An iterative method (AIM) is one of the numerical method, which is easy to apply and very time convenient for solving nonlinear differential equations. However, if we want to work in a large interval, sometimes it may be difficult to apply AIM. Therefore, a multistage AIM named Multistage Modified Iterative Method (MMIM) is introduced in this article to work in a large computational interval. The applicability of MMIM for increasing the solution domain of the given problems is construed in this article. Some problems are solved numerically using MMIM, which provides a better result in the extended interval as compared to AIM. Comparison tables and some graphs are included to demonstrate the results.

#### Keywords

- an iterative method
- multistage modified iterative method
- nonlinear integral equations
- ordinary differential equations

#### MSC 2010

- 34K05

Nonlinear differential and integral equations have been a key in the various fields of Engineering, Physics, Life Sciences, and Mathematics. There is no general method to solve the nonlinear differential equations. Also, it is not possible to convert a nonlinear differential equation into an appropriate form using variable transformations and sometimes not possible to get the close form analytical solution of the nonlinear differential equations. George Adomian developed a numerical method from 1970 to 1990. The important feature of the method is the deputation of ‘Adomian polynomials’, which acknowledge the approximation of nonlinear part without using perturbation and linearisation techniques. In articles [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], a number of renowned nonlinear problems have been solved by some other efficient numerical methods.

However, numerical and analytical methods are not limited in the fields of ODE(s) and PDE(s), but may have a crucial role in the field of nuclear magnetic resonance, astrophysics, cold plasma, porous media, and physics through fractional differential equations [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23]. In spite of these abovementioned methods, AIM [24] is getting a lot of attention in getting the solution for nonlinear differential equations. However, this method works for a small interval only. But to find a numerical solution for a large interval using AIM, we have to use large number of components. So, in most of the problems, after some iterations, either mathematical software declines to give the solution for a given problem or the iterations become so much complicated to move to successive steps of the solution. In Multistage Adomian Method [25], a comparatively large interval is split into a number of small subintervals and these small subintervals are used to find the solution.

The objective of this article is to solve a nonlinear problem for a large interval using AIM. In the present work, we introduce MMIM, developed by using AIM in each of subinterval separately. Therefore, MMIM is helpful to find the solution in a comparatively large interval in comparison to AIM.

The procedure given in this article can be extended to solve the various problems, like Riccati differential equation, Abel's differential equation, integral equations, partial differential equation and fractional differential equations.

In Section 2, the outline of AIM is provided. In Section 3, two-step modified iterative method is discussed. In Section 4, an analysis of the two-step modified iterative method has been discussed. MMIM is explained in Section 5. An absolute error formula is discussed in Section 6. Finally, some examples are solved to establish the competency of MMIM, in Section 7.

Consider a general functional equation:

The nonlinear terms can be decomposed as

This implies

Therefore, using Eq. (6) in Eq. (1),

If

Therefore using Banach fixed point theorem [26] a solution for Eq. (1) is obtained.

Consider, the general functional Eq. (1). In TSMIM, source term _{1} and _{2} (_{1} and _{2} may contain more than one term). Under this assumption, we set
_{0} and _{1}, a modified recursive scheme defined as follows,

When we analyse Eqs (5) and (9), it is observed that zeroth component _{0} is defined by the function _{1} as a part of _{2} is added to the definition of the component _{1} in Eq. (9). This change in _{0} may result as a lection in the computational work and accelerate the convergence of the solution. This slight variation in the definition of the components _{0} and _{1} may provide the solution in only two iterations.

It should be noted that the success of this method depends on proper choice of the part _{1} or _{2}, but it cannot be said that which part of _{1} or _{2}, would be appropriate to take as a zeroth component.

For the sake of convenience, let us consider Eq. (1)

In the study of TSMIM, the source term has been split into two parts _{1} and _{2}. Now, modified Eq. (1) can be written as

Using the recursive scheme Eq. (9) and choosing _{0} such that _{0} = _{1} =

If zeroth component does not satisfy the given problem, then the method discussed in article [24] can be used.

In most of the problems, the computational interval _{i}^{th}_{i}_{i+1} and for

Therefore, to solve the problem of type Eq. (1), initial guess of the solution should be known in the interval and the solution can be considered as
_{1}(_{1}). This first subinterval will be used to obtain an initial condition at _{1}, i.e., _{1}) = _{2} for the solution _{2}(_{1}, _{2}). Similarly the initial condition for the third interval at _{2} is _{2}) = _{3} and get _{3}(_{i}_{i−1}, _{i}_{i}

The length of the subinterval should be small to get the higher accuracy.

It is not necessary that the length of each subinterval to be equal, it can also be changed to achieve better results.

Since MMIM is a modification of AIM, and in each subinterval AIM will be used separately, therefore the absolute error formula for the solution _{i}_{i−1}, _{i}

In this section, some examples have been included for the verification of the MMIM. AIM and MMIM are applied separately to find the solution in a given interval. Further, solutions of the both methods are compared graphically and the comparative study is summarised in Tables 1 and 2, respectively.

