A Novel Third Order Numerical Method for Solving Volterra Integro-Differential Equations

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1 A Novel Third Order Numerical Method for Solving Volterra Integro-Differential Equations arxiv: v1 [math.na] 9 Apr 016 Sachin Bhalekar, Jayvant Patade 1 Department of Mathematics, Shivaji University, Kolhapur , India. sachin.math@yahoo.co.in, sbb maths@unishivaji.ac.in (Sachin Bhalekar, jayvantpatade1195@gmai.com (Jayvant Patade Abstract In this paper we introduce a numerical method for solving nonlinear Volterra integrodifferential equations. In the first step, we apply implicit trapezium rule to discretize the integral in given equation. Further, the Daftardar-Gejji and Jafari technique (DJM is used to find the unknown term on the right side. We derive existence-uniqueness theorem for such equations by using Lipschitz condition. We further present the error, convergence, stability and bifurcation analysis of the proposed method. We solve various types of equations using this method and compare the error with other numerical methods. It is observed that our method is more efficient than other numerical methods. Keywords: Volterra Integro-differential equations, Trapezium rule, Daftardar-Gejji and Jafari method, numerical solution, error, convergence, stability, bifurcation. 1 Introduction The equations involving derivative of a dependent variable are frequently used to model the natural phenomena. The derivative of such function at a pointt 0 can be approximated by using its values in a small neighborhood oft 0. Thus the derivative is a local operator which cannot model the memory and hereditary properties involved in the real world problems. To overcome this drawback, one has to include some nonlocal operator in the given equation. Volterra [1, ] suggested the use of an integral operator (which is nonlocal to model the problems in population dynamics. In his monograph [1], he discussed the theory of integral, integro-differential and functional equations. Existence-uniqueness, stability and applications of integro-differential equations (IDE is presented in a book by Lakshmikantham and Rao [3]. Existence theory of nonlinear IDEs is also discussed in [4]. Recently, Kostic [5] discussed the theory of abstract Volterra integro-differential equations. A-stable linear multi-step methods to solve Volterra IDEs (VIDE are proposed by Matthys in [6]. Brunner presented various numerical methods to solve VIDEs in [7]. Day [8] used trapezoidal rule to devise a numerical method to solve nonlinear VIDEs. Linz [9] derived fourth order numerical methods for such equations. Runge-Kutta method, predictor-corrector method and explicit multistep method are derived for IDEs by Wolfe and Phillips in [10]. Dehghan 1 Corresponding author 1

2 and Salehi [11] developed a numerical scheme based on the moving least square method for these equations. Singular IDEs in generalized Holder spaces are solved by using collocation and mechanical quadrature methods in [1]. Saeedi et al. [13] described the operational Tau method for solving nonlinear VIDEs of the second kind. The stability of numerical methods for VIDEs is studied in [14, 15]. A survey of numerical treatments of VIDEs is taken in [16]. The applications of integro-differential equations can be found in many fields such as mechanics and electromagnetic theory [17], nuclear reactor [18, 3], visco-elasticity [19, 0], heat conduction in materials with memory [1, ] and man-environment epidemics [3]. In the present work we utilize a decomposition method proposed by Daftardar-Gejji and Jafari (DJM [4] to generate a more accurate and faster numerical method for solving nonlinear VIDEs.The paper is organized as follows: The preliminaries are given in section. A new numerical method is presented in Section 3. Analysis of this numerical method is given in Section 4. Section 5 deals with different types of illustrative examples. In Section 6 we provide software package and conclusions are summarized in Section 7. Preliminaries.1 Basic Definitions and Results: In this section, we discuss some basic definitions and results[5, 3, 9]. Definition.1 Lety j be the approximation to the exact valuey(x j obtained by a given method with step-size h. Then a method is said to be convergent if and only if lim y(x j y j 0, j = 1, N. (.1 h 0 Definition. A method is said to be of order p if p is the largest number for which there exists a finite constant C such that Theorem.1 Consider the integral equation y(x j y j Ch p, j = 1, N. (. y(t = f(t+ 0 K(x,t,y(tdt, y( = y 0. (.3 Assume that (i f is continuous in0 x a, (ii K(x, t, y is continuous function for0 t x a and y <, (iii K(x, t, y satisfies a Lipschitz condition K(x,t,y 1 K(x,t,y L y 1 y (.4 for all0 t x a. Then the Eq. (.3 has a unique solution.

