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An interpolation-based method for solving Volterra integral equations

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Abstract

In this study, the second kind Volterra integral equations (VIEs) are considered. An algorithm based on the two-point Taylor formula as a special case of the Hermite interpolation is proposed to approximate the solution of such problems. The method can be applied for solving both the linear and nonlinear VIEs and systems of nonlinear VIEs. The convergence analysis and the error estimate of the method are described. A multistep form of the algorithm which is particularly beneficial in large intervals is also presented. The main advantage of the proposed algorithm is that it gives high accurate results in acceptable computational times. In order to indicate the validity of the method, it is employed for solving several illustrative examples. The efficiency of the method is confirmed through our study respecting the absolute errors and CPU times.

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References

  1. Baratella, P.: A Nyström interpolant for some weakly singular linear Volterra integral equations. J. Comput. Appl. Math. 231, 725–734 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bartoshevich, M.A.: A heat-conduction problem. J. Eng. Phys. 28, 240–244 (1975)

    Article  MathSciNet  Google Scholar 

  3. Ding, H.J., Wang, H.M., Chen, W.Q.: Analytical solution for the electroelastic dynamics of a nonhomogeneous spherically isotropic piezoelectric hollow sphere. Arch. Appl. Mech. 73, 49–62 (2003)

    Article  MATH  Google Scholar 

  4. Farengo, R., Lee, Y.C., Guzdar, P.N.: An electromagnetic integral equation: application to microtearing modes. Phys. Fluids 26, 3515–3523 (1983)

    Article  MATH  Google Scholar 

  5. Wazwaz, A.: Linear and Nonlinear Integral Equations, vol. 639. Springer, Berlin (2011)

    Book  MATH  Google Scholar 

  6. Rahman, M.: Integral Equations and Their Applications. WIT Press, Southampton (2007)

    MATH  Google Scholar 

  7. Ladopoulos, E.G.: Singular Integral Equations: Linear and Non-linear Theory and Its Applications in Science and Engineering. Springer, Berlin (2013)

    MATH  Google Scholar 

  8. Serov, V.S., Schürmann, H.W., Svetogorova, E.: Integral equation approach to reflection and transmission of a plane TE-wave at a (linear/nonlinear) dielectric film with spatially varying permittivity. J. Phys. A Math. Gen. 37, 3489 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  9. Yousefi, S.A.: Numerical solution of Abel’s integral equation by using Legendre wavelets. Appl. Math. Comput. 175, 574–580 (2006)

    MathSciNet  MATH  Google Scholar 

  10. Hansen, P.C., Jensen, T.K.: Large-Scale Methods in Image Deblurring. International Workshop on Applied Parallel Computing. Springer, Berlin (2006)

    Book  Google Scholar 

  11. Schürmann, H.W., Serov, V.S., Shestopalov, Y.V.: Reflection and transmission of a plane TE-wave at a lossless nonlinear dielectric film. Phys. D 158, 197–215 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hethcote, H.W., Lewis, M.A., Van Den Driessche, P.: An epidemiological model with a delay and a nonlinear incidence rate. J. Math. Biol. 27, 49–64 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  13. Van Den Driessche, P., Watmough, J.: A simple SIS epidemic model with a backward bifurcation. J. Math. Biol. 40, 525–540 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  14. Conte, D., Paternoster, B.: Multistep collocation methods for Volterra integral equations. Appl. Numer. Math. 59, 1721–1736 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Cherruault, Y., Zitoun, F.B.: A Taylor expansion approach using Faà di Bruno’s formula for solving nonlinear integral equations of the second and third kind. Kybernetes 38, 19 (2009)

    MATH  Google Scholar 

  16. Maleknejad, K., Mollapourasl, R., Shahabi, M.: On the solution of a nonlinear integral equation on the basis of a fixed point technique and cubic B-spline scaling functions. J. Comput. Appl. Math. 239, 346–358 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  17. Maleknejad, K., Najafi, E.: Numerical solution of nonlinear Volterra integral equations using the idea of quasilinearization. Commun. Nonlinear Sci. Numer. Simul. 16, 93–100 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Maleknejad, K., Rashidinia, J., Jalilian, H.: Non-polynomial spline functions and Quasi-linearization to approximate nonlinear Volterra integral equation. Filomat 32, 3947–3956 (2018)

