A novelty Multi-Step Associated with Laplace Transform Semi Analytic Technique for Solving Generalized Non-linear Differential Equations

: In this work, a novel technique to obtain an accurate solutions to nonlinear form by multi-step combination with Laplace-variational approach (MSLVIM) is introduced. Compared with the traditional approach for variational it overcome all difficulties and enable to provide us more an accurate solutions with extended of the convergence region as well as covering to larger intervals which providing us a continuous representation of approximate analytic solution and it give more better information of the solution over the whole time interval. This technique is more easier for obtaining the general Lagrange multiplier with reduces the time and calculations. It converges rapidly to exact formula with simply computable terms with few time. To investigate of this technique, selected examples to show the ability, validity, accurately and effectiveness .


Introduction:
The exact solutions in many problems epically for non-linear cases are difficult to obtain, therefore many approximate analytical methods appeared for this purpose [1][2][3][4] and also [5][6][7][8] as well as [9][10] . In this work, a novel technique, which combining a multistep associated with Laplace transformation based on variational approach (MSLVIM) is presented to obtain an accurate solutions for nonlinear quadratic form. Many researchers recently applied variational approach (VIM) for obtaining the analytic solutions with different types of equations including partial, algebraic, differential, diffusion, delay and integrodifferential equations [11][12][13][14][15] . The application of Laplace with VIM is found to find the approximate solution of telegraph equation 16 . The approximate solution of nonlinear gas dynamics with VIM Equation is conducted 17 . Laplace transformation implemented to make VIM easier for getting the analytic solution 18 . The analytic solution of timefractional Fornberg-Whitham Equation is obtained by VIM with Laplace transform and comparison with Adomian decomposition method is conducted 19 . A multi-step with homotopy is applied to obtain the analytic solution for solving fractional-order model for HIV 20 . Authors in 21 , implemented a multi-step to get the approximate solutions for different types of equations. A Laplace variational iteration approach strategy for solving differential equations is discussed 22 . Homotopy perturbation associated with Laplace transform and comparison with the variational iteration approach is considered 23 . Variational Iteration with the Laplace transform for Modified Fractional Derivatives with Nonsingular Kernel is presented 24 . Variational iteration method associated with the multistage for solving nonlinear system ordinary differential equations is studied 25 .
In this letter, a MSLVIM presents. The mainly thrust for this technique is to construct a correction functional and to overcome all difficulties which face us related to integration when one obtain the Lagrange multiplier and to get an accurate solution for large intervals with more extended of the convergence region which the traditional approach fail normally. It converges rapidly to exact formula with simply computable terms with reduce and few time. As well as covering to larger intervals which providing us a continuous representation of approximate analytic solution therefore it capable to give better information of the solution over results confirm the simplicity, suitability and effectiveness of this technique using only few terms of the iterative scheme. In the next section present Laplace-variational iteration method (LVIM).

LVIM for Nonlinear Differential Equations
Consider the general nonlinear differential equation where L , N are n-th linear and analytic nonlinear terms respectively, () qt is analytic known function.
Taking the Laplace transform to the both sides of Eq.1, one get By using the differential property of the Laplace transform for the linear operator , where () Ps is the Laplace transform to ( ). pt According to the VIM, the correctional function of Eq. 3 can be constructed as

Multi-Step LVIM for Non-linear Differential Equations
Like other analytical methods such as HAM, ADM and DTM, the VIM has some drawbacks. The series solutions are usually valid in any every small region and it converges slowly or completely diverges in some cases. To overcome this problem and shortcoming, multi-step methods presented in many works 1,20,21 .
In this work, a new multi-step technique based on a combination with Laplace transform with standard VIM presents. This technique does not require calculating the integral at each step of the process. Moreover and importantly, the analytic solution is valid for a long time interval.
T be the interval for which it is required to obtain the analytic solution of Eq.

Numerical Simulation
In this section, apply the MSLVIM technique as explains above on nonlinear differential equations to investigate the validation and efficiency for obtaining the analytic solutions.

Example 1
Consider the nonlinear differential equation

Conclusion:
In this letter, a new MSLVIM technique for solving generalized non-linear differential equations is proposed. Selected examples are conducted confirm that this technique an efficiency alternative and in certain cases an enhance and support of the VIM strategy. A multi-Step VIM is a reliable modification of the VIM which more improving an accurate solutions and extended of the convergence region for whole time interval compared to the series solution when the standard VIM failed for the larger time interval. This technique provide instant and visual symbol terms of approximately and analytically solutions for non-linear differential equations. A Laplace transform correction functional presents to enable us for expressing the integral in many cases in the form of detour. Laplace transform made the variational approach easier to tackling. A particularly for obtaining the general Lagrange multiplier which is very hard in some cases of non-linear form. A validity for the proposed technique has been shown successfully by applying it for non-linear form. Results indicate that this technique is effective and applicable to different types of non-linear problems in addition its reliable and promising compared with the existing approaches. Finally, we can use this technique for solving many types of fractional, partial and delay differential equations to get more an accurate results with more extended of the convergence region for future work.