Exact Solution of Non-Linear Volterra Integral Equation of First Kind Using Rishi Transform

Sudhanshu Aggarwal; Rishi  Kumar; Jyotsna Chandel

doi:10.48165/

Authors

Sudhanshu Aggarwal Assistant Professor, Department of Mathematics, National Post Graduate College, Barhalganj, Gorakhpur-273402, Uttar Pradesh, India
Rishi Kumar Research Scholar, Department of Mathematics, D.S. College, Aligarh (Dr. Bhimrao Ambedkar University, Agra), Uttar Pradesh 202001, India
Jyotsna Chandel Associate Professor, Department of Mathematics, D.S. College, Aligarh Agra), Uttar Pradesh 202001, India

DOI:

https://doi.org/10.48165/

Keywords:

Analytical Solution, Rishi Transform, Inverse Rishi Transform, Convolution, Volterra Integral Equation

Abstract

The problems of Engineering and Science can easily represent by developing their mathematical models in the terms of integral equations. Various analytical and numerical methods are available that can be used for solving integral equations of different kinds. In this paper, authors have considered recently developed integral transform “Rishi Transform” for obtaining the exact solution of non-linear Volterra integral equation of first kind (NLVIEFK). Four numerical problems have considered for demonstrating the complete procedure of determining the exact solution. Results of these problems depict that Rishi transform is very effective integral transform and it provides the exact solution of NLVIEFK without doing complicated calculation work.

References

A.M. Wazwaz (2011). Linear and nonlinear integral equations: Methods and applications, Higher Education Press, Beijing.

A. Jerri (1999). Introduction to integral equations with applications, Wiley, New York.

C. M. Dafermos (1970). An abstract Volterra equation with application to linear viscoelasticity, J. Differential Equations, 7, 554-589.

N. Distefano (1968). A Volterra integral equation in the stability of linear hereditary phenomena, J. Math. Anal. Appl., 23, 365-383.

W. Feller (1941). On the integral equation of renewal theory, Ann. Math. Stat., 12, 243-267.

P. L. Goldsmith (1967). The calculation of true particle size distributions from the sizes observed in a thin slice, Brit. J. Appl. Phys., 18, 813-830.

N. Levinson (1960). A nonlinear Volterra equation arising in the theory of superfluidity, J. Math. Anal. Appl., 1, 1-11.

S. P. Lin (1975). Damped vibration of a string, J. Fluid Mech., 72, 787-797.

W. R. Mann and F. Wolf, (1951). Heat transfer between solids and gases under nonlinear boundary conditions, Quart. Appl. Math., 9, 163-184.

J. R. Phillips (1966). Some integral equations in geometric probability, Biometrika, 53, 365-374.

T. G. Rogers and E. H. Lee (1964). The cylinder problem in viscoelastic stress analysis, Quart. Appl. Math., 22, 117-131.

C. C. Shilepsky (1974). The asymptotic behavior of an integral equation with application to Volterra’s population equation, J. Math. Anal. Appl., 48, 764-779.

F. J. S. Wang (1978). Asymptotic behavior of some deterministic epidemic models, SIAM J. Math. Anal., 9, 529-534.

K. E. Swick (1981). A nonlinear model for human population dynamics, SIAM J. Appl. Math., 40, 266- 278.

R. P. Kanwal (1997). Linear integral equations, Birkhauser, Boston.

R. Chauhan and S. Aggarwal (2019). Laplace transform for convolution type linear Volterra integral equation of second kind, Journal of Advanced Research in Applied Mathematics and Statistics, 4(3&4), 1- 7.

C. T. H. Baker (1978). Runge-Kutta methods for Volterra integral equations of the second kind, Lecture Notes in Mathematics 630, Springer-Verlag, Berlin.

H. Brunner, E. Hairer and S. P. Norsett (1982). Runge-Kutta theory for Volterra equations of the second kind, Math. Comp., 19, 147-163.

H. Brunner and M. D. Evans (1977). Piecewise polynomial collocation for Volterra integral equations of the second kind, J.Inst. Math. Appl., 20, 415-423.

M. E. A. EL Tom (1971). Application of spline functions in Volterra integral equations, J. Inst. Math. Appl., 8, 354-357.

