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- Open Access

# The new modified Ishikawa iteration method for the approximate solution of different types of differential equations

- Necdet Bildik
^{1}, - Yasemin Bakır
^{1}and - Ali Mutlu
^{1}Email author

**2013**:52

https://doi.org/10.1186/1687-1812-2013-52

© Bildik et al.; licensee Springer 2013

**Received:**1 September 2012**Accepted:**16 February 2013**Published:**12 March 2013

## Abstract

In this article, the new Ishikawa iteration method is presented to find the approximate solution of an ordinary differential equation with an initial condition. Additionally, some numerical examples with initial conditions are given to show the properties of the iteration method. Furthermore, the results of absolute errors are compared with Euler, Runge-Kutta and Picard iteration methods. Finally, the present method, namely the new modified Ishikawa iteration method, is seen to be very effective and efficient in solving different type of the problem.

**MSC:**65K15, 65L07, 65L06, 65L70.

## Keywords

- ordinary differential equation
- Euler method
- fixed point
- numerical analysis
- modified Ishikawa iteration
- Picard successive iteration method

## Introduction

Various kinds of numerical methods, especially iterative methods [1–3], were used to solve different types of differential equations. In recent years, there has been a growing interest in the treatment of iterative approximation of fixed point theory on normed linear spaces [4–13], Banach spaces [14–18] and Hilbert spaces [19, 20], respectively.

A new modified Ishikawa iteration method has been developed to find an approximate solution for different types of differential equations with initial conditions [13, 15–29] on metric spaces. The solutions are also obtained in terms of the Picard iteration. Also in Section 2, examples of these kinds of equations are solved using this new method which is called new modified Ishikawa iteration method and also the results are discussed and comparison using Euler [1, 2], Runge-Kutta [30, 31], and Picard iteration methods [1, 2, 27] is presented in tables and figures. Additionally, this approximated method can solve various different differential equations such as integral, difference, integro-differential and functional differential equations. Finally, numerical results shows that the new modified Ishikawa iteration method is more or less effective and also convenient for solving different types of differential equations.

Now, let us give some of the important theorems and definitions in order to solve linear and nonlinear differential equations using a contraction mapping.

## 1 Preliminaries

**Theorem 1.1** (Banach contraction principle)

*Let*$(X,d)$

*be a complete metric space and*$T:X\to X$

*be a contraction with the Lipschitzian constant*

*L*.

*Then*

*T*

*has a unique fixed point*$u\in X$.

*Furthermore*,

*for any*$x\in X$

*we have*

*with*

(*see* [13]).

*Proof*We first show uniqueness. Suppose there exist $x,y\in X$ with $T(x)=x$ and $T(y)=y$. Then

therefore $d(x,y)=0$.

*X*is complete, there exists $u\in X$ with ${lim}_{n\to \mathrm{\infty}}{T}^{n}(x)=u$. Moreover, the continuity of

*T*yields

*u*is a fixed point of

*T*. Finally, letting $m\to \mathrm{\infty}$ in (1.1) yields

□

**Corollary 1.2** *Let* $(X,d)$ *be a complete metric space and let* $B({x}_{0},r)=\{x\in X:d(x,{x}_{0})<r\}$, *where* ${x}_{0}\in X$ *and* $r>0$. *Suppose* $T:B({x}_{0},r)\to X$ *is a contraction* (*that is*, $d(T(x),T(y))\le Ld(x,y)$ *for all* $x,y\in B({x}_{0},r)$ *with* $0\le L<1$) *with* $d(T({x}_{0}),{x}_{0})<(1-L)r$. *Then* *T* *has a unique fixed point in* $B({x}_{0},r)$ (*see* [13]).

**Definition 1.3** If the sequence ${\{{x}_{n}\}}_{n=0}^{\mathrm{\infty}}$ provides the condition ${x}_{n+1}=T{x}_{n}$ for $n=0,1,2,\dots $ , then this is called the Picard iteration [1, 27].

**Definition 1.4**Let ${x}_{0}\in X$ be arbitrary. If the ${\{{x}_{n}\}}_{n=0}^{\mathrm{\infty}}$ sequence provides the condition

- (i)
$0\le {\alpha}_{n}\le {\beta}_{n}<1$, for all positive integers

*n*, - (ii)
${lim}_{n\to \mathrm{\infty}}{\beta}_{n}=0$,

- (iii)
${\sum}_{n\ge 0}{\alpha}_{n}{\beta}_{n}=\mathrm{\infty}$.

**Definition 1.5**If $\lambda \in [0,1]$, $\gamma \in [0,1]$ and ${y}_{0}\in X$,

*T*is defined as a contraction mapping with regard to Picard iteration and also the ${\{{y}_{n}\}}_{n=0}^{\mathrm{\infty}}$ sequence provides the conditions

then this is called a new modified Ishikawa iteration where $T={\int}_{{x}_{0}}^{x}F(t,{y}_{n}(t))\phantom{\rule{0.2em}{0ex}}dt$.

In order to illustrate the performance of the new modified Ishikawa iteration method in solving linear and nonlinear differential equations and justify the accuracy and efficiency of the method presented in this paper, we consider the following examples. In all examples, we used four types of iteration methods and the comparison is shown in figures and tables respectively.

## 2 Application of methods

**Example 2.1** Let us consider the differential equation ${y}^{\prime}=\sqrt{|y|}$ subject to the initial condition $y(0)=1$.

Firstly, we obtained the exact solution of the equation as $|y|=\frac{1}{4}{(x+2)}^{2}=1+x+\frac{{x}^{2}}{4}$.

So, $|T(x)-T(y)|\le \frac{2}{3}|x-y|$ is found. Thus *T* has a unique fixed point, which is the unique solution of the integral equation $T={\int}_{{x}_{0}}^{x}F(t,{y}_{n}(t))\phantom{\rule{0.2em}{0ex}}dt$ or the differential equation ${y}^{\prime}=\sqrt{|y|}$, $y(0)=1$.

are obtained.

Thus, ${y}_{3}=1.689999654$ is obtained for ${x}_{3}=0.6$.

*λ*and

*γ*. Now we may give Table 5 which is expressed that absolute error of Example 2.1 for different values of

*λ*and

*γ*with $x=0.2$, $x=0.4$ and $x=0.6$ respectively.

**The solutions obtained by the new modified Ishikawa iteration method for different values of**
λ
**and**
γ

x | λ = 0.5, γ = 0.5 | λ = 0.5, γ = 0.25 | λ = 0.25, γ = 0.5 | λ = 0.25, γ = 0.25 |
---|---|---|---|---|