3 Theorem. Assume thatf C[I R n,r n ],K C[I I R n,r n ] and s K(t,s,y(s dt N, for s x +a,y Ω = {φ C[I,R n ] : φ( = and φ(x y 0 b}. Then IVP (.3 possesses at least one solution.. Existence and Uniqueness Theorem In this subsection we prove existence and uniqueness theorem, which is generalization of Theorem.1. Theorem.3 Consider the Volterra integro-differential equation y (x = f(x,y(x+ with initial condition y( = y 0. Assume thatf andk are continuous and satisfy Lipschitz condition K(x,t,y(tdt, (.5 f(x,y 1 f(x,y L 1 y 1 y (.6 K(x,t,y 1 K(x,t,y L y 1 y (.7 for every x a, t a, y 1 <, y < and (.5 has unique solution. a > 0. Then the Eq. Proof: Integrating Eq.(.5 and usingy( = y 0, we get where y(x = y 0 + y(x = y 0 + y(x = y 0 + f(x,y(xdx+ ( f(x,y(x+ x 0 x G(x,y(x = f(x,y(x+ ( K(x,t,y(tdt dx K(x,t,y(tdt dx x 0 x G(x,y(xdx, (.8 K(x,t,y(tdt. The Eq. (.8 is of the form (.3. We have following observations (iy 0 is continuous because it is constant, (ii KernelGis continuous for0 x a, becausef andk are continuous in the same domain, 3

4 (iii G(x,y 1 G(x,y = f(x,y 1 (x+ K(x,t,y 1 (tdt f(x,y (x K(x,t,y (tdt f(x,y 1 (x f(x,y (x + K(x,t,y 1 (tdt K(x,t,y (tdt L 1 y 1 y +al y 1 y ( x a (L 1 +al y 1 y G satisfy a Lipschitz condition. All the conditions of the Theorem.1 are satisfies. Hence the Eq.(.5 has unique solution..3 Daftardar-Gejji and Jafari Method A new iterative method was introduced by Daftardar-Gejji and Jafari (DJM [4] in 006 for solving nonlinear functional equations. The DJM has been used to solve a variety of equations such as fractional differential equations [6], partial differential equations [7], boundary value problems [8, 9], evolution equations [30] and system of nonlinear functional equations [31]. The method is successfully employed to solve Newell-Whitehead-Segel equation [3], fractional-order logistic equation [33] and some nonlinear dynamical systems [34] also. Recently DJM has been used to generate new numerical methods [35, 36, 37] for solving differential equations. In this section we describe DJM which is very useful for solving the equations of the form u = g +L(u+N(u, (.9 where g is a given function, L and N are linear and nonlinear operators respectively. DJM provides the solution to Eq.(.9 in the form of series u = u i. (.10 where u 0 = g, i=0 u m+1 = L(u m +G m, m = 0,1,,, (.11 ( m ( m 1 and G m = N u m N u m,m 1. j=0 4 j=0

5 Thek-term approximate solution is given by k 1 u = u i (.1 for suitable integerk. The following convergence results for DJM are described in [38]. i=0 Theorem.4 If N is C ( in a neighborhood of u 0 and N (n (u 0 L, for any n and for some real L > 0 and u i M < 1, i = 1,,, then the series e n=0 G n is absolutely convergent ton and moreover, G n LM n e n 1 (e 1, n = 1,,. Theorem.5 If N is C ( and N (n (u 0 M e 1, n, then the series n=0 G n is absolutely convergent to N. 3 Numerical Method In this section we present a numerical method based on DJM to solve Volterra integro-differential equation. Integrating Eq. (.5 from x = x j to x = x j +h, we get y(x j +h = y(x j + j +h x j f(x,y(xdx+ j +h x j K(x,t,y(tdtdx (3.1 Applying trapezium formula [39] to evaluate integrals on right of Eq. (3.1, we get y(x j +h = y(x j + h f(x j,y j 4 (K(x j,,y 0 +K(x j,x j,y j +K(x j+1,,y 0 K(x j,x i,y i + K(x j+1,x i,y i + h f(x j+1,y j+1 4 K(x j+1,x j+1,y j+1 +O(h 3. (3. Ify j is an approximation toy(x j, then approximate solution is given by y j+1 = y j + h f(x j,y j 4 (K(x j,,y 0 +K(x j,x j,y j +K(x j+1,,y 0 K(x j,x i,y i + K(x j+1,x i,y i + h f(x j+1,y j+1 4 K(x j+1,x j+1,y j+1. (3.3 5