    Article  MathSciNet  Google Scholar 

  19. Mirzaee, F., Hoseini, S.F.: Hybrid functions of Bernstein polynomials and block-pulse functions for solving optimal control of the nonlinear Volterra integral equations. Indagationes Mathematicae 27, 835–849 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  20. Mirzaee, F., Hadadiyan, E.: Applying the modified block-pulse functions to solve the three-dimensional Volterr-Fredholm integral equations. Appl. Math. Comput. 265, 759–767 (2015)

    MathSciNet  MATH  Google Scholar 

  21. Wazwaz, A., Rach, R., Duan, J.S.: Adomian decomposition method for solving the Volterra integral form of the Lan-Emden equations with initial values and boundary conditions. Appl. Math. Comput. 219, 5004–5019 (2013)

    MathSciNet  MATH  Google Scholar 

  22. Vosughi, H., Shivanian, E., Abbasbandy, S.: A new analytical technique to solve Volterra’s integral equations. Math. Methods Appl. Sci. 34, 1243–1253 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kant, K., Nelakanti, G.: Approximation methods for second kind weakly singular Volterra integral equations. J. Comput. Appl. Math. 368, 112531 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  24. Assari, P., Dehghan, M.: A meshless local Galerkin method for solving Volterra integral equations deduced from nonlinear fractional differential equations using the moving least squares technique. Appl. Numer. Math. 143, 276–299 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  25. Assari, P., Adibi, H., Dehghan, M.: A meshless discrete Galerkin (MDG) method for the numerical solution of integral equations with logarithmic kernels. J. Comput. Appl. Math. 267, 160–181 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  26. Shamooshaky, M.M., Assari, P., Adibi, H.: The numerical solution of nonlinear Fredholm-Hammerstein integral equations of the second kind utilizing Chebyshev wavelets. J. Math. Comput. Sci. 10, 235–246 (2014)

    Article  Google Scholar 

  27. Avazzadeh, Z., Heydari, M., Chen, W., Loghmani, G.B.: Smooth solution of partial integrodifferential equations using radial basis functions. J. Appl. Anal. Comput. 4, 115–127 (2014)

    MathSciNet  MATH  Google Scholar 

  28. Assari, P., Dehghan, M.: The approximate solution of nonlinear Volterra integral equations of the second kind using radial basis functions. Appl. Numer. Math. 131, 140–157 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  29. Assari, P., Adibi, H., Dehghan, M.: The numerical solution of weakly singular integral equations based on the meshless product integration (MPI) method with error analysis. Appl. Numer. Math. 81, 76–93 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  30. Assari, P., Adibi, H., Dehghan, M.: A meshless method for solving nonlinear two-dimensional integral equations of the second kind on non-rectangular domains using radial basis functions with error analysis. J. Comput. Appl. Math. 239, 72–92 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  31. Assari, P., Adibi, H., Dehghan, M.: A numerical method for solving linear integral equations of the second kind on the non-rectangular domains based on the meshless method. Appl. Math. Model. 37, 9269–9294 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  32. Mirzaee, F., Bimesl, S.: An efficient numerical approach for solving systems of high-order linear Volterra integral equations. Sci. Iran. 21, 2250–2263 (2014)

    MATH  Google Scholar 

  33. Mirzaee, F., Hoseini, S.F.: Application of Fibonacci collocation method for solving Volterra-Fredholm integral equations. Appl. Math. Comput. 273, 637–644 (2016)

    MathSciNet  MATH  Google Scholar 

  34. Mirzaee, F., Bimesl, S.: A new Euler matrix method for solving systems of linear Volterra integral equations with variable coefficients. J. Egypt. Math. Soc. 22, 238–248 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  35. Mirzaee, F., Hadadiyan, E.: A new numerical method for solving two-dimensional Volterra-Fredholm integral equations. J. Appl. Math. Comput. 52, 489–513 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  36. Ahmadinia, M., Afshari, H.A., Heydari, M.: Numerical solution of Itô-Volterra integral equation by least squares method. Numer. Algorithms 84, 591–602 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  37. Assari, P., Adibi, H., Dehghan, M.: A meshless method based on the moving least squares (MLS) approximation for the numerical solution of two-dimensional nonlinear integral equations of the second kind on non-rectangular domains. Numer. Algorithms 67, 423–455 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  38. Saberirad, F., Karbassi, S.M., Heydari, M.: Numerical solution of nonlinear fuzzy Volterra integral equations of the second kind for changing sign kernels. Soft. Comput. 23, 11181–11197 (2019)

