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A Numerical Approach for the System of Nonlinear Variable-order Fractional Volterra Integral Equations

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Abstract

In this paper, a combination approach based on Bernoulli polynomials and Gauss-Jacobi quadrature formula is developed to solve the system of nonlinear variable-order fractional Volterra integral equations (V-O-FVIEs). For this, we extend the constant coefficient in the Gauss-Jacobi formula to the variable coefficient and used it in our method. The method converts the system of V-O-FVIEs into the corresponding nonlinear system of algebraic equations. In addition, we use Gronwall inequality and the collectively compact theory to prove the existence and uniqueness of the solution of the original equation and the approximate equation, respectively. The convergence analysis and the error estimation of proposed method are discussed. Finally, some numerical examples illustrate the effectiveness of the method.

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References

  1. Atkinson, K., Flores, J.: The discrete collocation method for nonlinear integral equations. IMA. J. Numer. Anal. 13, 195–213 (1993)

    MathSciNet  Google Scholar 

  2. Anselmi-Tamburini, U., Spinolo, G.: On the least-squares determinations of lattice dimensions: a modified singular value decomposition approach to ill-conditioned cases. J. Appl. Cryst. 26, 5–8 (1993)

    ADS  CAS  Google Scholar 

  3. Atangana, A., Baleanu, D.: Numerical solution of a kind of fractional parabolic equations via two difference schemes. Abstr. Appl. Anal. 828764, 8 (2013)

    MathSciNet  Google Scholar 

  4. Assari, P., Dehghan, M.: The numerical solution of two-dimensional logarithmic integral equations on normal domains using radial basis functions with polynomial precision. Eng. Comput. 33, 853–870 (2017)

    Google Scholar 

  5. Azodi, H.D., Yaghouti, M.R.: Bernoulli polynomials collocation for weakly singular Volterra integro-differential equations of fractional order. Filomat. 32, 3623–3635 (2018)

    MathSciNet  Google Scholar 

  6. Agarwal, P., El-Sayed, A.A., Tariboon, J.: Vieta-Fibonacci operational matrices for spectral solutions of variable-order fractional integro-differential equations. J. Comput. Appl. Math. Not Applicable, 113063 (2021)

  7. Brunner, H.: Collocation methods for Volterra integral and related functional differential equations. Cambridge University Press, Cambridge. 15, (2004)

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

    Google Scholar 

  9. Bhrawy, A.H., Tohidi, E., Soleymani, F.: A new Bernoulli matrix method for solving high-order linear and nonlinear Fredholm integro-differential equations with piecewise intervals. Appl. Math. Comput. 219, 482–497 (2012)

    MathSciNet  Google Scholar 

  10. Bhrawy, A.H., Zaky, M.A.: Numerical simulation for two-dimensional variable-order fractional nonlinear cable equation. Nonlinear Dyn. 80, 101–116 (2015)

    MathSciNet  Google Scholar 

  11. Baratella, P.: A nyström interpolant for some weakly singular nonlinear Volterra integral equations. J. Comput. Appl. Math. 237, 542–555 (2013)

    MathSciNet  Google Scholar 

  12. Bazm, S.: Bernoulli polynomials for the numerical solution of some classes of linear and nonlinear integral equations. J. Comput. Appl. Math. 275, 44–60 (2015)

    MathSciNet  Google Scholar 

  13. Costabile, F.A., Dell’accio, F.: Expansions over a rectangle of real functions in Bernoulli polynomials and applications. BIT Numer. Math. 41, 451–464 (2001)

    MathSciNet  Google Scholar 

  14. Chen, Y.M., Liu, L.Q., Li, B.F., Sun, Y.: Numerical solution for the variable order linear cable equation with Bernstein polynomials. Appl. Math. Comput. 238, 329–341 (2014)

    MathSciNet  Google Scholar 

  15. Chen, C., He, X.M., Huang, J.: Mechanical quadrature methods and their extrapolations for solving the first kind boundary integral equations of Stokes equation. Appl. Numer. Math. 96, 165–179 (2015)

