Laplace Adomian Decomposition Method for Solving the Nonlinear Volterra Integral Equation with Weakly Kernels

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1 Studie in Nonlinear Science (4): 9-4, ISSN -9 IDOSI Publication, Laplace Adomian Decompoition Method for Solving the Nonlinear Volterra Integral Equation with Weakly Kernel F.A. Hendi Department of Mathematic Faculty of Science, King Abdul Aziz Univerity, Jeddah, K.S.A. Abtract: A computational method (The Comb ined Laplace Adomian decompoition method) applied for olving linear and nonlinear Volterra integral equation with weakly kernel, we dicu ome eample, when the kernel take a Carleman and logarithmic form to demontrate the high accuracy and implicity of our method.a comparion with the Toeplitz matri method i made. MSC: (45D5) Key word: Adomian decompoition method nonlinear volterra integral equation laplace tranform carleman and logarithmic kernel INTRODUCTION In practical application one frequently encounter the Volterra integral equation of the econd kind with a weakly ingular kernel of the form y() = g() +λ p(,t)k(,y(t))dt () a where the kernel p(,t) i weakly ingular and the given function g() and k(,y(t)) are aumed to be ufficiently mooth in order to guarantee the eitence and uniquene of a olution y C[a,], [-]. Typical form of p(,t) are (i) p(,t) = t, <α< (ii) p(,t) = ln t For volterra equation with bounded kernel, the moothne of the kernel and of the forcing function g() determine the moothne of the olution on the cloed interval [a,] with >a. If we allow weakly ingular kernel, then the reulting olution are typically non-mooth at the initial point of the interval of integration, where their derivative become unbounded. Some reult concerning the behavior of the eact olution of equation of type () are given in [4, 5]. It i well known that the nonlinear Volterra integral equation () i uually handled by many method, uch a Toeplitz matri method [6] and other. The traditional Laplace method by itelf cannot be ued in thi cae becaue of the nonlinearity of thi equation, then from [7] there eit a method combining the Laplace tranform method with Adomian decompoition method for analytic treatment of nonlinear ingular integral equation that decribe heat tranfer. If p(,t) of() i a difference kernel p(-t). The nonlinear Volterra integral equation () can be epreed a: y() = g() +λ p( t)k(y(t))dt () o it i poible to overcome thi difficulty of nonlinearity by the powerful Adomian decompoition method. Correponding Author: F.A. Hendi, Department of Mathematic, Faculty of Science, King Abdul Aziz Univerity, Jeddah, K.S.A. 9

2 Studie in Nonlinear Sci., (4): 9-4, In thi paper we conider the nonlinear Volterra integral equation with weakly kernel of the form (i) and (ii). The combined algorithm i capable of handling the two type ingular Volterra integral equation, then we compare the reult with the numerical olution for the ingular Volterra integral equation by uing Toeplitz matri method [6]. CARLEMAN KERNEL We aume that the kernel p(,t) of () take the form (i), then nonlinear Volterra integral equation() can be epreed a: Applying the Laplace tranform to both ide of () give: y() = g() +λ t k(y(t))dt () { } { } Y() = G() +λl t L k(y(t)) (4) The Adomian decompoition method can be ued to handle (4). We repreent the linear term Y() by So Y() = Y() (5) n= n= n y() = y() (6) n The nonlinear term k(y(t)) will be repreented by the Adomian polynomial A n in the form: where A n, n are given by k(y()) = A() (7) n= n n i n n i λ= n! d i= n d A = [ k( λ y)],n =,,,... (8) λ where the o called Adomian polynomial A n can be evaluated for all form of nonlinearity. Subtituting (5) and (6) into (4) lead to { } The Adomian decompoition method introduce the recurive relation Yn = G() +λ L t L A() n (9) n= n= { } Y() = G(),Y k () α + =λ L A k(),k () Appling the invere Laplace tranform to the firt part of (9) give y (), that will define A. Uing A () will enable u to evaluate y (). A k () are the Adomian for the nonlinear term. The Adomian decompoition method aume that the linear term y() can be decompoed by the erie. The obtained erie olution may converge to an eact olution if uch a olution eit. Eample (Linear cae) [6]: Conider the nonlinear volterra integral equation with Carleman kernel

3 Studie in Nonlinear Sci., (4): 9-4, Table : y E y LA E LA y TM E TM.7 4.9E- 4.94E- 4.E E-.95E-.56.6E-.585E-.848E-.548E-.58E E-.775E-.58E E- 4.89E E E-.968E E-.64E E-.964E- 4.8E E-.4E-5..E-.66E-.658E E-.59E E E E E- 5.49E-6.6.6E-.649E E E-.58E-6.4.6E-.66E-.55E E- 8.6E-7. 4.E-4 4.8E-4.78E E-4.55E-7 () y() = +λ t k(y(t))dt where the eact olution i y() =, α = /, λ =.. By taking Laplace tranform we obtain Γ[ ] Y() = +λ Y[] () The Adomian decompoition method introduce the recurive relation Γ[ ] Y() =,Y k+ () =λ Y(),k k () o that, taking the invere Laplace tranform of both ide of relation () give 8 Γ = = =λ n n= y() y(),y(),y() Γ ( Γ ) λ y() = λ,y() = [ ( Γ ),y y + y + y Γ 4 We denote by y E to the eact olution and y LA, y TM to Laplace Adomian and Toeplitz matri olution.a comparion of the reult hown in Table. Eample (Nonlinear cae) [8] 4 y() = ( ) + ( t) y(t)dt (4) where the eact olution y() = /. By taking Laplace tranform we obtain π( + ) π Y() = + L 5 { y[t] } (5)

