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On the solution of the integro-differential equation with an integral boundary condition

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Abstract

In this paper, we present an existence of solution for a functional integro-differential equation with an integral boundary condition arising in chemical engineering, underground water flow and population dynamics, and other field of physics and mathematical chemistry. By using the techniques of noncompactness measures, we employ the basic fixed point theorems such as Darbo’s theorem to obtain the mentioned aim in Banach algebra. Then this paper presents a powerful numerical approach based on Sinc approximation to solve the equation. Then convergence of this technique is discussed by preparing a theorem which shows exponential type convergence rate and guarantees the applicability of that. Finally, some numerical examples are given to confirm efficiency and accuracy of the numerical scheme.

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References

  1. Ahmad, B., Sivasundaram, S.: Some existence results for fractional integro-differential equations with nonlinear conditions. Commun. Appl. Anal. 12, 107–112 (2008)

    MATH  MathSciNet  Google Scholar 

  2. Ahmad, B., Alghamdi, B.S.: Approximation of solutions of the nonlinear Duffing equation involving both integral and non-integral forcing terms with separated boundary conditions. Comput. Phys. Commun. 179(6), 409–416 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  3. Ahmad, B.: On the existence of T-periodic solutions for Duffing type integro-differential equations with p-Laplacian. Lobachevskii J. Math. 29(1), 1–4 (2008)

    MATH  MathSciNet  Google Scholar 

  4. Chang, Y.K., Nieto, J.J.: Existence of solutions for impulsive neutral integro-differential inclusions with nonlocal initial conditions via fractional operators. Numer. Func. Anal. Optim. 30(3–4), 227–244 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  5. Luo, Z., Nieto, J.J.: New results for the periodic boundary value problem for impulsive integro-differential equations. Nonlinear Anal. Theory Meth. Appl. 70(6), 2248–2260 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  6. Mesloub, S.: On a mixed nonlinear one point boundary value problem for an integro-differential equation. Boundary Value Problems, Article ID 814947, 8 pages (2008)

    Google Scholar 

  7. Nieto, J.J., Rodriyguez-Lopez, R.: New comparison results for impulsive integro-differential equations and applications. J. Math. Anal. Appl. 328(2), 1343–1368 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  8. Ahmad, B., Alsaedi, A., Alghamdi, B.S.: Analytic approximation of solutions of the forced Duffing equation with integral boundary conditions. Nonlinear Anal. Real World Appl. 9(4), 1727–1740 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  9. Ahmad, B., Alsaedi, A.: Existence of approximate solutions of the forced Duffing equation with discontinuous type integral boundary conditions. Nonlinear Anal. Real World Appl. 10(1), 358–367 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  10. Benchohra, M., Hamani, S., Nieto, J.J.: The method of upper and lower solutions for second order differential inclusions with integral boundary conditions. Rocky Mountain J. Math. 40(1), 13–26 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  11. Boucherif, A.: Second-order boundary value problems with integral boundary conditions. Nonlinear Theory Meth. Appl. 70(1), 364–371 (2009)

    Article  MATH  Google Scholar 

  12. Chang, Y.-K., Nieto, J.J., Li, W.-S.: On impulsive hyperbolic differential inclusions with nonlocal initial conditions. J. Optim. Theory Appl. 140(3), 431–442 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  13. Chang, Y.K., Nieto, J.J., Li, W.S.: Controllability of semilinear differential systems with nonlocal initial conditions in Banach spaces. J. Optim. Theory Appl. 142, 267–273 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  14. Feng, M., Du, B., Ge, W.: Impulsive boundary value problems with integral boundary conditions and one-dimensional p-Laplacian. Nonlinear Anal. Theory Meth. Appl. 70, 3119–3126 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  15. Yang, Z.: Existence of nontrivial solutions for a nonlinear Sturm–Liouville problem with integral boundary conditions. Nonlinear Anal. Theory Meth. Appl. 68(1), 216–225 (2008)

    Article  MATH  Google Scholar 

  16. Gallardo, J.M.: Second order differential operators with integral boundary conditions and generation of semigroups. Rocky Mt. J. Math. 30, 1265–1292 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  17. Karakostas, G.L., Tsamatos, P.C.: Multiple positive solutions of some Fredholm integral equations arisen from nonlocal boundary-value problems. Electron. J. Differ. Equat. 30, 1–17 (2002)