| $\begin{array}{l}{y}_{1}=1.2\\ {y}_{2}=1.1\\ {y}_{3}=1.15\\ {y}_{4}=1.125\\ {y}_{5}=1.1375\\ {y}_{6}=1.13125\\ {y}_{7}=1.134375\\ {y}_{8}=1.1328125\\ {y}_{9}=1.13359375\\ {y}_{10}=1.133203125\\ {y}_{11}=1.13398437\end{array}$ | $\begin{array}{l}{y}_{1}=1.2\\ {y}_{2}=1.05\\ {y}_{3}=1.125\\ {y}_{4}=1.06875\\ {y}_{5}=1.096875\\ {y}_{6}=1.07578125\\ {y}_{7}=1.086328125\\ {y}_{8}=1.078417968\\ {y}_{9}=1.082373046\\ {y}_{10}=1.079406738\\ {y}_{11}=1.080889892\end{array}$ | $\begin{array}{l}{y}_{1}=1.2\\ {y}_{2}=1.1\\ {y}_{3}=1.125\\ {y}_{4}=1.1125\\ {y}_{5}=1.115625\\ {y}_{6}=1.1140625\\ {y}_{7}=1.114453125\\ {y}_{8}=1.114257812\\ {y}_{9}=1.11430664\\ {y}_{10}=1.114282226\\ {y}_{11}=1.11428833\end{array}$ | $\begin{array}{l}{y}_{1}=1.2\\ {y}_{2}=1.05\\ {y}_{3}=1.0875\\ {y}_{4}=1.059375\\ {y}_{5}=1.06640625\\ {y}_{6}=1.06113281\\ {y}_{7}=1.062451171\\ {y}_{8}=1.061462402\\ {y}_{9}=1.061709594\\ {y}_{10}=1.0615242\\ {y}_{11}=1.061570548\end{array}$ |

| $\begin{array}{l}{y}_{1}=1.4\\ {y}_{2}=1.2\\ {y}_{3}=1.3\\ {y}_{4}=1.25\\ {y}_{5}=1.275\\ {y}_{6}=1.2625\\ {y}_{7}=1.26875\\ {y}_{8}=1.265625\\ {y}_{9}=1.2671875\\ {y}_{10}=1.26640625\\ {y}_{11}=1.266796874\end{array}$ | $\begin{array}{l}{y}_{1}=1.4\\ {y}_{2}=1.1\\ {y}_{3}=1.25\\ {y}_{4}=1.1375\\ {y}_{5}=1.19375\\ {y}_{6}=1.1515625\\ {y}_{7}=1.17265625\\ {y}_{8}=1.156835937\\ {y}_{9}=1.164746093\\ {y}_{10}=1.158813476\\ {y}_{11}=1.161779784\end{array}$ | $\begin{array}{l}{y}_{1}=1.4\\ {y}_{2}=1.2\\ {y}_{3}=1.25\\ {y}_{4}=1.225\\ {y}_{5}=1.23125\\ {y}_{6}=1.228125\\ {y}_{7}=1.22890625\\ {y}_{8}=1.228515624\\ {y}_{9}=1.228613281\\ {y}_{10}=1.228564452\\ {y}_{11}=1.22857666\end{array}$ | $\begin{array}{l}{y}_{1}=1.4\\ {y}_{2}=1.1\\ {y}_{3}=1.175\\ {y}_{4}=1.11875\\ {y}_{5}=1.1328125\\ {y}_{6}=1.122265624\\ {y}_{7}=1.124902343\\ {y}_{8}=1.122924804\\ {y}_{9}=1.123419189\\ {y}_{10}=1.1230484\\ {y}_{11}=1.123141097\end{array}$ |

| $\begin{array}{l}{y}_{1}=1.5\\ {y}_{2}=1.25\\ {y}_{3}=1.375\\ {y}_{4}=1.3125\\ {y}_{5}=1.34375\\ {y}_{6}=1.328125\\ {y}_{7}=1.3359375\\ {y}_{8}=1.33203125\\ {y}_{9}=1.333984375\\ {y}_{10}=1.333007812\\ {y}_{11}=1.333496093\end{array}$ | $\begin{array}{l}{y}_{1}=1.5\\ {y}_{2}=1.125\\ {y}_{3}=1.3125\\ {y}_{4}=1.171875\\ {y}_{5}=1.2421875\\ {y}_{6}=1.189453125\\ {y}_{7}=1.215820312\\ {y}_{8}=1.196044921\\ {y}_{9}=1.205932617\\ {y}_{10}=1.198516845\\ {y}_{11}=1.202224731\end{array}$ | $\begin{array}{l}{y}_{1}=1.5\\ {y}_{2}=1.25\\ {y}_{3}=1.3125\\ {y}_{4}=1.28125\\ {y}_{5}=1.2890625\\ {y}_{6}=1.28515625\\ {y}_{7}=1.286132812\\ {y}_{8}=1.285644531\\ {y}_{9}=1.285766601\\ {y}_{10}=1.285705566\\ {y}_{11}=1.285720825\end{array}$ | $\begin{array}{l}{y}_{1}=1.5\\ {y}_{2}=1.125\\ {y}_{3}=1.21875\\ {y}_{4}=1.1484375\\ {y}_{5}=1.166015625\\ {y}_{6}=1.152832031\\ {y}_{7}=1.156127929\\ {y}_{8}=1.153656005\\ {y}_{9}=1.154273986\\ {y}_{10}=1.1538105\\ {y}_{11}=1.153926372\end{array}$ |

**The solutions obtained by the new modified Ishikawa iteration method for different values of**
λ
**and**
γ

x | λ = 0.75, γ = 0.25 | λ = 0.25, γ = 0.75 | λ = 0.75, γ = 0.75 |
---|---|---|---|

| $\begin{array}{l}{y}_{1}=1.2\\ {y}_{2}=1.05\\ {y}_{3}=1.1625\\ {y}_{4}=1.078125\\ {y}_{5}=1.14140625\\ {y}_{6}=1.093945312\\ {y}_{7}=1.129541015\\ {y}_{8}=1.102844238\\ {y}_{9}=1.122866821\\ {y}_{10}=1.107849883\\ {y}_{11}=1.119112586\end{array}$ | $\begin{array}{l}{y}_{1}=1.2\\ {y}_{2}=1.15\\ {y}_{3}=1.1625\\ {y}_{4}=1.159375\\ {y}_{5}=1.16015625\\ {y}_{6}=1.159960937\\ {y}_{7}=1.160009765\\ {y}_{8}=1.159997558\\ {y}_{9}=1.16000061\\ {y}_{10}=1.159999847\\ {y}_{11}=1.160000038\end{array}$ | $\begin{array}{l}{y}_{1}=1.2\\ {y}_{2}=1.15\\ {y}_{3}=1.1875\\ {y}_{4}=1.178125\\ {y}_{5}=1.18515625\\ {y}_{6}=1.183398437\\ {y}_{7}=1.184716796\\ {y}_{8}=1.184387207\\ {y}_{9}=1.184634399\\ {y}_{10}=1.184572601\\ {y}_{11}=1.18458805\end{array}$ |

| $\begin{array}{l}{y}_{1}=1.4\\ {y}_{2}=1.1\\ {y}_{3}=1.325\\ {y}_{4}=1.15625\\ {y}_{5}=1.2828125\\ {y}_{6}=1.187890624\\ {y}_{7}=1.259082031\\ {y}_{8}=1.205688476\\ {y}_{9}=1.245733642\\ {y}_{10}=1.215699767\\ {y}_{11}=1.238225173\end{array}$ | $\begin{array}{l}{y}_{1}=1.4\\ {y}_{2}=1.3\\ {y}_{3}=1.325\\ {y}_{4}=1.31875\\ {y}_{5}=1.3203125\\ {y}_{6}=1.319921874\\ {y}_{7}=1.320019531\\ {y}_{8}=1.319995116\\ {y}_{9}=1.32000122\\ {y}_{10}=1.319999694\\ {y}_{11}=1.320000076\end{array}$ | $\begin{array}{l}{y}_{1}=1.4\\ {y}_{2}=1.3\\ {y}_{3}=1.375\\ {y}_{4}=1.35625\\ {y}_{5}=1.3703125\\ {y}_{6}=1.366796874\\ {y}_{7}=1.369433593\\ {y}_{8}=1.368774414\\ {y}_{9}=1.369268798\\ {y}_{10}=1.369145202\\ {y}_{11}=1.369176101\end{array}$ |

