\left(1\right)},\quad\quad n=0,\ 1,\ 2,\ \dots ,\ N-1\,. \tag{14}\] Equation (14) is called “Fadugba – Falodun Scheme”.

Remark 1. [9]

1. This method is a good candidate to be included in the family of linear explicit numerical methods of Runge-Kutta type.

2. FFS given by (14) has the convergence of second order accuracy.

3. Equation (14) is found to be conditionally stable with the region of linear stability.

4. The region of absolute stability for the numerical method (14) is defined by the region in the complex plane such that \(\left|1+z+\frac{z^{2} }{2} \right|<1\). The stability region is plotted in the Figure 1 below;

2.2. Improved Euler method

It is a known fact that the Euler’s method is not very ideal for a realistic computation. It is clear that a large number of steps are required to achieve a certain degree of accuracy. The derivation of the Improved Euler method is summarized as follows: Let us suppose that y\({}_{1}\) has been computed from the Euler’s method as, \[\label{14a} y_{1} \, \, =\, \, y_{0} \, \, +\, \, h\, y'\, (x_{0} )\, \, =\, \, y_{0} \, \, +\, \, h\, f\, \, (x_{0} ,\, \, y_{0} )\,, \tag{15}\] and let it be denoted as, \[\label{14b} y_{1}^{*} \, \, =\, \, y_{0} \, \, +\, \, h\, f\, \, (x_{0} ,\, \, y_{0} )\,. \tag{16}\] We compute \(y'(x)\, \, =\, \, \frac{dy}{dx} \, \, =\, \, f\, \, (x,\, \, y)\) at x = x\({}_{1}\), where \(y\, (x_{1} )\, \, =\, \, y_{1}^{*}\), i.e., \(y'(x_{1} )\, \, =\, \, \frac{dy}{dx} \, \, =\, \, f\, \, (x_{1} ,\, \, y_{1} )\) . The value of y\({}_{1}\) is then improved in the Euler’s formula by taking the average value of \(\frac{dy}{dx}\) for \((x_{0} ,\, \, y_{0} )\) and \((x_{1} ,\, \, y_{1} )\). Then the improved Euler’s may be written as,

\[\begin{aligned} \label{14c}y_1=y_0+\frac{h}{2}(k_1+k_2)\,, \end{aligned} \tag{17}\] where \[\begin{aligned} \label{14d}k_1=f(x_0,\ y_0),\quad k_2=f(x_0+h,y_0+hk_1)\,. \end{aligned} \tag{18}\] Continuing this way, the improved Euler method for the solution of Initial value problems in ordinary differential equation is given by \[\label{GrindEQ__15_} y_{n+1}=y_n+\frac{h}{2}(k_1+k_2)\,, \tag{19}\] where \[\label{GrindEQ__16_} k_1=f(x_n,\ y_n)\quad \text{and}\quad k_2=f(x_n+h,y_n+hk_1),\quad\quad n = 0, 1, 2, {\dots}, N-1. \tag{20}\]

3. Algorithms of the methods

This section presents the algorithms of the two methods to solve second order initial value problems in ordinary differential equations as follows:

3.1. Algorithm of the Fadugba-Falodun Scheme

Given the initial value problem of the form \[\label{GrindEQ__17_} \begin{cases} a\left(x\right)y^{''}+b\left(x\right)y^{'}+c\left(x\right)y=q(x),\\ y\left(x_0\right)={y_0,\ \ y}^{'}\left(x_0\right)=z_0\,.\end{cases} \tag{21}\] Equation (21) can be reduced to its first order equivalent as follows, \[\label{GrindEQ__18_} y^{'}=z,\ \ z^{'}=G\left(x,y,z\right),\ \ y\left(x_0\right)=y_0,\ \ z\left(x_0\right)=z_0\ \ \tag{22}\] where

\(G\left(x,y,z\right)=\frac{q\left(x\right)-C\left(x\right)y-b\left(x\right)z}{a(x)}.\)