    Google Scholar 

  18. Lomtatidze, A., Malaguti, L.: On a nonlocal boundary-value problems for second order nonlinear singular differential equations. Georgian Math. J. 7, 133–154 (2000)

    MATH  MathSciNet  Google Scholar 

  19. Bouziani, A., Merazga, N.: Solution to a semilinear pseudoparabolic problem with integral conditions. Electron. J. Differ. Equat. 115, 1–18 (2006)

    MathSciNet  Google Scholar 

  20. Rzepecki, B.: Measure of noncompactness and Krasnosel’skii’s fixed point theorem. Bull. Acad. Polon. Sci. Ser. Sci. Math. Astron. Phys. 24, 861–865 (1976)

    MATH  MathSciNet  Google Scholar 

  21. Banas, J., Goebel, K.: Measures of noncompactness in Banach spaces. Marcel Dekker, New York (1980)

    MATH  Google Scholar 

  22. Muhammad, M., Nurmuhammad, A., Mori, M., Sugihara, M.: Numerical solution of integral equations by means of the Sinc collocation method based on the double exponential transformation. J. Comput. Appl. Math. 177, 269–286 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  23. Maleknejad, K., Torabi, P., Mollapourasl, R.: Fixed point method for solving nonlinear quadratic Volterra integral equations. Comput. Math. Appl. 62, 2555–2566 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  24. Rasty, M., Hadizadeh, M.: A product integration approach based on new orthogonal polynomials for nonlinear weakly singular integral equations. Acta Applicandae Mathematicae 109(3), 861–873 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  25. Maleknejad, K., Nosrati, M.: The method of moments for solution of second kind Fredholm integral equations based on b-spline wavelets. Int. J. Comput. Math. 87(7), 1602–1616 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  26. Mohsen, A., El-Gamel, M.: A Sinc-collocation method for the linear Fredholm integro-differential equations. J. Appl. Math. Phys. 58, 380–390 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  27. Maleknejad, K., Nouri, K., Mollapourasl, R.: Existence of solutions for some nonlinear integral equations. Commun. Nonlinear Sci. Numer. Simul. 14, 2559–2564 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  28. Volk, W.: The iterated Galerkin methods for linear integro-differential equations. J. Comput. Appl. Math. 21, 63–74 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  29. Maleknejad, K., Mirzaee, F.: Using rationalized Haar wavelet for solving linear integral equations. Appl. Math. Comput. 160, 579–587 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  30. Avudainayagam, A., Vani, C.: Wavelet Galerkin method for integro-differential equations. Appl. Numer. Math. 32, 247–254 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  31. Shamloo, A.S., Shahkar, S., Madadi, A.: Numerical solution of the Fredholme-Volterra integral equation by the Sinc function. Am. J. Comput. Math. 2(2), 136–142 (2012)

    Article  Google Scholar 

  32. Maleknejad, K., Alizadeh, M., Mollapourasl, R.: Convergence of Sinc approximation for Fredholm integral equation with degenerate kernel. Kybernetes 41(3), 482–490 (2012)

    MathSciNet  Google Scholar 

  33. Stenger, F.: Numerical methods based on Sinc and analytic functions. Springer, New York (1993)

    Book  MATH  Google Scholar 

  34. Guo, D., Lakshmikantham, V., Liu, X.Z.: Nonlinear integral equations in abstract spaces. Kluwer, Boston (1996)

  35. Kaneko, H., Neamprem, K., Novaprateep, B.: Wavelet collocation method and multilevel augmentation method for Hammerstein quations. SIAM J. Sci. Comput. 34(1), A309–A338 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  36. Babolian, E., Shahsavaran, A.: Numerical solution of nonlinear Fredholm and Volterra integral equations of the second kind using Haar wavelets and collocation method. J. Sci. Tarbiat Moallem Univ. 7(3), 213–22 (2007)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran

    Khosrow Maleknejad, Iraj Najafi Khalilsaraye & Mahdieh Alizadeh

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  1. Khosrow Maleknejad
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  2. Iraj Najafi Khalilsaraye
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  3. Mahdieh Alizadeh
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Correspondence to Khosrow Maleknejad.

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Maleknejad, K., Najafi Khalilsaraye, I. & Alizadeh, M. On the solution of the integro-differential equation with an integral boundary condition. Numer Algor 65, 355–374 (2014). https://doi.org/10.1007/s11075-013-9709-8

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  • DOI: https://doi.org/10.1007/s11075-013-9709-8

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