| $\begin{array}{l}{y}_{1}=1.5\\ {y}_{2}=1.125\\ {y}_{3}=1.40625\\ {y}_{4}=1.1953125\\ {y}_{5}=1.353515625\\ {y}_{6}=1.234863281\\ {y}_{7}=1.323852539\\ {y}_{8}=1.257110595\\ {y}_{9}=1.307167053\\ {y}_{10}=1.269624709\\ {y}_{11}=1.297781467\end{array}$ | $\begin{array}{l}{y}_{1}=1.5\\ {y}_{2}=1.375\\ {y}_{3}=1.40625\\ {y}_{4}=1.3984375\\ {y}_{5}=1.400390625\\ {y}_{6}=1.399902343\\ {y}_{7}=1.400024414\\ {y}_{8}=1.399993896\\ {y}_{9}=1.400001525\\ {y}_{10}=1.399999618\\ {y}_{11}=1.400000095\end{array}$ | $\begin{array}{l}{y}_{1}=1.5\\ {y}_{2}=1.375\\ {y}_{3}=1.46875\\ {y}_{4}=1.4453125\\ {y}_{5}=1.462890625\\ {y}_{6}=1.458496093\\ {y}_{7}=1.461791992\\ {y}_{8}=1.460968017\\ {y}_{9}=1.461585998\\ {y}_{10}=1.461431503\\ {y}_{11}=1.461470126\end{array}$ |

**The solutions obtained by the new modified Ishikawa iteration method for different values of**
λ
**and**
γ

x | λ = 0.75, γ = 0.25 | λ = 0.25, γ = 0.75 | λ = 0.75, γ = 0.75 |
---|---|---|---|

| $\begin{array}{l}{y}_{1}=1.6\\ {y}_{2}=1.15\\ {y}_{3}=1.4875\\ {y}_{4}=1.234375\\ {y}_{5}=1.42421875\\ {y}_{6}=1.281835937\\ {y}_{7}=1.388623046\\ {y}_{8}=1.308532714\\ {y}_{9}=1.368600463\\ {y}_{10}=1.323549651\\ {y}_{11}=1.35733776\end{array}$ | $\begin{array}{l}{y}_{1}=1.6\\ {y}_{2}=1.45\\ {y}_{3}=1.4875\\ {y}_{4}=1.478125\\ {y}_{5}=1.48046875\\ {y}_{6}=1.479882812\\ {y}_{7}=1.480029296\\ {y}_{8}=1.479992675\\ {y}_{9}=1.48000183\\ {y}_{10}=1.479999541\\ {y}_{11}=1.480000114\end{array}$ | $\begin{array}{l}{y}_{1}=1.6\\ {y}_{2}=1.45\\ {y}_{3}=1.5625\\ {y}_{4}=1.534375\\ {y}_{5}=1.55546875\\ {y}_{6}=1.550195312\\ {y}_{7}=1.55415039\\ {y}_{8}=1.553161621\\ {y}_{9}=1.553903197\\ {y}_{10}=1.553717803\\ {y}_{11}=1.553764151\end{array}$ |

| $\begin{array}{l}{y}_{1}=2\\ {y}_{2}=1.25\\ {y}_{3}=1.8125\\ {y}_{4}=1.390625\\ {y}_{5}=1.70703125\\ {y}_{6}=1.469726562\\ {y}_{7}=1.647705078\\ {y}_{8}=1.514221191\\ {y}_{9}=1.614334106\\ {y}_{10}=1.539249419\\ {y}_{11}=1.595562934\end{array}$ | $\begin{array}{l}{y}_{1}=2\\ {y}_{2}=1.75\\ {y}_{3}=1.8125\\ {y}_{4}=1.796875\\ {y}_{5}=1.80078125\\ {y}_{6}=1.799804687\\ {y}_{7}=1.800048828\\ {y}_{8}=1.799987792\\ {y}_{9}=1.800003051\\ {y}_{10}=1.799999236\\ {y}_{11}=1.80000019\end{array}$ | $\begin{array}{l}{y}_{1}=2\\ {y}_{2}=1.75\\ {y}_{3}=1.9375\\ {y}_{4}=1.890625\\ {y}_{5}=1.92578125\\ {y}_{6}=1.916992187\\ {y}_{7}=1.923583984\\ {y}_{8}=1.921936035\\ {y}_{9}=1.923171996\\ {y}_{10}=1.922863006\\ {y}_{11}=1.922940253\end{array}$ |

| $\begin{array}{l}{y}_{1}=2.5\\ {y}_{2}=1.375\\ {y}_{3}=2.21875\\ {y}_{4}=1.5859375\\ {y}_{5}=2.060546875\\ {y}_{6}=1.704589843\\ {y}_{7}=1.971557617\\ {y}_{8}=1.771331786\\ {y}_{9}=1.921501159\\ {y}_{10}=1.808874128\\ {y}_{11}=1.893344401\end{array}$ | $\begin{array}{l}{y}_{1}=2.5\\ {y}_{2}=2.125\\ {y}_{3}=2.21875\\ {y}_{4}=2.1953125\\ {y}_{5}=2.201171875\\ {y}_{6}=2.199707031\\ {y}_{7}=2.200073242\\ {y}_{8}=2.199981688\\ {y}_{9}=2.200004577\\ {y}_{10}=2.199998854\\ {y}_{11}=2.200000285\end{array}$ | $\begin{array}{l}{y}_{1}=2.5\\ {y}_{2}=2.125\\ {y}_{3}=2.40625\\ {y}_{4}=2.3359375\\ {y}_{5}=2.388671875\\ {y}_{6}=2.375488281\\ {y}_{7}=2.385375976\\ {y}_{8}=2.382904053\\ {y}_{9}=2.384757994\\ {y}_{10}=2.384294509\\ {y}_{11}=2.38441038\end{array}$ |

**The solutions obtained by the new modified Ishikawa iteration method for different values of**
λ
**and**
γ

x | λ = 0.5, γ = 0.5 | λ = 0.5, γ = 0.25 | λ = 0.25, γ = 0.5 | λ = 0.25, γ = 0.25 |
---|---|---|---|---|

| $\begin{array}{l}{y}_{1}=1.6\\ {y}_{2}=1.3\\ {y}_{3}=1.45\\ {y}_{4}=1.375\\ {y}_{5}=1.4125\\ {y}_{6}=1.39375\\ {y}_{7}=1.403125\\ {y}_{8}=1.3984375\\ {y}_{9}=1.40078125\\ {y}_{10}=1.399609375\\ {y}_{11}=1.400195312\end{array}$ | $\begin{array}{l}{y}_{1}=1.6\\ {y}_{2}=1.15\\ {y}_{3}=1.375\\ {y}_{4}=1.20625\\ {y}_{5}=1.290625\\ {y}_{6}=1.22734375\\ {y}_{7}=1.258984375\\ {y}_{8}=1.235253905\\ {y}_{9}=1.24711914\\ {y}_{10}=1.238220214\\ {y}_{11}=1.242669677\end{array}$ | $\begin{array}{l}{y}_{1}=1.6\\ {y}_{2}=1.3\\ {y}_{3}=1.375\\ {y}_{4}=1.3375\\ {y}_{5}=1.346875\\ {y}_{6}=1.3421875\\ {y}_{7}=1.343359375\\ {y}_{8}=1.342773437\\ {y}_{9}=1.342919921\\ {y}_{10}=1.342846679\\ {y}_{11}=1.34286499\end{array}$ | $\begin{array}{l}{y}_{1}=1.6\\ {y}_{2}=1.15\\ {y}_{3}=1.2625\\ {y}_{4}=1.178125\\ {y}_{5}=1.19921875\\ {y}_{6}=1.183398437\\ {y}_{7}=1.187353515\\ {y}_{8}=1.184387206\\ {y}_{9}=1.185128783\\ {y}_{10}=1.1845726\\ {y}_{11}=1.184711646\end{array}$ |

