**The Combined Laplace-Adomian Method for Weakly-Singular Kernel Volterra Integro-Differential Equations**

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The Combined Laplace-Adomian Method for Weakly-Singular Kernel Volterra Integro-Differential Equations

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Abstraction: In this paper, the combined method of the Laplace Transform Method and the Adomian Decomposition Method is used to work out the weakly-singular Volterra integro-differential equations. In order to research the rapid decay of the weakly-singular Volterra integro-differential equations, the pade estimate is used. The consequences prove that the method is a powerful and efficient technique in order to set up series solutions for the remarkable sort of differential equations.

**Keywords:**Volterra integro-differential equations, Laplace Transform Method, Adomian Decomposition Method, Linear and nonlinear weakly-singular equations

**1. Introduction**

The weakly-singular meat Volterra Integro differential equation is written as:

Where,is the unknown map andis the meat of the built-in equation. The mapsandare normally assumed to be uninterrupted or square built-in can be obtain on.

The first order linear and nonlinear decrepit remarkable Volterra integro differential equations are whenand 2nd order additive weakly Voltera integro differential equation are when. The integrand is weakly-singular and is uninterrupted at the remarkable point. Kernelis remarkable at.

Eq. ( 1.1 ) is of import equation in many physical applications such as neutron diffusion and biological species coexisting together with increasing and diminishing rates of generating.

Wazwaz studied nonlinear Volterra integro–differential equations by uniting the Laplace transform–Adomian decomposition method [ 1 ] . Brunner has obtained numerical solution of nonlinear Volterra integro-differential equations [ 2 ] . Contea and Preteb used Fast collocation methods for Volterra built-in equations of whirl type [ 3 ] . The application of spectral Jacobi-collocation methods to a certain category of decrepit remarkable Volterra built-in equations is presented by Ma et Al. [ 4 ] . A numerical solution of decrepit remarkable Volterra built-in equations including the Abels equations by the 2nd Chebyshev ripple method is presented by Zhuand wang [ 5 ] . Yi and Huang [ 6 ] presented CAS ripple method for work outing the fractional integro-differential equation with a decrepit remarkable meat. Bernstein series solution of a category of additive integro-differential equations with decrepit remarkable meat is presented by IAYiket Al. [ 7 ] . Application of the homotopy disturbance method for nonlinear differential equations is presented by Nourazar et Al. [ 8-10 ] .

In the present research work, a combined signifier of the Laplace transform method with the Adomian decomposition method is used in order to set up series solutions for the weakly-singular meats Volterra integro-differential equations.

The thought of combined Laplace-Adomian method is presented in Section 2. Application of the combined Laplace-Adomian method to work out the weakly-singular meats Volterra integro-differential equations is presented in Section 3.

**2. The combined Laplace-Adomian method**

To exemplify the basic thought of the combined Laplace-Adomian method, see the nonlinear Volterra built-in equation:

At first we apply the Laplace transform to the both side of Eq. ( 2.1 ) ,

For managing and turn toing the nonlinear term, the adomian decomposition and the adomian multinomials can be used. At first we represent the additive termat the left side by an infinite series of constituents obtained by:

And likewise,

where the constituentswill be determined recursively. The nonolinear termat the right side of Eq. ( 2.2 ) will be represent by an infinite series of Adomian multinomialsin the signifier as,

Whereare defined by,

We can measure the alleged adomian multinomialsfor all signifiers of nonlinearity. By presuming that the nonlinear map isthe adomian multinomials are achieved by,

Substituting Eq. ( 2.4 ) and Eq. ( 2.5 ) in to Eq. ( 2.2 ) we have,

By utilizing the adomian decomposition method the recursive relation can be shown as,

By using reverse Laplace transform to the first portion of Eq. ( 2.9 ) , we can obtain. Then,can be defined by utilizing.Besides, can be found by utilizing. the finding ofleads us to cipheringthat will let us to findand so on. This computations lead to the complete finding of the constituent of.

Obtaining the series solution can take us to happening exact solution of job if such a solution exists. Otherwise this series solution can be tested by pade estimates.

**3. The Volterra integro-differential equation**

The second-order Volterra Integro-differential equation is considered in order to show capableness and effectivity of the algorithm to work out the equation.