6 Equation (3.3 is of the form (.9, where u = y j+1, g = y j + h f(x j,y j 4 (K(x j,,y 0 +K(x j,x j,y j +K(x j+1,,y 0 K(x j,x i,y i + K(x j+1,x i,y i N(u = h f(x j+1,y j+1 4 K(x j+1,x j+1,y j+1. Applying DJM to equation (3.3, we obtain 3-term solution as That is u = u 0 +u 1 +u = u 0 +N(u 0 +N(u 0 +u 1 N(u 0 = u 0 +N(u 0 +u 1 = u 0 +N(u 0 +N(u 0. y j+1 = y j + h f(x j,y j 4 (K(x j,,y 0 +K(x j,x j,y j +K(x j+1,,y 0 K(x j,x i,y i + K(x j+1,x i,y i +N (y j f(x j,y j 4 (K(x j,,y 0 +K(x j,x j,y j +K(x j+1,,y 0 ( K(x j,x i,y i + K(x j+1,x i,y i +N y j + h f(x j,y j 4 (K(x j,,y 0 +K(x j,x j,y j +K(x j+1,,y 0 K(x j,x i,y i + K(x j+1,x i,y i. (3.4 6

7 or y j+1 = y j + h f(x j,y j 4 (K(x j,,y 0 +K(x j,x j,y j +K(x j+1,,y 0 K(x j,x i,y i + K(x j+1,x i,y i + h ( f x j+1,y j + h f(x j,y j 4 (K(x j,,y 0 +K(x j,x j,y j +K(x j+1,,y 0 K(x j,x i,y i + K(x j+1,x i,y i ( f x j+1,y j + h f(x j,y j 4 (K(x j,,y 0 +K(x j,x j,y j +K(x j+1,,y 0 K(x j,x i,y i + K(x j+1,x i,y i + (x h 4 K j+1,x j+1,y j f(x j,y j 4 (K(x j,,y 0 +K(x j,x j,y j +K(x j+1,,y 0 K(x j,x i,y i + K(x j+1,x i,y i + (x h 4 K j+1,x j+1,y j f(x j,y j 4 (K(x j,,y 0 +K(x j,x j,y j +K(x j+1,,y 0 K(x j,x i,y i + K(x j+1,x i,y i ( f x j+1,y j + h f(x j,y j 4 (K(x j,,y 0 +K(x j,x j,y j +K(x j+1,,y 0 K(x j,x i,y i + K(x j+1,x i,y i + (x h 4 K j+1,x j+1,y j f(x j,y j 4 (K(x j,,y 0 +K(x j,x j,y j +K(x j+1,,y 0 K(x j,x i,y i + K(x j+1,x i,y i (3.5 7

8 If we set M 1 = y j + h f(x j,y j 4 (K(x j,,y 0 +K(x j,x j,y j +K(x j+1,,y 0 K(x j,x i,y i + K(x j+1,x i,y i, (3.6 M = M 1 + h f(x j+1,m 1 4 K(x j+1,x j+1,m 1 (3.7 then equation (3.5 becomes y j+1 = M 1 + h f(x j+1,m 4 K(x j+1,x j+1,m. (3.8 4 Analysis of Numerical Method 4.1 Error Analysis Theorem 4.1 The numerical method (3.8 is of third order. Proof: Supposey j+1 is an approximation to y(x j+1. By using Eq.(3. and (3.8, we obtain y(x j+1 y j+1 = h f(x j+1,y j+1 4 K(x j+1,x j+1,y j+1 +O(h 3 Using Eq.(3.7 and (3.8, we get ( h y(x j+1 y j+1 L 1 4 L h f(x j+1,m h 4 K(x j+1,x j+1,m h f(x j+1,y j+1 f(x j+1,m 4 K(x j+1,x j+1,y j+1 K(x j+1,x j+1,m +O(h 3 ( h L 1 4 L y j+1 M +O(h 3. M 1 + h f(x j+1,m 4 K(x j+1,x j+1,m M 1 h f(x j+1,m 1 h 4 K(x j+1,x j+1,m 1 +O(h 3 ( ( h L 1 4 L h f(x j+1,m h f(x j+1,m 1 + h 4 K(x j+1,x j+1,m h 4 K(x j+1,x j+1,m 1 +O(h 3. 8