| $\begin{array}{l}{y}_{1}=2\\ {y}_{2}=1.5\\ {y}_{3}=1.75\\ {y}_{4}=1.625\\ {y}_{5}=1.6875\\ {y}_{6}=1.65625\\ {y}_{7}=1.671875\\ {y}_{8}=1.6640625\\ {y}_{9}=1.66796875\\ {y}_{10}=1.666015625\\ {y}_{11}=1.666992187\end{array}$ | $\begin{array}{l}{y}_{1}=2\\ {y}_{2}=1.25\\ {y}_{3}=1.625\\ {y}_{4}=1.34375\\ {y}_{5}=1.484375\\ {y}_{6}=1.37890625\\ {y}_{7}=1.431640625\\ {y}_{8}=1.392089843\\ {y}_{9}=1.411865234\\ {y}_{10}=1.39703369\\ {y}_{11}=1.404449462\end{array}$ | $\begin{array}{l}{y}_{1}=2\\ {y}_{2}=1.5\\ {y}_{3}=1.625\\ {y}_{4}=1.5625\\ {y}_{5}=1.578125\\ {y}_{6}=1.5703125\\ {y}_{7}=1.572265625\\ {y}_{8}=1.571289062\\ {y}_{9}=1.571533203\\ {y}_{10}=1.571411132\\ {y}_{11}=1.57144165\end{array}$ | $\begin{array}{l}{y}_{1}=2\\ {y}_{2}=1.25\\ {y}_{3}=1.4375\\ {y}_{4}=1.296875\\ {y}_{5}=1.33203125\\ {y}_{6}=1.305664062\\ {y}_{7}=1.312255859\\ {y}_{8}=1.307312011\\ {y}_{9}=1.308547973\\ {y}_{10}=1.307621001\\ {y}_{11}=1.307852744\end{array}$ |

| $\begin{array}{l}{y}_{1}=2.5\\ {y}_{2}=1.75\\ {y}_{3}=2.125\\ {y}_{4}=1.9375\\ {y}_{5}=2.03125\\ {y}_{6}=1.984375\\ {y}_{7}=2.0078125\\ {y}_{8}=1.99609375\\ {y}_{9}=2.001953125\\ {y}_{10}=1.999023437\\ {y}_{11}=2.000488281\end{array}$ | $\begin{array}{l}{y}_{1}=2.5\\ {y}_{2}=1.375\\ {y}_{3}=1.9375\\ {y}_{4}=1.515625\\ {y}_{5}=1.7265625\\ {y}_{6}=1.568359375\\ {y}_{7}=1.647460937\\ {y}_{8}=1.588134764\\ {y}_{9}=1.617797851\\ {y}_{10}=1.595550535\\ {y}_{11}=1.606674193\end{array}$ | $\begin{array}{l}{y}_{1}=2.5\\ {y}_{2}=1.75\\ {y}_{3}=1.9375\\ {y}_{4}=1.84375\\ {y}_{5}=1.8671875\\ {y}_{6}=1.85546875\\ {y}_{7}=1.858398437\\ {y}_{8}=1.856933593\\ {y}_{9}=1.857299804\\ {y}_{10}=1.857116698\\ {y}_{11}=1.857162475\end{array}$ | $\begin{array}{l}{y}_{1}=2.5\\ {y}_{2}=1.375\\ {y}_{3}=1.65625\\ {y}_{4}=1.4453125\\ {y}_{5}=1.498046875\\ {y}_{6}=1.458496093\\ {y}_{7}=1.468383788\\ {y}_{8}=1.460968016\\ {y}_{9}=1.462821959\\ {y}_{10}=1.461431501\\ {y}_{11}=1.461779116\end{array}$ |

**Absolute error of Example 2.1 for different values of**
λ
**and**
γ
**(**
$\mathit{x}\mathbf{=}\mathbf{0.2}$
**,**
$\mathit{x}\mathbf{=}\mathbf{0.4}$
**and**
$\mathit{x}\mathbf{=}\mathbf{0.6}$
**respectively)**

x = 0.2 | x = 0.4 | x = 0.6 | |
---|---|---|---|

| 0.12398437 | 0.173203126 | 0.289804688 |

| 0.129110108 | 0.278220216 | 0.447330323 |

| 0.09571167 | 0.21142334 | 0.34713501 |

| 0.148429452 | 0.316858903 | 0.505288354 |

| 0.090887414 | 0.201774827 | 0.33266224 |

| 0.049999962 | 0.119999924 | 0.209999886 |

| 0.02541195 | 0.070823899 | 0.136235849 |

Picard | 0.0003 | 0.002328 | 0.007369875 |

Runge-Kutta | 0.000000435 | 0.000000081 | 0.00000046 |

Euler | 0.01 | 0.020910977 | 0.032659931 |

**Corollary 2.1** *If we compare the approximate solution with the different values of* *λ* *and* *γ*, *then the conclusion may be indicated using Table *1, *Table *2, *Table *3 *and Table *4 *as follows*.

*The best approximation is obtained taking the different values of* *λ* *and* *γ* *and using the new modified Ishikawa iteration method for* $x=0.2$, $x=0.4$ *and* $x=0.5$ *getting* ($\lambda =0.25$, $\gamma =0.25$; $\lambda =0.5$, $\gamma =0.25$; $\lambda =0.25$, $\gamma =0.5$; $\lambda =0.75$, $\gamma =0.25$; $\lambda =0.5$, $\gamma =0.5$; $\lambda =0.25$, $\gamma =0.75$; $\lambda =0.75$, $\gamma =0.75$) *respectively*.

*Similarly*, *we calculated the solution for* $x=0.6$, $x=1$ *and* $x=1.5$ *then the approximation is found more sensitive taking* ($\lambda =0.25$, $\gamma =0.25$; $\lambda =0.5$, $\gamma =0.25$; $\lambda =0.25$, $\gamma =0.5$; $\lambda =0.75$, $\gamma =0.25$; $\lambda =0.5$, $\gamma =0.5$; $\lambda =0.25$, $\gamma =0.75$; $\lambda =0.75$, $\gamma =0.75$) *respectively*.

**Corollary 2.2** *Absolute error of the modified Ishikawa iteration method is computed taking different values of* *λ* *and* *γ* ($x=0.2$, $x=0.4$ *and* $x=0.6$), *which is not more effective than Runge*-*Kutta*, *Picard and Euler iteration methods*.

**Example 2.2**

Firstly, we obtained the exact solution of the equation as $y=2{e}^{x}-{x}^{2}-2x-2$.

Using Theorem 1.1 and Corollary 1.2, since $T={\int}_{{x}_{0}}^{x}F(t,{y}_{n}(t))\phantom{\rule{0.2em}{0ex}}dt$, then *T* has a unique fixed point, which is the unique solution of the differential equation ${y}^{\prime}=y+{x}^{2}$ with the initial condition $y(0)=0$.

