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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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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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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DOI: https://doi.org/10.1007/s11075-023-01630-w
Keywords
- Bernoulli polynomial
- Variable-order fractional integral equation
- Gauss-Jacobi quadrature formula
- Convergence analysis