9 Using Eq.(3.6 and (3.7,we get ( h y(x j+1 y j+1 L 1 4 L h f(x j+1,m 1 4 K(x j+1,x j+1,m +O(h 3 ( (1 h 3 L 1 + h 4 L 1 f(x j+1,m 1 + h 4 K(x j+1,x j+1,m +O(h. The numerical method (3.8 is of third order. Corollary 1 The numerical method (3.8 is convergent. Proof: By Theorem 4.1 and definition (.1 the numerical method (3.8 is convergent. 4. Stability Analysis of Numerical Method Consider the test equation y = αy +β y(tdt. (4.1 Applying numerical method (3.8 to this equation, we get where y j+1 = M 1 + hα M β 4 M, (4. M 1 = h j 1 β y 0 +h β y i + ( M = 1+ hα β 4 ( 1+ hα + 3h β y j, (4.3 4 M 1. (4.4 If we set u = hα and v = h β then Eq. (4. can be written as ( v y j+1 = + uv 4 + u v 8 + v 8 + uv 8 + v3 y 0 3 ( + v + uv + u v j v 4 + uv 4 + v (1+u+ u + u3 3uv +v u v 16 + v 4 + 7uv 3 + 3v3 64 y i y j. (4.5 Simplifying Eq.(4.5, we get y j = b 1 y j 1 +b y j, (4.6 9

10 where and b 1 = +u+ u + u3 3uv +v u v 16 + v 4 + 7uv 3 + 3v3 64, (4.7 b = 1 u u u3 8 uv 4 u v 16 + uv 3 + v3 64. (4.8 The characteristic equation [40] of the difference Eq. (4.6 is The characteristic roots are given by r 1 = b 1 + b 1 +4b r b 1 r b = 0. (4.9, r = b 1 b 1 +4b. (4.10 The zero solution of system (4.6 is asymptotically stable if r 1 < 1 and r < 1. Thus the stability region of numerical method (3.8 is given by r 1 < 1 and r < 1 as shown in Fig.1. Fig.1 Stability region for numerical method (3.8 10

11 4.3 Bifurcations Analysis We discuss bifurcation analysis for the following test equation given in [41] y = e α(x t y(tdt. (4.11 As discussed in [41], Eq.(4.11 with convolution kernel and with exponentially fading memory is practically important. The qualitative behavior of the Eq.(4.11 is described as below [41]: Table 1: Bifurcation in IDE (4.11. Region Qualitative behavior α Solutions are converging to zero without oscillation 0 < α < Solutions are damped oscillations converging to zero < α < 0 Divergent solutions exhibiting unbounded oscillations α Solutions become unbounded without oscillations Now we apply the numerical scheme (3.8 to Eq.(4.11 and discuss the bifurcations. where j 1 y j+1 = a 0 e αhj y 0 a 1 e αh(j 1 y i +a y j, (4.1 Eq. (4.11 can be written as where a 0 = (1 h h h4 (1+e αh, 16 a 1 = (1 h h 4 + h4 (1+e αh, 16 a = (1 (1 h 4 + h4 h 16 4 e αh. y j+ b 1 y j+1 +b y j = 0, (4.13 b 1 = e αh +a, b = (a 1 +a e αh. The characteristic equation of the difference Eq. (4.13 is r b 1 r +b = 0. (4.14 The roots of this equation are r = b 1 ± b 1 4b. (

