Total views : 275

### Chebyshev-Legendre Spectral Approach in Fractional Integro-Differential Equations

#### Affiliations

• Faculty of Mathematical Sciences and Computer, Kharazmi University, Taleghani Avenue, Tehran, Iran, Islamic Republic of

#### Abstract

Objectives: In the current research we implement Chebyshev-Legendre spectral approach for a class of linear fractional integro-differential equations (FIDE) including those of Fredholm and Volterra types via initial conditions or two-point boundary value conditions. Methods: An equations derived that characterizes the approximation solution by using the shifted Legendre and shifted Chebyshev collocation points. By providing various numerical instances, the precision and efficiency of the developed algorithms were evaluated. We show that for a class of equations, we obtain the exact solutions. Results: Some particular linear solvers were presented for the linear and nonlinear FIDE with initial conditions by applying downshifted Legendre polynomials. The fractional derivatives were defined in the Caputo sense. Furthermore, a novel method developed by executing downshifted Legendre procedure in union with the downshifted Chebyshev collocation way for the solution of FIDE. Based on our research, the presented research is the first investigation regarding the Chebyshev-Legendre spectral approach for determining FIDE with initial conditions. Conclusion: In the current paper, A numerical algorithm was proposed to determine the overall equations, utilizing Gauss-collocation features and estimated the solution directly applying the downshifted Legendre polynomials.

#### Keywords

Chebyshev-Legendre Spectral Approach, FIDE, Spectral Method

#### Full Text:

|  (PDF views: 215)

#### References

• Chow TS. Fractional dynamics of interfaces between soft-nanoparticles and rough substrates. Physics Letter A.2005; 342(1-2):148-55.
• Ortigueira M. Introduction to fraction linear systems. Part 2: discrete-time case. IEE Proc., Vis. Image Signal Process.2000; 147(1):7-78.
• Wittayakiattilerd W, Chonwerayuth A. Fractional IntegroDifferential Equations of Mixed Type with Solution Operator and Optimal Controls. Journal of Mathematics Research. 2011; 3(3):140.
• Sajeer K, Rodrigues P. Novel Approach of Implementing Speech Recognition using Neural Networks for Information Retrieval. Indian Journal of Science & Technology.2015; 8(33):34-56.
• Rawashdeh EA. Numerical solution of fractional integrodifferential equations by collocation method. Applied Mathematics and Computation. 2006; 176(1):1-6.
• Huang XL, Yulin Z, Xiang-Yang D. Approximate solution of fractional integro-differential equations by Taylor expansion method. Computers and Mathematics with Applications.2011; 62(3):1112-34.
• Xindong Z, Bo T, Yinnian H. Homotopy analysis method for higher-order fractional integro-differential equations.Computers and Mathematics with Applications. 2011; 62(8):3194-203.
• Shaher M, Muhammad A. Numerical methods for fourth order fractional integro-differential equations. Applied Mathematics and Computation. 2006; 182(1):754-60.
• Li Y. Solving a nonlinear fractional diﬀerential equation using Chebyshev wavelets Commun. Nonlinear Sci. Numer.Simulat. 2010; 15(9):2284-92.
• Doha EH, Bhrawy AH, Ezz-Eldien SS. A shifted Legendre spectral method for fractional-order multi-point boundary value problems. Advances in Difference Equations. 2012; 34:34-67.
• Doha EH, Bhrawy AH, Ezz-Eldien SS. Efficient Chebyshev spectral methods for solving multi-term fractional orders differential equations. Applied Mathematical Modeling.2011; 35:5662-72.
• Saadatmandi A, Dehghan M. A new operational matrix for solving fractional-order differential equations. Signal Process.2006; 84:2602-10.
• Khader M, Sweilam H. A Chebyshev pseudo-spectral method for solving fractional order integro-differential equation. The ANZIAM Journal of Australian Mathematical Society. 2010; 51:464-75.
• Karimi S, Ataei A. Operational Tau approximation for a general class of fractional integro-differential equations. J.Comp. and Appl. Math. 2012; 30:655-74.

### Refbacks

• There are currently no refbacks.