Thus ${y}_{3}=0.08423766727440010142666666666667$ is obtained for ${x}_{3}=0.6$.

*λ*and

*γ*. Now we may give Table 10 which is expressed that absolute error of Example 2.2 for different values of

*λ*and

*γ*with $x=0.2$, $x=0.4$ and $x=0.6$ respectively.

**The solutions obtained by the new modified Ishikawa iteration method for different values of**
λ
**and**
γ

x | λ = 0.5, γ = 0.5 | λ = 0.5, γ = 0.25 | λ = 0.25, γ = 0.5 | λ = 0.25, γ = 0.25 |
---|---|---|---|---|

| $\begin{array}{l}{y}_{1}=0.002666666667\\ {y}_{2}=0.0013333\\ {y}_{3}=0.002\\ {y}_{4}=0.001666664\\ {y}_{5}=0.0018333328\\ {y}_{6}=0.001749999864\\ {y}_{7}=0.001791666568\\ {y}_{8}=0.001770833216\\ {y}_{9}=0.001781249888\\ {y}_{10}=0.001776041312\\ {y}_{11}=0.00177864572\end{array}$ | $\begin{array}{l}{y}_{1}=0.002666666667\\ {y}_{2}=0.00066666664\\ {y}_{3}=0.00166666528\\ {y}_{4}=0.00091666432\\ {y}_{5}=0.00129166648\\ {y}_{6}=0.00101041644\\ {y}_{7}=0.001151041456\\ {y}_{8}=0.001045572688\\ {y}_{9}=0.001098307072\\ {y}_{10}=0.00105875628\\ {y}_{11}=0.001078531672\end{array}$ | $\begin{array}{l}{y}_{1}=0.002666666667\\ {y}_{2}=0.001333333333\\ {y}_{3}=0.001666666664\\ {y}_{4}=0.001499999992\\ {y}_{5}=0.001541666664\\ {y}_{6}=0.001520833328\\ {y}_{7}=0.001526041656\\ {y}_{8}=0.001523437488\\ {y}_{9}=0.001522786448\\ {y}_{10}=0.001523111968\\ {y}_{11}=0.001523030584\end{array}$ | $\begin{array}{l}{y}_{1}=0.002666666667\\ {y}_{2}=0.000666666664\\ {y}_{3}=0.001166666666\\ {y}_{4}=0.000791666664\\ {y}_{5}=0.000885416664\\ {y}_{6}=0.00081510416\\ {y}_{7}=0.00083268228\\ {y}_{8}=0.000819498688\\ {y}_{9}=0.000822794584\\ {y}_{10}=0.000820322656\\ {y}_{11}=0.00082094064\end{array}$ |

| $\begin{array}{l}{y}_{1}=0.021333333\\ {y}_{2}=0.010666666\\ {y}_{3}=0.016\\ {y}_{4}=0.013333333\\ {y}_{5}=0.014666662\\ {y}_{6}=0.013999998\\ {y}_{7}=0.014333332\\ {y}_{8}=0.014166665\\ {y}_{9}=0.014249999\\ {y}_{10}=0.014208332\\ {y}_{11}=0.014229165\end{array}$ | $\begin{array}{l}{y}_{1}=0.021333333\\ {y}_{2}=0.00533333312\\ {y}_{3}=0.013333332\\ {y}_{4}=0.007333331456\\ {y}_{5}=0.010333333\\ {y}_{6}=0.00808333152\\ {y}_{7}=0.009208331648\\ {y}_{8}=0.008364581504\\ {y}_{9}=0.008786456576\\ {y}_{10}=0.00847005024\\ {y}_{11}=0.008628253376\end{array}$ | $\begin{array}{l}{y}_{1}=0.021333333\\ {y}_{2}=0.010666666\\ {y}_{3}=0.013333333\\ {y}_{4}=0.011999999\\ {y}_{5}=0.012333333\\ {y}_{6}=0.012166666\\ {y}_{7}=0.012208333\\ {y}_{8}=0.012187499\\ {y}_{9}=0.012182291\\ {y}_{10}=0.012184895\\ {y}_{11}=0.012184244\end{array}$ | $\begin{array}{l}{y}_{1}=0.021333333\\ {y}_{2}=0.00533333312\\ {y}_{3}=0.009333333331\\ {y}_{4}=0.006333333312\\ {y}_{5}=0.007083333312\\ {y}_{6}=0.00652083328\\ {y}_{7}=0.00666145824\\ {y}_{8}=0.006555989504\\ {y}_{9}=0.006582356672\\ {y}_{10}=0.006562581248\\ {y}_{11}=0.00656752512\end{array}$ |

| $\begin{array}{l}{y}_{1}=0.041666666\\ {y}_{2}=0.020833333\\ {y}_{3}=0.03125\\ {y}_{4}=0.026041625\\ {y}_{5}=0.028645825\\ {y}_{6}=0.027343747\\ {y}_{7}=0.02799479\\ {y}_{8}=0.027669269\\ {y}_{9}=0.027832029\\ {y}_{10}=0.027750649\\ {y}_{11}=0.027791339\end{array}$ | $\begin{array}{l}{y}_{1}=0.041666666\\ {y}_{2}=0.010416666\\ {y}_{3}=0.026041664\\ {y}_{4}=0.014322913\\ {y}_{5}=0.020182288\\ {y}_{6}=0.015787756\\ {y}_{7}=0.017985022\\ {y}_{8}=0.016337073\\ {y}_{9}=0.017161048\\ {y}_{10}=0.016543066\\ {y}_{11}=0.016852057\end{array}$ | $\begin{array}{l}{y}_{1}=0.041666666\\ {y}_{2}=0.020833333\\ {y}_{3}=0.026041666\\ {y}_{4}=0.023437499\\ {y}_{5}=0.024088541\\ {y}_{6}=0.02376302\\ {y}_{7}=0.0238444\\ {y}_{8}=0.02380371\\ {y}_{9}=0.023793538\\ {y}_{10}=0.023798624\\ {y}_{11}=0.023797352\end{array}$ | $\begin{array}{l}{y}_{1}=0.041666666\\ {y}_{2}=0.010416666\\ {y}_{3}=0.018229166\\ {y}_{4}=0.012369791\\ {y}_{5}=0.013834635\\ {y}_{6}=0.012736025\\ {y}_{7}=0.01301066\\ {y}_{8}=0.012804667\\ {y}_{9}=0.012856165\\ {y}_{10}=0.012817541\\ {y}_{11}=0.012827197\end{array}$ |