12 Theorem 4. [40] All solutions of the difference equation y j+ +p 1 y j+1 +p y j = 0, (4.16 converge to zero (i.e.,the zero solution is asymptotically stable if and only ifmax{ r 1, r } < 1, wherer 1,r are the roots of the characteristic equationr +p 1 r +p = 0 of Eq. (4.16. Define α 0 = 1 ( h Log 1+ h6, 64 ( α 1 = 1 ( h h Log 3h 4 +8h 6 h 8 +8h 104+8h 6 h 8 ( 4+h and (16 4h +h 4 ( α = 1 h Log ( h +3h 4 8h 6 +h 8 +8h 104+8h 6 h 8 ( 4+h. (16 4h +h 4 Using Theorem 4. we have the following observations: Case 1: The characteristics roots of the Eq. (4.15 are real and distinct r 1 = 1 (1+e hα h 1 e hα h + h e hα h 4 h e hα h 6 + ( ( 4e hα 1+ h6 + 1+e 64 hα h 1 e hα h + h e hα h 4 h e hα h 6 r = 1 (1+e hα h 1 e hα h + h e hα h 4 h e hα h 6 ( ( 4e hα 1+ h6 + 1+e 64 hα h 1 e hα h + h e hα h 4 h e hα h 6, when Case 1.1: α > α 1. In this case the characteristic roots are of magnitude less than unity (cf. Region I in Fig.. Therefore, all the solutions of difference Eq. (4.13 converge to zero without oscillations. Case 1.: α < α. In this case the characteristic roots are of magnitude greater than unity. This is shown in Fig. as Region IV. In this region, all the solutions will tend to infinity without oscillations. Case : The characteristics roots are real and equal r = r 1 = r = 1 (1+e hα h 1 e hα h + h e hα h 4 h e hα h 6, 1

13 when Case.1: α = α 1. In this case the characteristic root r (0,1. Hence the solutions of Eq. (4.13 tend to zero without oscillations as in case 1.1. This case is shown in Fig. as boundary line of Region I and Region II. Case.: α = α. In this case the characteristic root r > 1 and the solutions are unbounded (cf. boundary line of Region III and Region IV in Fig.. Case 3: The characteristics roots are complex r 1 = 1 (1+e hα h 1 e hα h + h e hα h 4 h e hα h 6 +i 4e hα ( 1+ h6 64 ( 1+e hα h 1 e hα h + h e hα h 4 h e hα h 6 r = 1 (1+e hα h 1 e hα h + h e hα h 4 h e hα h 6 ( ( i 4e hα 1+ h6 1+e 64 hα h 1 e hα h + h e hα h 4 h e hα h 6, when α < α < α 1. Note that r 1 = r. Case 3.1: If α 0 < α < α 1 then r 1 < 1. The solutions of Eq. (4.13 are oscillatory and converge to zero (Region II in Fig.. Case 3.: If α = α 0 then r 1 = 1 and bifurcation occurs. This is shown by the boundary line of Region II and Region III in Fig.. Case 3.3: Ifα < α < α 0 then r 1 > 1. This implies that the solutions of Eq. (4.13 are oscillatory and diverge to infinity (Region III in Fig.. 13

14 Fig.: Bifurcation digram for Eq. (4.13. Note: It is clear from Table 1 and Fig. that the bifurcations in numerical scheme (4.13 coincide with those in IDE (4.11 forh=0. 5 Illustrative examples Example 5.1 [8, 10] Consider the Volterra integro-differential equation y (x = 1+x y(x+ x(1+xe t(x t y(tdt, y(0 = 1. (5.1 The exact solution of Eq.(5.1 is y(x = e x. We compare errors obtained using different numerical methods given in [8, 10], viz. third order Runge-Kutta method, predictor-corrector method, multi-step method and Day method with our method for h = 0.1 and h = 0.05 in Table and Table 3 respectively. It is observed that the error in our method is very less as compared with other numerical methods. Note that the error values in Table and Table 3 for all other methods are taken from [8, 10]. 14

15 Table : Comparison of numerical results of example 6.1 with h=0.1 x Our method Runge-Kutta Predictor- Multi-step Day method O(h 3 O(h 3 Corrector O(h 3 O(h Table 3: Comparison of numerical results of example 6.1 with h=0.05 x Our method Runge-Kutta Predictor- Multi-step Day method O(h 3 O(h 3 Corrector O(h 3 O(h Example 5. [11] Consider the nonlinear Volterra integro-differential equation y (x = x 1 sin(x4 + 0 x tcos(x y(tdt, y(0 = 0. (5. The exact solution of Eq.(5. isy(x = x. We compare the maximum absolute errors obtained using meshless method given in [11] and CPU times used for different values of n in the interval [0,1]. It is observed that the error in our method is very less as compared with linear meshless method. Further, our method is more time efficient than both linear and quadratic meshless method. 15