    MathSciNet  Google Scholar 

  16. Doha, E.H., Abdelkawy, M.A., Amin, A.Z.M., Baleanu, D.: Spectral technique for solving variable-order fractional Volterra integro-differential equations. Numer. Meth. Part. D. E. 34, 1659–1677 (2018)

    MathSciNet  Google Scholar 

  17. Engl, H.W., Hanke, M., Neubauer, A.: Regularization of Inverse Problems, Springer, Berlin. 375, (1996)

  18. El-Sayed, A.A., Agarwal, P.: Numerical solution of multiterm variable-order fractional differential equations via shifted Legendre polynomials. Math. Methods Appl. Sci. 41, 3978–3991 (2019)

    MathSciNet  Google Scholar 

  19. Hansen, P.C., O’Leary, D.P.: The use of the L-curve in the regularization of discrete ill-posed problems. SIAM. J. Sci. Comput. 14(6), 1487–1503 (1993)

    MathSciNet  Google Scholar 

  20. Huang, J., Lv, T., Li, Z.C.: Mechanical quadrature methods and their splitting extrapolations for boundary integral equations of first kind on open arcs. Appl. Numer. Math. 59(12), 2908–2922 (2009)

    MathSciNet  Google Scholar 

  21. Heydari, M.H.: A computational method for a class of systems of nonlinear variable-order fractional quadratic integral equations. Appl. Numer. Math. 153, 164–178 (2020)

    MathSciNet  Google Scholar 

  22. Krylov, V.I.: Approximate Calculation of Integrals. Dover publications, Mineola, New York (1962)

    Google Scholar 

  23. Kovalnogov, V.N., Fedorov, R.V., Khakhalev, Y.A., Simos, T.E., Tsitouras, C.: A neural network technique for the derivation of Runge-Kutta pairs adjusted for scalar autonomous problems. Mathematics. 9, 1842 (2021)

    Google Scholar 

  24. Lv, T., Huang, J.: High precision algorithm for integral equation. China Science Publishing. (2013)

  25. Li, H., Huang, J.: High-accuracy quadrature methods for solving boundary integral equations of axisymmetric elasticity problems. Comput. Math. Appl. 71(1), 459–469 (2016)

    MathSciNet  Google Scholar 

  26. Liu, H.Y., Huang, J., Pan, Y.B., Zhang, J.P.: Barycentric interpolation collocation methods for solving linear and nonlinear high-dimensional Fredholm integral equations. J. Comput. Appl. Math. 327, 141–154 (2018)

    MathSciNet  Google Scholar 

  27. Liu, H.Y., Huang, J., Zhang, W., Ma, Y.Y.: Meshfree approach for solving multi-dimensional systems of Fredholm integral equations via barycentric Lagrange interpolation. Appl. Math. Comput. 346, 295–304 (2019)

    MathSciNet  Google Scholar 

  28. Meerschaert, M.M., Tadjeran, C.: Finite difference approximations for fractional advection dispersion equations. J. Comput. Appl. Math. 172, 65–77 (2004)

    ADS  MathSciNet  Google Scholar 

  29. Magin, R.L.: Fractional calculus models of complex dynamics in biological tissues. Comput. Math. Appl. 59, 1586–1593 (2010)

    MathSciNet  Google Scholar 

  30. Matinfar, M., Taghizadeh, E., Pourabd, M.: Application of moving least squares algorithm for solving systems of Volterra integral equations. Int. J. Nonlin. Sci. Num. 22,(2021) https://doi.org/10.1515/ijnsns-2016-0100

  31. Nagy, A.M., Sweilam, N.H., El-Sayed, A.A.: New operational matrix for solving multi-term variable order fractional differential equations. J. Comput. Nonlinear Dynam. 13, 011001–011007 (2018)

    Google Scholar 

  32. Pan, Y.B., Huang, J.: Extrapolation method for solving two-dimensional Volterral integral equations of the second kind. Appl. Math. Comput. Not Applicable, 124784 (2020)

  33. Rena, Q.W., Tian, H.J.: Numerical solution of the static beam problem by Bernoulli collocation method. Appl. Math. Model. 40, 8886–8897 (2016)