L. Garey (1972). Implicit methods for Volterra integral equations of the second kind, Proc. Second Manitoba Conference on Numerical Mathematics, 167-177.

A. Goldfine (1977). Taylor series methods for the solution of Volterra integral and integro-differential equations, Math. Comp., 31, 691-707.

P. J. Van Der Houwen and H. J. J. Te Riele (1981). Backward differentiation type formulas for Volterra integral equations of the second kind, Numer. Math., 37, 205-217.

P. J. Van Der Houwen, P. H. M. Wolkenfelt and C. T. H. Baker (1981). Convergence and stability analysis for modified Runge-Kutta methods in the numerical treatment of second kind Volterra equations, IMA J. Numer. Anal., 1, 303-328.

H. S. Hung (1970). The numerical solution of differential and integral equations by spline functions, Tech. Summary Rept. 1053, Math. Res. Center, Univ. Wisconsin, Madison.

P. Linz (1967). The numerical solution of Volterra integral equations by finite difference methods, Tech. Summary Rept. 825, Math. Res. Center, Univ. Wisconsin, Madison.

S. Aggarwal, R. Chauhan and N. Sharma (2018). A new application of Kamal transform for solving linear Volterra integral equations, International Journal of Latest Technology in Engineering, Management & Applied Science, 7(4), 138-140.

S. Aggarwal, R. Chauhan and N. Sharma (2018). A new application of Mahgoub transform for solving linear Volterra integral equations, Asian Resonance, 7(2), 46-48.

S. Aggarwal, N. Sharma and R. Chauhan (2018). Solution of linear Volterra integral equations of second kind using Mohand transform, International Journal of Research in Advent Technology, 6(11), 3098-3102.

S. Aggarwal, N. Sharma and R. Chauhan (2018). A new application of Aboodh transform for solving linear Volterra integral equations, Asian Resonance, 7(3), 156-158.

S. Aggarwal, A. Vyas and S.D. Sharma (2020). Primitive of second kind linear Volterra integral equation using Shehu transform, International Journal of Latest Technology in Engineering, Management & Applied Science, 9(8), 26-32.

S. Aggarwal, K. Bhatnagar and A. Dua (2020). Method of Taylor’s series for the primitive of linear second kind non-homogeneous Volterra integral equations, International Journal of Research and Innovation in Applied Science, 5(5), 32-35.

G. K. Watugula (1993). Sumudu transform: A new integral transform to solve differential equations and control engineering problems, International Journal of Mathematical Education in Science and Technology, 24(1), 35-43.

Z. H. Khan and W. A. Khan (2008). Natural transform-properties and applications, NUST Journal of Engineering Science, 1, 127-133.

T. M. Elzaki (2011). The new integral transforms “Elzaki Transform”, Global Journal of Pure and Applied Mathematics, 1, 57-64.

K. S. Aboodh (2013). The new integral transforms “Aboodh Transform”, Global Journal of Pure and Applied Mathematics, 9(1), 35-43.

M. A. M. Mahgoub (2016). The new integral transform “Mahgoub Transform”, Advances in Theoretical and Applied Mathematics, 11(4), 391-398.

K. Abdelilah and S. Hassan (2016). The new integral transforms “Kamal Transform”, Advances in Theoretical and Applied Mathematics, 11(4), 451-458.

Zafar, Zain Ul Abadin (2016). ZZ transform method, International Journal of Advanced Engineering and Global Technology, 4(1), 1605-1611.

M. Mohand and A. Mahgoub (2017)., The new integral transform “Mohand Transform”, Advances in Theoretical and Applied Mathematics, 12(2), 113 – 120.

L. S. Sadikali (2018). Introducing a new integral transform: Sadik transform, American International Journal of Research in Science, Technology, Engineering & Mathematics, 22(1), 100-102.

S. Maitama and W. Zhao (2019). New integral transform: Shehu transform a generalization of Sumudu and Laplace transform for solving differential equations, International Journal of Analysis and Applications, 17(2), 167-190.

Mohand M. Abdelrahim Mahgoub (2019). The new integral transform ''Sawi Transform'', Advances in Theoretical and Applied Mathematics, 14(1), 81-87.

L. M. Upadhyaya (2019). Introducing the Upadhyaya integral transform, Bulletin of Pure and Applied Sciences, 38E (Math & Stat.) (1), 471-510.