16 Table 4: Maximum absolute errors and CPU times used for different values of n. n Our method Time Linear Meshless Time Quadratic Meshless Time method method Example 5.3 Consider the VIDE [11] y (x = 1 x + xe x + 0 xte y (t dt, y(0 = 0. (5.3 The exact solution of Eq.(5.3 isy(x = x. We compare the maximum absolute errors obtained using meshless method given in [11] and CPU time used for different values of n in the interval [0, 1]. It is observed that our method is more accurate as compared with linear meshless and more time efficient than both linear and quadratic meshless method. Table 5: Maximum absolute errors and CPU times used for different values of n. n Our method Time Linear Meshless Time Quadratic Meshless Time method method We compare our solution with exact solution for h = 0.01 in Fig.3. It is observed that our solution coincides with exact solution. 16

17 Example 5.4 Consider the VIDE Fig.3: Comparison of solutions of Eq.(5.3 forh = y (x = 1+ 0 e t y (tdt, y(0 = 1. (5.4 The exact solution of Eq.(5.4 is y(x = e x. We compare our solution with exact solution for h = 0.01 in Fig.4. It is observed that our solution is well in agreement with exact solution. 6 Software Package Fig.4: Comparison of solutions of Eq.(5.4 forh = In this section we provide software package based on Mathematica-10 to solve VIDEs using numerical method. This software is used for solving VIDEs of the form (5.3. One has to provide the value of f(x,y and K(x,t,y in first two windows, initial condition y 0, step size h and number of 17

18 stepsnin third, fourth and fifth windows respectively. The last window gives required solution curve. We solve Eq.(5. using this software package as shown in Fig.5. f(x,y= x- Sin x4 NNM1D K(x,t,y= tx Cos x y y 0 = 0 h= Conclusions n= y x Fig.5: Software Package A new method proposed in this paper is a combination of trapezium rule and DJM. The DJM is used to find the approximation to the unknown term on the right side of the difference formula obtained by using trapezium rule. This combination is proved to be very efficient. The error in this third order method is very less as compared to other similar methods. However, the time taken for the computation was very less. The stability and bifurcation analysis also show the power of this method. An ample number of examples show the wide range of applicability of this method. Acknowledgements: S. Bhalekar acknowledges CSIR, New Delhi for funding through Research Project [5(045/15/EMR- II]. 18

19 References [1] V. Volterra, Theory of Functionals and of Integral and Integro-differential Equations, Dover, New York, [] V. Volterra, Lecons sur la theorie mathematique de la luttle pour la vie Gauthier - Villars, Paris, [3] V. Lakshmikantham and M. R. M. Rao, Theory of Integro-Differential Equations, Gordon and Breach Science Publishers, Amsterdam, [4] D. O Regan and M. Meehan, Existence Theory for Nonlinear Integral and Integrodifferential Equations, Kluwer Academic, Dordrecht, [5] M. Kostic, Abstract Volterra Integro-Differential Equations, CRC Press, Boca Raton, 015. [6] J. Matthys, A-Stable Linear Multistep Methods for Volterra Integro-Differential Equations, Numer. Math. 7, ( [7] H. Brunner, High-order methods for the numerical solution of Volterra integro-differential equations, J. Comput. Appl. Math., 15 ( [8] J. T. Day, Note on the numerical solution of integro-differential equations. The Computer Journal, 9(4, ( [9] P. Linz, A method for solving nonlinear Volterra integral equations of the second kind, Mathematics of Computation, 3(107 ( [10] M. A. Wolfe, and G. M. Phillips, Some methods for the solution of non-singular Volterra integro-differential equations, The Computer Journal, 11(3, ( [11] M. Dehghan, R. Salehi, The numerical solution of the non-linear integro-differential equations based on the meshless method, J. Comput. Appl. Math., 01, 36(9, [1] I. Caraus, F. Al Faqih Approximate solution of singular integro-differential equations in generalized Holder spaces, Numer. Algor., 45(1-4 ( [13] L. Saeedi, A. Tari and S. H. M. Masuleh, Numerical solution of a class of the nonlinear Volterra integro-differential equations, J. Appl. Math. & Informatics, 31 ( [14] H. Brunner and J. D. Lambert, Stability of numerical methods for Volterra integrodifferential equations, Computing 1 ( [15] C. T. H. Baker, A. Makroglou and E. Short, Regions of stability in the numerical treatment of Volterra integro-differential equations, SIAM J. Numer. Anal., 16(6 (