    MathSciNet  Google Scholar 

  34. Rahimkhani, P., Ordokhani, Y., Babolian, E.: Fractional-order Bernoulli functions and their applications in solving fractional Fredholm-Volterra integro-differential equations. Appl. Numer. Math. 122, 66–81 (2017)

    MathSciNet  Google Scholar 

  35. Soon, C.M., Coimbra, F.M., Kobayashi, M.H.: The variable viscoelasticity oscillator. Ann. Phys. 14, 378–389 (2005)

    Google Scholar 

  36. Sun, H.G., Chen, W., Chen, Y.Q.: Variable-order fractional differential operators in anomalous diffusion modeling. Physica A. 388, 4586–4592 (2009)

    ADS  CAS  Google Scholar 

  37. Sun, H.G., Chen, W., Li, C., Chen, Y.Q.: Fractional differential models for anomalous diffusion. Physica A. 389, 2719–2724 (2010)

    ADS  CAS  Google Scholar 

  38. Shen, J., Tang, T., Wang, L.L.: Spectral Methods. Springer Series in Computational Mathematics. Heidelberg: Springer. (2011)

  39. Sun, L.J., Hou, J., Xing, C.J., Fang, Z.W.: A robust Hammerstein-Wiener model identification method for highly nonlinear systems. Processes. 10(12), 2664 (2022)

    Google Scholar 

  40. Tikhonov, A., Arsenin, V.Y.: Solutions of Ill-Posed Problems. Wiley, New York (1977)

    Google Scholar 

  41. Tadjeran, C., Meerschaert, M.M., Scheffler, H.P.: A second order accurate numerical approximation for the fractional diffusion equation. J. Comput. Phys. 213, 205–213 (2006)

    ADS  MathSciNet  Google Scholar 

  42. Toutounian, F., Tohidi. E.: A new Bernoulli matrix method for solving second order linear partial differential equations with the convergence analysis. Appl. Math. Comput. 223, 298-310 (2013)

  43. Tohidi, E., Shirazian, M.: Numerical solution of linear HPDEs via Bernoulli operational matrix of differentiation and comparison with Taylor matrix method. Math. Sci. Lett. 1, 61–70 (2012)

    Google Scholar 

  44. Tohidi, E., Ezadkhah, M.M., Shateyi, S.: Numerical solution of nonlinear fractional Volterra integro-differential equations via Bernoulli polynomials. Abstr. Appl. Anal. (2014). https://doi.org/10.1155/2014/162896

    Article  MathSciNet  Google Scholar 

  45. Taghizadeh, E., Matinfar, M.: Modified numerical approaches for a class of Volterra integral equations with proportional delays. Comput. Appl. Math. 38, (2019)

  46. Taherpour, V., Nazari, M., Nemat, A.: A new numerical Bernoulli polynomial method for solving fractional optimal control problems with vector components. Comput. Methods Differ. Equ. 9, 446–466 (2021)

    MathSciNet  Google Scholar 

  47. Umarov, S., Steinberg, S.: Variable order differential equations and diffusion with changing modes. Z. Anal. Anwend. 28, 431–450 (2009)

    MathSciNet  Google Scholar 

  48. Wang, H., Zheng, X.: Wellposedness and regularity of the variable-order time-fractional diffusion equations. J. Math. Anal. Appl. 475, 1778–1802 (2019)

    MathSciNet  Google Scholar 

  49. Wang, Y.F., Huang, J., Zhang, L., Deng, T.: A combination method for solving multi-dimensional systems of Volterra integral equations with weakly singular kernels, Numer. Algorithms. 91,? 473-504 (2022)

  50. Ye, H.P., Gao, J.M., Ding, Y.S.: A generalized Gronwall inequality and its application to a fractional differential equation. J. Math. Anal. Appl. 328, 1075–1081 (2007)

    MathSciNet  Google Scholar 

  51. Yang, Y., Heydari, M.H., Avazzadeh, Z., Atangana, A.: Chebyshev wavelets operational matrices for solving nonlinear variable-order fractional integral equations. Adv. Differ. Equ-ny. 611, (2020)

