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A Novel Numerical Method for Solving Volterra Integro-Differential Equations

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Abstract

In this paper we introduce a numerical method for solving nonlinear Volterra integro-differential equations. In the first step, we apply implicit trapezium rule to discretize the integral in given equation. Further, the Daftardar-Gejji and Jafari technique is used to find the unknown term on the right side. We derive existence-uniqueness theorem for such equations by using Lipschitz condition. We further present the error, convergence, stability and bifurcation analysis of the proposed method. We solve various types of equations using this method and compare the error with other numerical methods. It is observed that our method is more efficient than other numerical methods.

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Acknowledgements

S. Bhalekar acknowledges the Science and Engineering Research Board (SERB), New Delhi, India for the Research Grant (Ref. MTR/2017/000068) under Mathematical Research Impact Centric Support (MATRICS) Scheme.

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Authors and Affiliations

  1. Department of Mathematics, Shivaji University, Kolhapur, 416004, India

    Jayvant Patade & Sachin Bhalekar

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  1. Jayvant Patade
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  2. Sachin Bhalekar
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Correspondence to Sachin Bhalekar.

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Appendix

Appendix

We provide software package based on Mathematica-10 to solve VIDEs using numerical method. This software is used for solving VIDEs of the form (38). One has to provide the value of f(x, y) and K(x, t, y) in first two windows, initial condition \(y_0\), step size h and number of steps n in third, fourth and fifth windows respectively. The last window gives required solution curve. We solve Eq. (37) using this software package as shown in following Fig. 

figure a

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Patade, J., Bhalekar, S. A Novel Numerical Method for Solving Volterra Integro-Differential Equations. Int. J. Appl. Comput. Math 6, 7 (2020). https://doi.org/10.1007/s40819-019-0762-4

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  • DOI: https://doi.org/10.1007/s40819-019-0762-4

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