For, the Eq. ( 3.1 ) is written as:

Where

Using the Laplace transform to both side of Eq. ( 3.2 ) we have,

Therefore we can compose Eq. ( 14 ) as,

Or equivalently,

The adomian decomposition method admit to utilize of,

Whereare the adomian multinomials for the nonlinear term.

The adomian method assumes that the additive termcan be determined by the series,

The nonlinear termcan be showed by the series,

Some of the adomian multinomials forare as,

By utilizing the return relation in the Eq. ( 3.6 ) and utilizing the reverse Laplace transform ofwe have,

.

.

.

The series solution is obtained by,

It can be concluded that this series solution admits the first status

The mathematical construction of u ( ten ) can be tested by pade estimates. Besides, it can be shown that pade estimates give consequences smaller mistake bounds than estimate by multinomials.

For obtaining pade estimates we foremost sethence we have,

By utilizing maple bundle the [ 3,3 ] and [ 4,4 ] pade estimates are given as,

The graphs of the Pade approximants for [ 3,3 ] and [ 4,4 ] is shown in Fig. 1, where the upper graph is for [ 3*,*3 ] .

The graph explores the rapid decay of the weakly-singular Volterra Integro equations. Besides, the 2nd statusis justified.

**Fig. 1.**The pade estimate for [ 3,3 ] and [ 4,4 ] of,

**4. Decision**

In the present research work, developed method of combined signifier of the Laplace transform method with the Adomian decomposition method is used to set up series solutions of the weakly-singular meats Volterra integro-differential equations. Besides, the rapid convergence to the exact solutions is numerically demonstrated by pade estimates. The consequences present cogency and great potency of the method as a powerful algorithm in order to build the series solution of remarkable meats differential equations.

**5. Mentions**

[ 1 ] A. M. Wazwaz, The combined Laplace transform–Adomian decomposition method for managing nonlinear Volterra integro–differential equations, Applied Mathematics and Computation, 2010, 216 ( 4 ) , 1304-1309.

[ 2 ] H. Brunner, On the numerical solution of nonlinear Volterra integro-differential equations, BIT Numerical Mathematics, 1973, 13 ( 4 ) , 381–390.

[ 3 ] D. Conte, I. D. Prete, Fast collocation methods for Volterra built-in equations of whirl type, Journal of computational and applied mathematics, 2006, 196 ( 2 ) , 652-663.

[ 4 ] X. Ma, C. Huang, X. Niu, Convergence analysis of spectral collocation methods for a category of decrepit remarkable Volterra built-in equations, Applied Mathematics and Computation, 2015, 250, 131-144.

[ 5 ] L. Zhu, Y. Wang, Numerical solutions of Volterra built-in equation with decrepit remarkable meats utilizing SCW method, Applied Mathematics and Computation, 2015, 260, 63-70.

[ 6 ] M. Yi, J. Huang, CAS ripple method for work outing the fractional integro-differential equation with a decrepit singular meat, International Journal of Computer Mathematics ahead-of-print, 2014, 1-14.

[ 7 ] O. R. IAYik, S. Mehmet, G. Zekeriya, Bernstein series solution of a category of additive integro-differential equations with decrepit remarkable meat, Applied Mathematics and Computation, 2011, 217 ( 16 ) , 7009-7020.

[ 8 ] S. S. Nourazar, M. Soori, A. Nazari-Golshan, On The Exact Solution of Newell-Whitehead-Segel Equation Using the Homotopy Perturbation Method, Australian Journal of Basic and Applied Sciences, 2011, 5 ( 8 ) , 1400-1411.

[ 9 ] S. S. Nourazar, M. Soori, A. Nazari-Golshan, On the exact solution of Burgers-Huxley equation utilizing the homotopy disturbance method, Journal of Applied Mathematics and Physics, 2015, 3 ( 3 ) , 285-294.

[ 10 ] S. S. Nourazar, M. Soori, A. Nazari-Golshan, On the Homotopy Perturbation Method for the Exact Solution of Fitzhugh–Nagumo Equation, International Journal of Mathematics and Computation, 2015, 27 ( 1 ) , 32-43.

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