  52. Zhuang, P., Liu, F., Anh, V., Turner, I.: Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term. Siam. J. Numer. Anal. 47(3), 1760–1781 (2009)

    MathSciNet  Google Scholar 

  53. Zheng, X., Wang, H.: An error estimate of a numerical approximation to a hidden-memory variable-order space-time fractional diffusion equation. Siam. J. Numer. Anal. 58, 2492–2514 (2020)

    MathSciNet  Google Scholar 

  54. Zheng, X.: Approximate inversion for Abel integral operators of variable exponent and applications to fractional Cauchy problems. Fract. Cacl. Appl. Anal. (2022). https://doi.org/10.1007/s13540-022-00071-

    Article  Google Scholar 

  55. Zheng, X.C.: Numerical approximation for a nonlinear variable-order fractional differential equation via a collocation method. Math. Comput. Simulat. 195, 107–118 (2022)

    MathSciNet  Google Scholar 

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Acknowledgements

The authors thank the reviewers for their comments, which have significantly improved the presentation.

Funding

The authors would like to acknowledge the support provided by the National Natural Science Foundation of China (12101089) and the Natural Science Foundation of Sichuan Province (2022NSFSC1844).

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Authors and Affiliations

  1. School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan, 611731, China

    Yifei Wang & Jin Huang

  2. School of Mathematics, Chengdu Normal University, Chengdu, 611130, China

    Hu Li

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  1. Yifei Wang
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All authors wrote the main manuscript text, which includes Sections 1–7. All authors reviewed the manuscript.

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Correspondence to Jin Huang.

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Appendix. The MATLAB codes of Example 1

Appendix. The MATLAB codes of Example 1

The algorithm in this article is written by MATLAB code. We give the main code, where the code of Bernoulli polynomials is written by (2.3), and the codes of the quadrature weights and quadrature points are detailed in [38].

Main code

clc;clear;

N=8;n=4;

format long

x=1/(2*(N+1)):1/(N+1):(2*N+1)/(2*(N+1));

x=x’;

F1=zeros(N+1,1);

F2=zeros(N+1,1);

U0=zeros(N+1,1);

V0=zeros(N+1,1);

K1=assemble_K111(N,U0,n,x);

K4=assemble_K114(N,V0,n,x);

for i=1:N+1

E(i,:)=One_dimensional_Bernoulli_polynomials(N,x(i));

F1(i)=tan(x(i))-atan(x(i))-2/gamma(3+alpha1(x(i)))*sin(x(i))*x(i)

⌃(2+alpha1(x(i)));

F2(i)=tan(x(i))+atan(x(i))-24/gamma(5+alpha2(x(i)))*x(i)⌃(7+alpha2(x(i)));

end

A=[E -E;E E];

F=[F1+K1;F2+K4];

Y=A\(\setminus \)F;

U1=Y(1:N+1);

V1=Y(N+2:2*(N+1));

mm=0;

e=1e-10;

while norm(U1-U0,’inf’)>e and norm(V1-V0,’inf’)>e

U0=U1;V0=V1;

mm=mm+1;

K1=assemble_K111(N,U0,n,x);

K4=assemble_K114(N,V0,n,x);

F=[F1+K1;F2+K4];

Y=A\(\setminus \)F;

U1=Y(1:N+1);

V1=Y(N+2:2*(N+1));

if mm>100 break; end

end

x0=0:0.1:1;

x0=x0’;

m=length(x0);

Y1=tan(x0);

Y2=atan(x0);

for i=1:m

    B1(i,:)=One_dimensional_Bernoulli_polynomials(N,x0(i));

end

y1=B1*U1;

y2=B1*V1;

error1=abs(y1-Y1)

error2=abs(y2-Y2)

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Wang, Y., Huang, J. & Li, H. A Numerical Approach for the System of Nonlinear Variable-order Fractional Volterra Integral Equations. Numer Algor 95, 1855–1877 (2024). https://doi.org/10.1007/s11075-023-01630-w

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