4 Studie in Nonlinear Sci., (4): 9-4, Table : y E y LA E LA E E E E E E E E E E-7 The Adomian decompoition method admit the ue of where the Adomian polynomial are: π( + ) π Y(),Y () L A (),k 5 k+ k = = { } (6) A() = y,a() = y y,a () = ( y + 4y y ) (7) and o on for other Adomian polynomial. Uing the recurrence relation (7), we find y() = ( ),y() = ( ) 5 8 = y() ( ),y y y y 7445 (8) Comparing the reult with the eact olution given in Table. LOGARITHMIC KERNEL If the kernel p(,t) of () take the form (ii), then nonlinear Volterra integral equation() can be epreed a: Applying the Laplace tranform to both ide of (9) give: y() = g() +λ ln t k(y(t))dt (9) γ+ ln() Y() G() ( )Lk(y(t)) = +λ { } () where γ i Euler' contant. Uing the ame method we hall find at the end the required olution by the invere of Laplace tranform. Eample (Linear cae) [6] y() = +λ ln(t )y(t)dt < t< ()

5 Studie in Nonlinear Sci., (4): 9-4, Table : y E y LA E LA y TM E TM.7 4.9E E-.569E- 4,964E- 6.47E-.56.6E-.75E E-4.689E-.95E E-.7595E E E-.68E-.4.96E-.9574E-.957E E E-5..E E-.8E-5.8E-.87E E- 6.98E- 6.79E E E-6.6.6E-.5969E-.898E-6.69E-.959E-6.4.6E E-.8E-6.65E-.54E-6. 4.E E E-7 4.4E-4.976E-7 where the eact olution y() =, λ =.. By taking Laplace tranform, we obtain γ+ ln() Y() ( )Lk(y(t)) The Adomian decompoition method introduce the recurive relation = +λ { } () o that γ+ ln Y() =,Y k+ () =λ Y(), k k λ y() =,y() = [ + 6ln], 8 λ y() = [45 π ln + 7(ln) ],y y + y + y () Comparing thi reult by the Toeplitz matri method given in Table. Eample 4 (Nonlinear cae)[6] (4) y() = +λ ln( t)y(t)dt where the eact olution y() =, λ = By taking Laplace tranform,we obtain γ+ ln() Y() ( )Lk(y(t)) The Adomian decompoition method introduce the recurive relation = +λ { } (5) o that γ+ ln Y() =,Y k+ () =λ Y(), k k (6) 7 y() =,y() = [ 6 4 ln] 98 y() = π 977 ln + 58(ln),y y +λ y + y ( ) 8 Reult in Table 4.

6 Studie in Nonlinear Sci., (4): 9-4, Table 4: y E y LA E LA y TM E TM..E E E E-.E E E-.864E-7 6.4E-.656E-9.6.6E E-.5958E-8.6E-.9E-.4.6E-.6E-.667E-9.6E- 9.E-. 4.E-4 4.E E- 4.E-4 4.E- CONCLUSION In thi work we howed the accuracy and implicity of Combined Laplace-Adomian method applied to linear and nonlinear Volterra integral equation with weakly kernel. Thi method preent a ueful way to develop computational method for olving thee kind of nonlinear ingular integral equation. We made a comparion with Toeplitz matri method for olving the ame problem. REFERENCES. Atkinon, K.E., 974. An eitence theorem for Abel integral equation. SIAM J. Math. Anal., 5: Baker, C.T.H., 977. The numerical treatment of integral equation. Clarendon Pre Oford.. Cahlon, B., 98. Numerical Solution of nonlinear Volterra integral equation. J. Comput. Appl. Math., 7: Logan, J.E., 976. The approimate olution of Volterra integral equation of the econd kind. PhD. Thei, Univerity of Iowa, Iowa City. 5. Lubich, Ch., 98. Runge-Kutta theory for Volterra and Able integral equation of the econd kind. Math. Comp., 4: Hendi, F.A. and Sh. Al-Hazmi,. The Nonlinear Volterra Integral Equation with Weakly Kernal and Toeplitz Matri Method. IJRRAS, (): Wazwaz, A. and M.S. Mehanna,. The Combined Laplace-Adomian method for handling ingular integral equation of Heat tranfer. I.J. of Nonlinear Science, (): Brunner, H., A. Peda and G. Vainikko, 997. The piecewie polynomial collocation method for nonlinear weakly ingular Volterra equation. Helinki Univerity of technology, reearch report, A9: Epoo, Finland. 4