**The solutions obtained by the new modified Ishikawa iteration method for different values of**
λ
**and**
γ

x | λ = 0.75, γ = 0.25 | λ = 0.25, γ = 0.75 | λ = 0.75, γ = 0.75 |
---|---|---|---|

| $\begin{array}{l}{y}_{1}=0.002666666667\\ {y}_{2}=0.000666666664\\ {y}_{3}=0.002166666664\\ {y}_{4}=0.001041666664\\ {y}_{5}=0.001885416664\\ {y}_{6}=0.00125260416\\ {y}_{7}=0.001727213536\\ {y}_{8}=0.001371256504\\ {y}_{9}=0.001638224272\\ {y}_{10}=0.00143799844\\ {y}_{11}=0.001588167816\end{array}$ | $\begin{array}{l}{y}_{1}=0.002666666667\\ {y}_{2}=0.002\\ {y}_{3}=0.002166666664\\ {y}_{4}=0.002125\\ {y}_{5}=0.002135416664\\ {y}_{6}=0.002132812496\\ {y}_{7}=0.002133463536\\ {y}_{8}=0.002133300776\\ {y}_{9}=0.002133341464\\ {y}_{10}=0.002133331288\\ {y}_{11}=0.002133333832\end{array}$ | $\begin{array}{l}{y}_{1}=0.002666666667\\ {y}_{2}=0.002\\ {y}_{3}=0.0025\\ {y}_{4}=0.002375\\ {y}_{5}=0.00246875\\ {y}_{6}=0.002445312496\\ {y}_{7}=0.002462890624\\ {y}_{8}=0.002458496088\\ {y}_{9}=0.002461791984\\ {y}_{10}=0.002406096801\\ {y}_{11}=0.002461585992\end{array}$ |

| $\begin{array}{l}{y}_{1}=0.021333333\\ {y}_{2}=0.005333333312\\ {y}_{3}=0.017333333\\ {y}_{4}=0.008333333312\\ {y}_{5}=0.015083333\\ {y}_{6}=0.010020833\\ {y}_{7}=0.0138177708\\ {y}_{8}=0.010970052\\ {y}_{9}=0.013105794\\ {y}_{10}=0.011503987\\ {y}_{11}=0.012705342\end{array}$ | $\begin{array}{l}{y}_{1}=0.021333333\\ {y}_{2}=0.016\\ {y}_{3}=0.017333333\\ {y}_{4}=0.017\\ {y}_{5}=0.017083333\\ {y}_{6}=0.017062499\\ {y}_{7}=0.017067708\\ {y}_{8}=0.017066406\\ {y}_{9}=0.017066731\\ {y}_{10}=0.01706665\\ {y}_{11}=0.01706667\end{array}$ | $\begin{array}{l}{y}_{1}=0.021333333\\ {y}_{2}=0.016\\ {y}_{3}=0.02\\ {y}_{4}=0.019\\ {y}_{5}=0.01975\\ {y}_{6}=0.019562499\\ {y}_{7}=0.019703124\\ {y}_{8}=0.019667968\\ {y}_{9}=0.019694335\\ {y}_{10}=0.019687744\\ {y}_{11}=0.019692687\end{array}$ |

| $\begin{array}{l}{y}_{1}=0.041666666\\ {y}_{2}=0.010416666\\ {y}_{3}=0.033854166\\ {y}_{4}=0.016276041\\ {y}_{5}=0.029459635\\ {y}_{6}=0.01957194\\ {y}_{7}=0.026987711\\ {y}_{8}=0.021425882\\ {y}_{9}=0.025597254\\ {y}_{10}=0.022468725\\ {y}_{11}=0.024815122\end{array}$ | $\begin{array}{l}{y}_{1}=0.041666666\\ {y}_{2}=0.03125\\ {y}_{3}=0.033854166\\ {y}_{4}=0.033203125\\ {y}_{5}=0.033365885\\ {y}_{6}=0.033325195\\ {y}_{7}=0.033335367\\ {y}_{8}=0.033335367\\ {y}_{9}=0.03333346\\ {y}_{10}=0.033333301\\ {y}_{11}=0.033333341\end{array}$ | $\begin{array}{l}{y}_{1}=0.041666666\\ {y}_{2}=0.03125\\ {y}_{3}=0.0390625\\ {y}_{4}=0.037109375\\ {y}_{5}=0.038574218\\ {y}_{6}=0.038208007\\ {y}_{7}=0.038482666\\ {y}_{8}=0.038414001\\ {y}_{9}=0.038465499\\ {y}_{10}=0.038452625\\ {y}_{11}=0.038462281\end{array}$ |

**The solutions obtained by the new modified Ishikawa iteration method for different values of**
λ
**and**
γ

x | λ = 0.5, γ = 0.5 | λ = 0.5, γ = 0.25 | λ = 0.25, γ = 0.5 | λ = 0.25, γ = 0.25 |
---|---|---|---|---|

| $\begin{array}{l}{y}_{1}=0.072\\ {y}_{2}=0.036\\ {y}_{3}=0.054\\ {y}_{4}=0.044999928\\ {y}_{5}=0.049499985\\ {y}_{6}=0.047249996\\ {y}_{7}=0.048376997\\ {y}_{8}=0.047812496\\ {y}_{9}=0.048093746\\ {y}_{10}=0.047953121\\ {y}_{11}=0.048023434\end{array}$ | $\begin{array}{l}{y}_{1}=0.072\\ {y}_{2}=0.017999992\\ {y}_{3}=0.044999996\\ {y}_{4}=0.024749993\\ {y}_{5}=0.034874994\\ {y}_{6}=0.027281243\\ {y}_{7}=0.031078119\\ {y}_{8}=0.028230462\\ {y}_{9}=0.02965429\\ {y}_{10}=0.028586419\\ {y}_{11}=0.029120355\end{array}$ | $\begin{array}{l}{y}_{1}=0.072\\ {y}_{2}=0.036\\ {y}_{3}=0.044999999\\ {y}_{4}=0.040499999\\ {y}_{5}=0.041624999\\ {y}_{6}=0.04062499\\ {y}_{7}=0.041203124\\ {y}_{8}=0.041132812\\ {y}_{9}=0.041115234\\ {y}_{10}=0.041124023\\ {y}_{11}=0.041121825\end{array}$ | $\begin{array}{l}{y}_{1}=0.072\\ {y}_{2}=0.017999999\\ {y}_{3}=0.031499999\\ {y}_{4}=0.021374999\\ {y}_{5}=0.023906249\\ {y}_{6}=0.022007812\\ {y}_{7}=0.022482421\\ {y}_{8}=0.022126464\\ {y}_{9}=0.022215453\\ {y}_{10}=0.022148711\\ {y}_{11}=0.022165397\end{array}$ |

| $\begin{array}{l}{y}_{1}=0.333333\\ {y}_{2}=0.1666666\\ {y}_{3}=0.25\\ {y}_{4}=0.208333\\ {y}_{5}=0.2291666\\ {y}_{6}=0.218749983\\ {y}_{7}=0.223958321\\ {y}_{8}=0.221354152\\ {y}_{9}=0.222656236\\ {y}_{10}=0.222005194\\ {y}_{11}=0.222330715\end{array}$ | $\begin{array}{l}{y}_{1}=0.333333\\ {y}_{2}=0.08333333\\ {y}_{3}=0.208333316\\ {y}_{4}=0.114583304\\ {y}_{5}=0.16145831\\ {y}_{6}=0.126302055\\ {y}_{7}=0.143880182\\ {y}_{8}=0.130696586\\ {y}_{9}=0.137288384\\ {y}_{10}=0.132344535\\ {y}_{11}=0.134816459\end{array}$ | $\begin{array}{l}{y}_{1}=0.333333\\ {y}_{2}=0.1666666\\ {y}_{3}=0.20833333\\ {y}_{4}=0.187499999\\ {y}_{5}=0.192708333\\ {y}_{6}=0.190104166\\ {y}_{7}=0.190755207\\ {y}_{8}=0.190429686\\ {y}_{9}=0.190348306\\ {y}_{10}=0.190388996\\ {y}_{11}=0.190378823\end{array}$ | $\begin{array}{l}{y}_{1}=0.333333\\ {y}_{2}=0.083333333\\ {y}_{3}=0.1458333333\\ {y}_{4}=0.098958333\\ {y}_{5}=0.110677083\\ {y}_{6}=0.10188802\\ {y}_{7}=0.104085285\\ {y}_{8}=0.102437336\\ {y}_{9}=0.102849323\\ {y}_{10}=0.102540332\\ {y}_{11}=0.10261758\end{array}$ |

