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In this paper, a reliable algorithm based on mixture of new integral transform and homotopy perturbation method is proposed to solve a nonlinear differential-difference equation arising in nanotechnology. Continuum hypothesis on nanoscales is invalid, and a differential-difference model is considered as an alternative approach to describing discontinued problems. The technique finds the solution without any discretization or restrictive assumptions and avoids the round-off errors. Comparison of the approximate solution with the exact one reveals that the method is very effective. It provides more realistic series solutions that converge very rapidly for nonlinear real physical problems.

Non-linear phenomena that appear in many areas of scientific fields such as solid state physics, plasma physics, fluid mechanics, population models and chemical kinetics, can be modeled by nonlinear partial differential equations. In many different fields of science and engineering, it is important to obtain exact or numerical solution of the nonlinear partial differential equations. Searching of exact and numerical solution of nonlinear equations in science and engineering is still quite problematic that needs new methods for finding the exact and approximate solutions. Most of new nonlinear equations do not have a precise analytic solution; so, numerical methods have largely been used to handle these equations. There are also analytic techniques for nonlinear equations. Some of the classic analytic methods are Lyapunov’s artificial small parameter method [

According to E-infinity theory [

where

where the subscript n in Equation (1) represents the

In this paper, we will study numerically Equation (2) using the mixture of new integral transform and homotopy perturbation method. It is worth mentioning that the proposed method is an elegant combination of the new integral transform, the homotopy perturbation method and He’s polynomials. The advantage of this technique is its capability of combining two powerful approaches for obtaining exact and approximate ana- lytical solutions for nonlinear equations. This method provides the solutions in terms of convergent series with easily computable components in a direct way without using linearization, perturbation or restrictive assumptions.

A new integral transform is derived from the classical Fourier integral. A new integral transform [

A new integral transform is defined for functions of exponential order. We consider functions in the set F defined as:

For a given function in the set F, the constant M must be finite number,

A new integral transform denoted by the operator

For further detail and properties of this transform, see [

To illustrate the basic idea of this method, we consider a general nonlinear partial differential equation with the initial conditions of the form:

where D is the second order linear differential operator

Taking the new integral transform on both sides of Equation (5), we get

Using the differentiation property of new integral transform and above initial conditions (see Appendix), we have

Now, applying new integral transform on both sides of Equation (8), we get

where

According to homotopy perturbation method, we have [

Now, by substituting

and

for some He’s polynomials

in Equation (10), we get

which is the mixture of the new integral transform and the homotopy perturbation method using He’s polyno- mials. Comparing the coefficient of like powers of p, the following approximations are obtained.

Then the solution is:

In this section, we apply the mixture of new integral transform and homotopy perturbation method to solve (2), subject to the initial condition

where d is an arbitrary constant.

Applying the new integral transform on both sides of (2) subject to initial condition (16), we have

The inverse new integral transform implies that

Applying the homotopy perturbation method, we get

where

Comparing the coefficients of like powers of p, we have

Therefore the approximate solution is

In this section, we calculate the numerical results of

n | Approximate Solution | Exact Solution | Absolute Error | Percentage Error |
---|---|---|---|---|

−15 | −0.086621367 | −0.08590324656 | 0.000718120 | 0.008359643 |

−5 | −0.030409180 | −0.02909509651 | 0.001314083 | 0.045165119 |

−4 | −0.020846123 | −0.01973559643 | 0.001110527 | 0.056270231 |

−3 | −0.010832958 | −0.00999922801 | 0.000833730 | 0.084132741 |

3 | 0.047197969 | 0.04600622678 | 0.001191742 | 0.025903933 |

4 | 0.054848015 | 0.05347954390 | 0.001368471 | 0.025588683 |

5 | 0.061662700 | 0.06019410003 | 0.001468600 | 0.024397739 |

15 | 0.093802552 | 0.09322206784 | 0.000580484 | 0.006226896 |

In this paper, we have successfully proposed the mixture of new integral transform and homotopy perturbation method for solving discontinued problems arising in nanotechnology. The result shows that the given method is a powerful and efficient technique in finding exact and approximate solutions for nonlinear differential equa- tions. Also, it can be observed that there is good agreement between the results obtained using the present method and the exact solution. It is worth mentioning that the method is capable of reducing the volume of the computational work as compared to the classical methods while still maintaining the high accuracy of the numerical result; the size reduction amounts to an improvement of the performance of the approach.

Kunjan Shah,Twinkle Singh, (2015) The Mixture of New Integral Transform and Homotopy Perturbation Method for Solving Discontinued Problems Arising in Nanotechnology. Open Journal of Applied Sciences,05,688-695. doi: 10.4236/ojapps.2015.511068

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