    Article  MATH  Google Scholar 

  39. De Hoog, F., Weiss, R.: Implicit Runge-Kutta methods for second kind Volterra integral equations. Numer. Math. 23, 199–213 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  40. Jiang, W., Chen, Z.: Solving a system of linear Volterra integral equations using the new reproducing kernel method. Appl. Math. Comput. 219, 10225–10230 (2013)

    MathSciNet  MATH  Google Scholar 

  41. Saffarzadeh, M., Loghmani, G.B., Heydari, M.: An iterative technique for the numerical solution of nonlinear stochastic Itô-Volterra integral equations. J. Comput. Appl. Math. 333, 74–86 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  42. Saffarzadeh, M., Heydari, M., Loghmani, G.B.: Convergence analysis of an iterative numerical algorithm for solving nonlinear stochastic Itô-Volterra integral equations with m-dimensional Brownian motion. Appl. Numer. Math. 146, 182–198 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  43. Saffarzadeh, M., Heydari, M., Loghmani, G.B.: Convergence analysis of an iterative algorithm to solve system of nonlinear stochastic Itô-Volterra integral equations. Math. Methods Appl. Sci. 43, 5212–5233 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  44. Eshkuvatov, K., Hameed, H.H., Taib, B.M., Nik Longcd, N.M.A.: General 2\(\times \)2 system of nonlinear integral equations and its approximate solution. J. Comput. Appl. Math. 361, 528–546 (2019)

  45. Babolian, E., Mordad, M.: A numerical method for solving systems of linear and nonlinear integral equations of the second kind by hat basis functions. Comput. Math. Appl. 62, 187–198 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  46. Mirzaee, F., Hadadiyan, E.: Numerical solution of Volterra-Fredholm integral equations via modification of hat functions. Appl. Math. Comput. 280, 110–123 (2016)

    MathSciNet  MATH  Google Scholar 

  47. Mirzaee, F., Hadadiyan, E.: Numerical solution of optimal control problem of the non-linear Volterra integral equations via generalized hat functions. IMA J. Math. Control Inform. 34, 889–904 (2017)

    MathSciNet  MATH  Google Scholar 

  48. Mirzaee, F., Hadadiyan, E.: Using operational matrix for solving nonlinear class of mixed Volterra-Fredholm integral equations. Math. Methods Appl. Sci. 40, 3433–3444 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  49. Mirzaee, F., Hadadiyan, E.: Application of two-dimensional hat functions for solving space-time integral equations. J. Appl. Math. Comput. 51, 453–486 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  50. Heydari, M., Avazzadeh, Z., Loghmani, G.B.: Chebyshev cardinal functions for solving Volterra-Fredholm integrodifferential equations using operational matrices. Iran. J. Sci. Technol. (Sci.) 36, 13–24 (2012)

    MathSciNet  Google Scholar 

  51. Brunner, H.: Volterra Integral Equations: An Introduction to Theory and Applications, vol. 30. Cambridge University Press, Cambridge (2017)

    Book  MATH  Google Scholar 

  52. Günerhan, H., Khodadad, F.S., Rezazadeh, H., Khater, M.M.: Exact optical solutions of the\( (2+1)\) dimensions Kundu-Mukherjee-Naskar model via the new extended direct algebraic method. Mod. Phys. Lett. B 2050225, 2050225 (2020)

    Article  MathSciNet  Google Scholar 

  53. Attia, R.A., Lu, D., Ak, T., Khater, M.M.: Optical wave solutions of the higher-order nonlinear Schrödinger equation with the non-Kerr nonlinear term via modified Khater method. Mod. Phys. Lett. B 34, 2050044 (2020)

    Article  Google Scholar 

  54. Ali, A.T., Khater, M.M., Attia, R.A., Abdel-aty, A.H., Lu, D.: Abundant numerical and analytical solutions of the generalized formula of Hirota-Satsuma coupled KdV system. Chaos Solitons Fract. 131, 109473 (2020)

    Article  MathSciNet  Google Scholar 

  55. Khater, M.M., Attia, R.A., Abdel-Aty, A.H., Abdou, M.A., Eleuch, H., Lu, D.: Analytical and semi-analytical ample solutions of the higher-order nonlinear Schrödinger equation with the non-Kerr nonlinear term. Res. Phys. 16, 103000 (2020)

