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Solving Volterra’s population growth model of arbitrary order using the generalized fractional order of the Chebyshev functions

Abstract

Volterra’s model for population growth in a closed system includes an integral term to indicate accumulated toxicity in addition to the usual terms of the logistic equation, that occurs in ecology. In this paper, a new numerical approximation is introduced for solving this model of arbitrary (integer or fractional) order. The proposed numerical approach is based on the generalized fractional order Chebyshev orthogonal functions of the first kind and the collocation method. Accordingly, we employ a collocation approach, by computing through Volterra’s population model in the integro-differential form. This method reduces the solution of a problem to the solution of a nonlinear system of algebraic equations. To illustrate the reliability of this method, we compare the numerical results of the present method with some well-known results in order to show that the new method is efficient and applicable.

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Acknowledgments

The authors are very grateful to reviewers and editor for carefully reading the paper and for their comments and suggestions which have improved the paper.

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Correspondence to Kourosh Parand.

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Parand, K., Delkhosh, M. Solving Volterra’s population growth model of arbitrary order using the generalized fractional order of the Chebyshev functions. Ricerche mat. 65, 307–328 (2016). https://doi.org/10.1007/s11587-016-0291-y

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• DOI: https://doi.org/10.1007/s11587-016-0291-y

Keywords

• Fractional order of the Chebyshev functions
• Volterra’s population model
• Mathematical ecology
• Collocation method
• Integro-differential equation

• 65M70
• 74S25
• 92D40