An approximate solution to the initial value problem (22) at the equally space point \(x_0,x_1,\dots ,x_n\) is given by \[\label{GrindEQ__20_} y_{n+1}=y_n+hz_n+(h+(e^{\mathrm{-h}}-1){z_n}^{'}\,, \tag{23}\] or \[\label{21a} y_{n+1}=y_n+hf_n+(h+(e^{\mathrm{-h}}-1)){f_n}^{(\mathrm{1})}\,, \tag{24}\] with \[ f_n=z_n,\tag{25}\] \[ {f_n}^{(1)}={z_n}^{'}\tag{26}\] \[z_{n+1}=z_n+hG_n+(h+(e^{\mathrm{-h}}-1)){G_n}^{(1)},\tag{27} \] with \[G_n=G\left(x_n,y_n,z_n\right)={z_n}^{'}\] and \[{z_n}^{''}={G_n}^{(1)}=G^{(1)}\left(x_n,y_n,z_n\right) \] \[\label{GrindEQ__23_} x_{n+1}=x_0+\left(n+1\right)h \tag{28}\] or \[\label{GrindEQ__24_} x_{n+1}=x_n+h,\ \ n=0,\ 1,\ 2,\dots ,\ N-1 \tag{29}\] where \[h,\ x_o,\ y_o\ \mathrm{and}\ z_o\mathrm{\ are\ given}\ .\]

Algorithm of the improved Euler method

Consider the initial value problem of second order ordinary differential equation given by (21). The approximate solution to (21) via the improved Euler method at the equally spaced point \(x_0,\ x_1,\ \dots ,\ x_n\ \mathrm{is\ given\ by}\) \(y_{n+1}=y_n+\frac{h}{2}(k_1+k_2)\), where \[\label{GrindEQ__26_} \ k_1=\ z_n,\ \ k_2=\ z_n+h{G(x}_n,y_n{,\ z}_n)\ \tag{30}\] \[\label{GrindEQ__27_} z_{n+1}=z_n+\frac{h}{2}\left[{G(x}_n,y_n{,\ z}_n\right)+{G(x}_n+h,y_n{,\ z}_n)] \tag{31}\] \[\label{GrindEQ__28_} x_{n+1}=x_n+h,\ n=0,1,\dots ,N-1 \tag{32}\] where \({h,x}_0,y_0\ \mathrm{and}{\ z}_0\ \mathrm{are\ given}.\)

4. Illustrative Example

In this section, one illustrative example is presented to discuss the efficiency and accuracy of the two methods. To do this, all the necessary computations have been carried out via MATLAB Version (R 2014a). The discussion of results is also presented.

Consider a second order initial value problem of the form \[\label{GrindEQ__29_} y^{''}+4y=2.5sin3x, {y\left(0\right)=y}^{'}\left(0\right)=0 \tag{33}\] At \(x_n=nh,\ n=0,\ 1,\ 2,\ \dots ,\ 10,\ \mathrm{with\ step\ length}\ \ \)

\(h=0.1.\ \ \)The exact solution of (33) is given by \[\label{GrindEQ__30_} y\left(x\right)=\frac{3}{4}{\mathrm{sin} \left(2x\right)\ }-\frac{1}{2}\mathrm{sin}\mathrm{}(3x) \tag{34}\] The comparative analysis of the results generated via FFS and IEM in the context of the Exact Solution (ES) is shown in Table 1. The absolute errors generated/incurred via the methods at each node of the computation were shown in Table 2.

5. Conclusion

In this paper, we have compared the Fadugba-Falodun Scheme (FFS) with the well-known existing method, denoted as “IEM,” for solving the initial value problem of second-order ordinary differential equations. To assess the accuracy and performance of these methods in the context of the exact solution, we have solved an illustrative example. Absolute errors at each nodal point were computed using MATLAB (R2014a). The results presented in Table 1 indicate that the results obtained with FFS closely align with those of the exact solution. Furthermore, the table demonstrates a perfect agreement between FFS and the exact solution (ES). In Table 1, it is noticeable that there is no significant difference between the results obtained with FFS and ES, in contrast to the performance of IEM. These results suggest that FFS is a promising alternative method for solving initial value problems of second-order ordinary differential equations. In future studies, the method can be further explored for solving higher-order ordinary differential equations, particularly those with singularity points.