| $\begin{array}{l}{y}_{1}=1.125\\ {y}_{2}=0.5625\\ {y}_{3}=0.84375\\ {y}_{4}=0.703123875\\ {y}_{5}=0.773437275\\ {y}_{6}=0.738281192\\ {y}_{7}=0.755859333\\ {y}_{8}=0.747070263\\ {y}_{9}=0.751464796\\ {y}_{10}=0.749267529\\ {y}_{11}=0.750366163\end{array}$ | $\begin{array}{l}{y}_{1}=1.125\\ {y}_{2}=0.2812499888\\ {y}_{3}=0.703124941\\ {y}_{4}=0.38654901\\ {y}_{5}=0.544921796\\ {y}_{6}=0.426269435\\ {y}_{7}=0.485595614\\ {y}_{8}=0.441100977\\ {y}_{9}=0.463348296\\ {y}_{10}=0.446662805\\ {y}_{11}=0.455005549\end{array}$ | $\begin{array}{l}{y}_{1}=1.125\\ {y}_{2}=0.5625\\ {y}_{3}=0.703124998\\ {y}_{4}=0.632812496\\ {y}_{5}=0.650390623\\ {y}_{6}=0.64160156\\ {y}_{7}=0.643798823\\ {y}_{8}=0.64270019\\ {y}_{9}=0.642425532\\ {y}_{10}=0.64256274\\ {y}_{11}=0.642528527\end{array}$ | $\begin{array}{l}{y}_{1}=1.125\\ {y}_{2}=0.281249998\\ {y}_{3}=0.492187499\\ {y}_{4}=0.333984373\\ {y}_{5}=0.373535155\\ {y}_{6}=0.343872067\\ {y}_{7}=0.351287836\\ {y}_{8}=0.345726009\\ {y}_{9}=0.347116465\\ {y}_{10}=0.34607362\\ {y}_{11}=0.346334332\end{array}$ |

**The solutions obtained by the new modified Ishikawa iteration method for different values of**
λ
**and**
γ

x | λ = 0.75, γ = 0.25 | λ = 0.25, γ = 0.75 | λ = 0.75, γ = 0.75 |
---|---|---|---|

| $\begin{array}{l}{y}_{1}=0.072\\ {y}_{2}=0.017999999\\ {y}_{3}=0.058499999\\ {y}_{4}=0.028124999\\ {y}_{5}=0.050906249\\ {y}_{6}=0.033820312\\ {y}_{7}=0.046634765\\ {y}_{8}=0.037023925\\ {y}_{9}=0.044232055\\ {y}_{10}=0.038825957\\ {y}_{11}=0.042880531\end{array}$ | $\begin{array}{l}{y}_{1}=0.072\\ {y}_{2}=0.054\\ {y}_{3}=0.058499999\\ {y}_{4}=0.057375\\ {y}_{5}=0.057656249\\ {y}_{6}=0.057585937\\ {y}_{7}=0.057603515\\ {y}_{8}=0.05759912\\ {y}_{9}=0.057600219\\ {y}_{10}=0.057599944\\ {y}_{11}=0.057600013\end{array}$ | $\begin{array}{l}{y}_{1}=0.072\\ {y}_{2}=0.054\\ {y}_{3}=0.0675\\ {y}_{4}=0.064125\\ {y}_{5}=0.06665625\\ {y}_{6}=0.066023437\\ {y}_{7}=0.066498046\\ {y}_{8}=0.066379394\\ {y}_{9}=0.066468383\\ {y}_{10}=0.066446136\\ {y}_{11}=0.066462821\end{array}$ |

| $\begin{array}{l}{y}_{1}=0.333333\\ {y}_{2}=0.083333333\\ {y}_{3}=0.2708333\\ {y}_{4}=0.130208333\\ {y}_{5}=0.235677083\\ {y}_{6}=0.15657552\\ {y}_{7}=0.215901692\\ {y}_{8}=0.171407063\\ {y}_{9}=0.204778034\\ {y}_{10}=0.179749805\\ {y}_{11}=0.198520977\end{array}$ | $\begin{array}{l}{y}_{1}=0.333333\\ {y}_{2}=0.25\\ {y}_{3}=0.270833333\\ {y}_{4}=0.265625\\ {y}_{5}=0.266927083\\ {y}_{6}=0.266601562\\ {y}_{7}=0.266682942\\ {y}_{8}=0.266662597\\ {y}_{9}=0.266667683\\ {y}_{10}=0.266666411\\ {y}_{11}=0.266666729\end{array}$ | $\begin{array}{l}{y}_{1}=0.333333\\ {y}_{2}=0.25\\ {y}_{3}=0.3125\\ {y}_{4}=0.296875\\ {y}_{5}=0.30859375\\ {y}_{6}=0.305664062\\ {y}_{7}=0.307861328\\ {y}_{8}=0.307312011\\ {y}_{9}=0.307723998\\ {y}_{10}=0.307621001\\ {y}_{11}=0.307698249\end{array}$ |

| $\begin{array}{l}{y}_{1}=1.175\\ {y}_{2}=0.281249998\\ {y}_{3}=0.914062498\\ {y}_{4}=0.439453123\\ {y}_{5}=0.795410155\\ {y}_{6}=0.52844238\\ {y}_{7}=0.72866821\\ {y}_{8}=0.578498837\\ {y}_{9}=0.691125864\\ {y}_{10}=0.606655591\\ {y}_{11}=0.670008297\end{array}$ | $\begin{array}{l}{y}_{1}=1.125\\ {y}_{2}=0.84375\\ {y}_{3}=0.914062498\\ {y}_{4}=0.896484375\\ {y}_{5}=0.900878905\\ {y}_{6}=0.899780271\\ {y}_{7}=0.900054929\\ {y}_{8}=0.899986264\\ {y}_{9}=0.90000343\\ {y}_{10}=0.899999137\\ {y}_{11}=0.90000021\end{array}$ | $\begin{array}{l}{y}_{1}=1.125\\ {y}_{2}=0.84375\\ {y}_{3}=1.0546875\\ {y}_{4}=1.001953125\\ {y}_{5}=1.041503906\\ {y}_{6}=1.031616209\\ {y}_{7}=1.039031982\\ {y}_{8}=1.037178037\\ {y}_{9}=1.038568493\\ {y}_{10}=1.038220878\\ {y}_{11}=1.03848159\end{array}$ |

**Absolute error of Example 2.2 for different values of**
λ
**and**
γ
**(**
$\mathit{x}\mathbf{=}\mathbf{0.4}$
**and**
$\mathit{x}\mathbf{=}\mathbf{0.6}$
**respectively)**

x = 0.2 | x = 0.4 | x = 0.6 | |
---|---|---|---|

| 0.0010268706 | 0.00942023 | 0.036214166 |

| 0.001726984648 | 0.015021141 | 0.055117245 |

| 0.001282485736 | 0.011465151 | 0.043115775 |

| 0.00198457568 | 0.017081869 | 0.017774779 |

| 0.001217348504 | 0.010944053 | 0.041357069 |

| 0.000672182488 | 0.006582725 | 0.026637587 |

| 0.000343930328 | 0.003956708 | 0.017774779 |

Picard | 0.00000000522 | 0.000000684 | 0.000012 |

Runge-Kutta | 0.000998849654 | 0.001220066 | 0.001491752 |

Euler | 0.00280551632 | 0.015649395 | 0.0426376 |

**Corollary 2.3** *If we compare the approximate solution with the different values of* *λ* *and* *γ*, *then the conclusion may be indicated using Table *6, *Table *7, *Table *8 *and Table *9 *as follows*.