20 [16] H. Brunner, A survey of recent advances in the numerical treatment of Volterra integral and integro-differential equations, J. Comput. Appl. Math., 8 ( [17] F. Bloom, Ill-posed Problems for Integro-differential Equations in Mechanics and Electromagnetic Theory, SIAM, Philadelphia, [18] W. E. Kastenberg and P.L.Chambre, On the stability of nonlinear space dependent reactor kinetics, Nucl. Sci. Eng. 31(1968, [19] MacCamy, R.C., A model for one dimensional nonlinear viscoelasticity, Q. Appl. Math. 35 ( [0] A. Narain and D. Joseph, Classification of linear viscoelastic solids based on a failure criterion, J. Elasticity, 14(1984, [1] W. Hrusa, Global existence and asymptotic stability for semilinear hyperbolic Volterra equation with large initial data, SI AM J. Math. Anal. 16(1985, [] J. Pruss, Positivity and regularity of hyperbolic Volterra equations in Banach spaces, Math. Ann. 79(1987, [3] V. Capasso, Asymptotic stability for an integrodifferential reaction diffusion system, J. Math. Anal. Appl. 103(1984, [4] V. Daftardar-Gejji, H. Jafari, An iterative method for solving non linear functional equations, J. Math. Anal. Appl. 316 ( [5] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, in: North-Holland Mathematics Studies, vol. 04, 006. [6] V. Daftardar-Gejji, S. Bhalekar, Solving fractional diffusion-wave equations using the New Iterative Method, Frac. Calc. Appl. Anal. 11 ( [7] S. Bhalekar, V. Daftardar-Gejji, New iterative method: Application to partial differential equations, Appl. Math. Comput. 03 ( [8] V. Daftardar-Gejji, S. Bhalekar, Solving fractional boundary value problems with Dirichlet boundary conditions, Comput. Math. Appl. 59 ( [9] S. T. Mohyud-Din, A. Yildirim, M. M. Hosseini, An iterative algorithm for fifth-order boundary value problems, World Appl. Sci. J. 8 ( [30] S. Bhalekar, V. Daftardar-Gejji, Solving evolution equations using a new iterative method, Numer. Methods Partial Differential Equations 6 (

21 [31] S. Bhalekar, V. Daftardar-Gejji, Solving a system of nonlinear functional equations using revised new iterative method, Int. J. Computa. Math. Sci. 6 ( [3] J. Patade, S. Bhalekar, Approximate analytical solutions of Newell-Whitehead-Segel equation using a new iterative method, World J. Modell. Simul. 11 ( ( [33] S. Bhalekar, V. Daftardar-Gejji, Solving fractional order logistic equation using a new iterative method, Int. J. Differ. Equ. 010 (01 (Art. ID [34] S. Bhalekar, V. Daftardar-Gejji, Numeric-analytic solutions of dynamical systems using a new iterative method, J. Appl. Nonlin. Dyn. 1 ( [35] V. Daftardar-Gejji, Y. Sukale, S Bhalekar, A new predictor-corrector method for fractional differential equations, Appl. Math. Comput. 44 ( [36] V. Daftardar-Gejji, Y. Sukale, S Bhalekar, Solving fractional delay differential equations: A new approach, Fract. Calc. Appl. Anal. 16 ( [37] J. Patade, S. Bhalekar, A new numerical method based on Daftardar-Gejji and Jafari technique for solving differential Equations, World J. Modell. Simul. 11 ( [38] S. Bhalekar, V. Daftardar-Gejji, Convergence of the new iterative method, Int. J. Differ. Equ. 011 (011. [39] M. K. Jain, Numerical methods for scientific and engineering computation, New Age International, New Delhi 01. [40] S. Elaydi, An introduction to difference equations, Springer Science & Business Media, 005. [41] J. T. Edwards, N. J. Ford, J. A. Roberts, Bifurcations in numerical methods for Volterra integro-differential equations, Int. J. bifurcation and chaos. 13(11(