    Google Scholar 

  56. Park, C., Khater, M.M., Attia, R.A., Alharbi, W., Alodhaibi, S.S.: An explicit plethora of solution for the fractional nonlinear model of the low-pass electrical transmission lines via Atangana-Baleanu derivative operator. Alexandr. Eng. J. 59, 1205–1214 (2020)

    Article  Google Scholar 

  57. Park, C., Khater, M.M., Abdel-Aty, A.H., Attia, R.A., Lu, D.: On new computational and numerical solutions of the modified Zakharov-Kuznetsov equation arising in electrical engineering. Alexandr. Eng. J. 59, 1099–1105 (2020)

    Article  Google Scholar 

  58. Phillips, G.M.: Explicit forms for certain Hermite approximations. BIT Numer. Math. 13, 177–180 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  59. Costabile, F.A., Napoli, A.: Solving BVPs using two-point Taylor formula by a symbolic software. J. Comput. Appl. Math. 210, 136–148 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  60. Costabile, F.A., Napoli, A.: Collocation for high order differential equations with two-points Hermite boundary conditions. Appl. Numer. Math. 87, 157–167 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  61. Karamollahi, N., Loghmani, G.B., Heydari, M.: Dual solutions of the nonlinear problem of heat transfer in a straight fin with temperature-dependent heat transfer coefficient. Int. J. Numer. Methods Heat Fluid Flow (2020). https://doi.org/10.1108/HFF-04-2020-0201

  62. Sugiyama, S.: Stability problems on difference and functional-differential equations. Proc. Jpn. Acad. 45, 526–529 (1969)

    MathSciNet  MATH  Google Scholar 

  63. Granas, A., Dugundji, J.: Fixed Point Theory. Springer, Berlin (2013)

    MATH  Google Scholar 

  64. Fazeli, S., Somayyeh, G., Hojjati, S.: Shahmorad: Super implicit multistep collocation methods for nonlinear Volterra integral equations. Math. Comput. Model. 55, 590–607 (2012)

    Article  MATH  Google Scholar 

  65. Erfanian, M.: The approximate solution of nonlinear integral equations with the RH wavelet bases in a complex plane. Int. J. Appl. Comput. Math. 4, 31 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  66. Saberi-Nadjafi, J., Mehrabinezhad, M., Diogo, T.: The Coiflet-Galerkin method for linear Volterra integral equations. Appl. Math. Comput. 221, 469–483 (2013)

    MathSciNet  MATH  Google Scholar 

  67. Berenguer, M.I., et al.: Biorthogonal systems for solving Volterra integral equation systems of the second kind. J. Comput. Appl. Math. 235, 1875–1883 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  68. Katani, R., Shahmorad, S.: Block by block method for the systems of nonlinear Volterra integral equations. Appl. Math. Model. 34, 400–406 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  69. Sekar, Chandra Guru, R., Murugesan, K. : STWS approach for Hammerstein system of nonlinear Volterra integral equations of the second kind. Int. J. Comput. Math. 94, 1867–1878 (2017)

  70. Abdi, A., Hojjati, G., Jackiewicz, Z., Mahdi, H.: A new code for Volterra integral equations based on natural Runge-Kutta methods. Appl. Numer. Math. 143, 35–50 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  71. Abdi, A., Hosseini, S.A., Podhaisky, H.: Numerical methods based on the Floater-Hormann interpolants for stiff VIEs. Numer. Algorithms 85, 867–886 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  72. Hairer, E., Lubich, C., Schlichte, M.: Fast numerical solution of nonlinear Volterra convolution equations. SIAM J. Sci. Stat. Comput. 6, 532–541 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  73. Capobianco, G., Conte, D., Del Prete, I., Russo, E.: Fast Runge-Kutta methods for nonlinear convolution systems of Volterra integral equations. BIT Numer. Math. 47, 259–275 (2007)

    Article  MathSciNet  MATH  Google Scholar 

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  1. Department of Mathematics, Yazd University, Yazd, Iran

    Nasibeh Karamollahi, Mohammad Heydari & Ghasem Barid Loghmani

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  1. Nasibeh Karamollahi
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  2. Mohammad Heydari
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Karamollahi, N., Heydari, M. & Loghmani, G.B. An interpolation-based method for solving Volterra integral equations. J. Appl. Math. Comput. 68, 909–940 (2022). https://doi.org/10.1007/s12190-021-01547-4

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