An Appropriate Numerical Method for Solving Nonlinear Volterra-Fredholm Integral Equations

Roghayeh Katani 1

Abstract


This paper is concerned with the numerical solution of the mixed Volterra-Fredholm integral equations by using a version of the block by block method. This method efficient for linear and nonlinear equations and it avoids the need for spacial starting values. The convergence is proved and finally performance of the method is illustrated by means of some significative examples.


Keywords


Volterra-Fredholm integral equations; Romberg quadrature rule; numerical solution.

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References


Brunner H. On the numerical solution of nonlinear Volterra-Fredholm integral equation by collocation methods, SIAM J. Numer. Anal. 1990; 27: 987-1000.

Brunner H, Messina E. Time-steping methods for Volterra-Fredholm integral equations, Rend. Mat. VII 2003; 23: 329-342.

Cardone A, Messina E, Russo E. A fast iterative method for discretized Volterra- Fredholm integral equations, J. Comput. Appl. Math. 2006; 189: 568-579.

Dastjerdi HL, Maalek Ghainia FM, Hadizadehb M. A meshless approximate solution of mixed Volterra-Fredholm integral equations, J. Comput. Math. 2012; 1-12.

Diekmann O. Thresholds and travelling waves for the geographical spread of infection, J. Math. Biol. 1978; 6: 109-130.

Gouyandeh Z, T. Allahviranloo, A. Armand, Numerical solution of nonlinear Volterra Fredholm-Hammerstein integral equations via Tau-collocation method with convergence analysis, J. Comput. Appl. Math. 2016; 308: 435-446.

Guoqiang H. Asymptotic error expansion for the Nystrom method for a nonlinear Volterra- Fredholm integral equation, J. Comput. Appl. Math. 1995; 59: 49-59.

Hatia L. On approximate solving of the Fourier problems, Demonstratio Math. 1979; 12: 913-922.

Katani R, Shahmorad S. The block by block method with Romberg quadrature for the solution of nonlinear Volterra integral equations on the large intervals, Ukrain. Math. J. 2012; 64(7): 1050-1063.

Kauthen JP. Continuous time collocation methods for Volterra-Fredholm integral equa-tions, Numer. Math. 1989; 56: 409-424.

Maleknejad K, Hadizadeh M. A new computational method for Volterra-Fredholm integral equations, Comput. Math. Appl. 1999; 37: 1-8.

Nemati S, Lima P, Ordokhani Y. Numerical method for the mixed Volterra-Fredholm integral equations using hybrid Legendre functions, Conference Applications of Mathematics, Prague 2015.

Pachpatte BG. On mixed Volterra-Fredholm type integral equations, Indian J. Math. Biol. 1977; 4: 337-351.

Thieme HR. A model for the spatio spread of an epidemic, J. Math. Biol. 1977; 4: 337-351.

Wazwaz AM. A reliable treatment for mixed Volterra-Fredholm integral equations, Appl. Math. Comput. 2002; 127: 405-414.

Yalsinbas S. Taylor polynomial solution of nonlinear Volterra-Fredholm integral equations, Appl. Math. Comput. 2002; 127: 195-206.

Yildirim A. Homotopy perturbation method for the mixed Volterra-Fredholm integral equations, Chaos Solitons Fractals 2009; 2: 2760-2764.

Yousefi S, Razzaghi SM. Legendre wavelets method for the nonlinear Volterra- Fredholm integral equations, Math. Comput. Simul. 2005; 70: 1-8.




DOI: http://dx.doi.org/10.24294/ijmss.v1i2.492

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