Comparative study of AIM and MMIM

0.4 | 0.16 | 3.49×10^{−9} |
3.496×10^{−9} |

0.8 | 0.64 | 5.833×10^{−5} |
5.833×10^{−5} |

1.2 | 1.44 | 1.761×10^{−2} |
1.671×10^{−5} |

1.6 | 2.56 | 0.989 | 5.644×10^{−4} |

2.0 | 4.0 | 5.71 | 1.009×10^{−3} |

Comparative study of AIM and MMIM

0.4 | 1.4 | 2.878×10^{−3} |
4.475×10^{−6} |

0.8 | 1.8 | 0.147 | 1.260×10^{−5} |

1.2 | 2.2 | 1.203 | 2.865×10^{−5} |

1.6 | 2.6 | 3.393 | 5.664×10^{−5} |

2.0 | 3.0 | 5.401 | 1.012×10^{−4} |

^{2} in [0, 2].

Therefore, we can apply TSMIM.

Let the solution of Eq. (14) exists in a converging series form,

Hence, the required solution would be

It should be noted that, if we take the 0th component as the solution of the given problem, the next component becomes zero. But if we do not take 0th component as the solution of the given problem, we shall apply AIM.

The graph of the solution Eq. (15) at ^{2} are shown in Figure 1.

Figure 1 shows a comparison between exact solution ‘_ _’ and computed solution ‘__’.

It can be illustrated from Figure 1 that the computed solution coincide with the exact solution in the interval [0, 1]. The behaviour of the exact solution changes rapidly for

First, we find solution in the interval [0, 1) by taking sum of the five components, _{1}(_{10} + _{11} + _{12} + _{13} + _{14}, where

The graph of the solution _{1}(_{10} + _{11} + _{12} + _{13} + _{14} in the interval [0, 1) is shown in Figure 2 The end point of the first interval [0, 1) will work as the initial point for the second interval [1, 1.1), such that

Therefore, solution in the interval [1, 1.1) is given by,

The graph of the solution _{2}(_{20} + _{21} + _{22} + _{23} + _{24} for the interval is shown in Figure 3.

Take end point of the interval [1, 1.1) as the initial point for the interval [1.1, 1.2), such that

Since, solution for the interval [1.1, 1.2) will be given by,

The graph of the solution _{3} = _{30} + _{31} + _{32} + _{33} + _{34} for the interval [1.1, 1.2) is shown in Figure 4.

Take end point of the interval [1.1, 1.2), as the initial point for the interval [1.2, 1.3), such that

Then, the solution in the interval [1.2, 1.3) will be given by,

Therefore, the graph of the solution _{4} = _{40} + _{41} + _{42} + _{43} + _{44} for the interval [1.2, 1.3) is shown in Figure 5.

Proceeding in similar way up to interval [1.9, 2], with length of the subinterval 0.1 and after combining or merging all subintervals, we get solution for the interval [0, 2] in Figure 6, which is quite similar to the graph of exact solution, therefore for more clarity we observe Table 1. It is clear from Table 1 that the solution using MMIM is more promising than AIM.

Here, we see that the solution of the problem Eq. (17) is not inbuilt under the given problem. Therefore, we use AIM.

The graph of the computed solution using AIM,

First, we find solution for the interval [0, 0.3) by taking sum of the five components _{1} = _{10} + _{11} + _{12} + _{13} + _{14}. Using Eq. (5), the graph of the solution in the interval [0, 0.3) is shown in Figure 8.

The end point of the interval [0, 0.3) will work as the initial point for second interval [0.3, 0.4) such that

Solution for the interval [0.3, 0.4) is given by,

The graph of the solution _{2} = _{20} + _{21} + _{22} + _{23} + _{24} for the interval [0.3, 0.4) is shown in Figure 9.

In the similar manner, the solution of the given problem Eq. (16) can be calculated up to interval [1.9, 2] with length of the subinterval 0.1. Combining and merging all the solution graphs, the graph of the solution for the interval [0, 2] is shown in Figure 10. In Figure 10, which is quite similar to the graph of exact solution, therefore for more clarity we observe Table 2. It is clear from Table 1 that the solution using MMIM is more promising than AIM.

In our work, a multistage iterative method as a modification in AIM is introduced, which is known as MMIM. MMIM is the extension of AIM, although the solution given by AIM is limited for a very small interval. Therefore MMIM is useful to solve the functional equations for a comparatively large interval by integrating AIM. The numerical examples and associated graphs and tables have been defined and constructed in this article. Figures 2–5 are the graphs of the solution of Eq. (14) for the subintervals [0, 1) to [1.2, 1.3). These graphs show the behaviour of the numerical solution calculated using AIM separately in their respective subintervals and the procedure is continued up to the subinterval [1.9, 2]. Then all the solution graphs are merged and the final graph is drawn as in Figure 6 with the concept of MMIM. We can observe with the help of Figure 1 that for the interval [0, 2], MMIM converges to the exact solution more accurately than the AIM, and further the efficiency of the method is shown in Table 1.

In Table 1, a comparison between absolute error between AIM and MMIM in the computational interval [0, 2] at different points are shown when used them separately. Therefore with the help of solution graphs and table of Eq. (14), we can state that MMIM is more accurate than AIM.

Similar results are followed in all the figures and table in Example 2. Therefore, MMIM is simple and easy to apply, requires less computational complexity and also provides more reliable results.

#### Comparative study of AIM and MMIM

0.4 | 1.4 | 2.878×10^{−3} |
4.475×10^{−6} |

0.8 | 1.8 | 0.147 | 1.260×10^{−5} |

1.2 | 2.2 | 1.203 | 2.865×10^{−5} |

1.6 | 2.6 | 3.393 | 5.664×10^{−5} |

2.0 | 3.0 | 5.401 | 1.012×10^{−4} |

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