*The best approximation is obtained taking the different values of* *λ* *and* *γ* *and using the modified Ishikawa iteration method for* $x=0.2$, $x=0.4$ *and* $x=0.5$ *getting* ($\lambda =0.25$, $\gamma =0.25$; $\lambda =0.5$, $\gamma =0.25$; $\lambda =0.25$, $\gamma =0.5$; $\lambda =0.75$, $\gamma =0.25$; $\lambda =0.5$, $\gamma =0.5$; $\lambda =0.25$, $\gamma =0.75$; $\lambda =0.75$, $\gamma =0.75$) *respectively*.

*Similarly*, *we calculated the solution for* $x=0.6$, $x=1$ *and* $x=1.5$ *then the approximation is found more sensitive taking* ($\lambda =0.25$, $\gamma =0.25$; $\lambda =0.5$, $\gamma =0.25$; $\lambda =0.25$, $\gamma =0.5$; $\lambda =0.75$, $\gamma =0.25$; $\lambda =0.5$, $\gamma =0.5$; $\lambda =0.25$, $\gamma =0.75$; $\lambda =0.75$, $\gamma =0.75$) *respectively*.

**Corollary 2.4** *Absolute error of the modified Ishikawa iteration method is computed taking different values of* *λ* *and* *γ* ($x=0.2$, $x=0.4$ *and* $x=0.6$), *which is not more effective than Picard*, *Runge*-*Kutta and Euler iteration methods*.

**Example 2.3**

Using Theorem 1.1 and Corollary 1.2, since $T={\int}_{{x}_{0}}^{x}F(t,{y}_{n}(t))\phantom{\rule{0.2em}{0ex}}dt$, then *T* has a unique fixed point, which is the unique solution of the differential equation ${y}^{\prime}=2x(y+1)$ with the initial condition $y(0)=0$.

Thus ${y}_{3}=0.433321409$ is obtained for ${x}_{3}=0.6$.

*λ*and

*γ*. Now we may give Table 15 which is expressed that absolute error of Example 2.3 for different values of

*λ*and

*γ*with $x=0.2$, $x=0.4$ and $x=0.6$ respectively.

**The solutions obtained by the new modified Ishikawa iteration method for differential values of**
λ
**and**
γ

x | λ = 0.5, γ = 0.5 | λ = 0.5, γ = 0.25 | λ = 0.25, γ = 0.5 | λ = 0.25, γ = 0.25 |
---|---|---|---|---|

| $\begin{array}{l}{y}_{1}=0.04\\ {y}_{2}=0.02\\ {y}_{3}=0.03\\ {y}_{4}=0.025\\ {y}_{5}=0.0275\\ {y}_{6}=0.02625\\ {y}_{7}=0.026875\\ {y}_{8}=0.0265625\\ {y}_{9}=0.02671875\\ {y}_{10}=0.026640625\\ {y}_{11}=0.026679707\end{array}$ | $\begin{array}{l}{y}_{1}=0.04\\ {y}_{2}=0.01\\ {y}_{3}=0.025\\ {y}_{4}=0.01375\\ {y}_{5}=0.019375\\ {y}_{6}=0.01515625\\ {y}_{7}=0.017265625\\ {y}_{8}=0.015683593\\ {y}_{9}=0.016474609\\ {y}_{10}=0.015881347\\ {y}_{11}=0.016177978\end{array}$ | $\begin{array}{l}{y}_{1}=0.04\\ {y}_{2}=0.02\\ {y}_{3}=0.025\\ {y}_{4}=0.0225\\ {y}_{5}=0.023125\\ {y}_{6}=0.0228125\\ {y}_{7}=0.022890625\\ {y}_{8}=0.022851562\\ {y}_{9}=0.022861328\\ {y}_{10}=0.022856445\\ {y}_{11}=0.022857666\end{array}$ | $\begin{array}{l}{y}_{1}=0.04\\ {y}_{2}=0.01\\ {y}_{3}=0.0175\\ {y}_{4}=0.011875\\ {y}_{5}=0.01328125\\ {y}_{6}=0.012226562\\ {y}_{7}=0.012490234\\ {y}_{8}=0.01229248\\ {y}_{9}=0.012341918\\ {y}_{10}=0.01230484\\ {y}_{11}=0.012314119\end{array}$ |

| $\begin{array}{l}{y}_{1}=0.16\\ {y}_{2}=0.08\\ {y}_{3}=0.12\\ {y}_{4}=0.1\\ {y}_{5}=0.11\\ {y}_{6}=0.105\\ {y}_{7}=0.1075\\ {y}_{8}=0.10625\\ {y}_{9}=0.106875\\ {y}_{10}=0.1065625\\ {y}_{11}=0.106718829\end{array}$ | $\begin{array}{l}{y}_{1}=0.16\\ {y}_{2}=0.04\\ {y}_{3}=0.1\\ {y}_{4}=0.055\\ {y}_{5}=0.0775\\ {y}_{6}=0.060625\\ {y}_{7}=0.0690625\\ {y}_{8}=0.062734374\\ {y}_{9}=0.065898437\\ {y}_{10}=0.06352539\\ {y}_{11}=0.064711913\end{array}$ | $\begin{array}{l}{y}_{1}=0.16\\ {y}_{2}=0.08\\ {y}_{3}=0.1\\ {y}_{4}=0.09\\ {y}_{5}=0.0925\\ {y}_{6}=0.09125\\ {y}_{7}=0.0915625\\ {y}_{8}=0.091406249\\ {y}_{9}=0.091445312\\ {y}_{10}=0.091445312\\ {y}_{11}=0.091430664\end{array}$ | $\begin{array}{l}{y}_{1}=0.16\\ {y}_{2}=0.04\\ {y}_{3}=0.07\\ {y}_{4}=0.0475\\ {y}_{5}=0.053125\\ {y}_{6}=0.048906249\\ {y}_{7}=0.049960937\\ {y}_{8}=0.049169921\\ {y}_{9}=0.049367675\\ {y}_{10}=0.04921936\\ {y}_{11}=0.049256479\end{array}$ |

| $\begin{array}{l}{y}_{1}=0.25\\ {y}_{2}=0.125\\ {y}_{3}=0.1875\\ {y}_{4}=0.15625\\ {y}_{5}=0.171875\\ {y}_{6}=0.1640625\\ {y}_{7}=0.16796875\\ {y}_{8}=0.166015625\\ {y}_{9}=0.166992187\\ {y}_{10}=0.166503906\\ {y}_{11}=0.166748171\end{array}$ | $\begin{array}{l}{y}_{1}=0.25\\ {y}_{2}=0.0625\\ {y}_{3}=0.15625\\ {y}_{4}=0.0859375\\ {y}_{5}=0.12109375\\ {y}_{6}=0.094726562\\ <\end{array}$ |