# A New Integral Transform and Applications: HK-Transform

## DOI:

https://doi.org/10.21271/ZJPAS.36.5.4## Keywords:

HK-Transform; Inverse HK-Transform; new integral transform; convolution property; HK-operator.## Abstract

In this paper, we propose a new integral transform called the Transform and employ it to analytically solve the Volterra integral equations (VIE.) of the first kind. To achieve this First we derive the transform of basic algebraic and transcendental mathematical functions. The fundamental characteristics of the transform, which can be used to solve a variety of functional equations, including, we then discuss the fundamental properties, such as ordinary differential equations and integral equations, then we discover the exact solution for general first class V.I.E. To illustrate the applicability of the transform, numerical problems are examined and handled carefully step-by-step. The results show that the proposed Transform new integral transform produces accurate solutions for first-kind VIE. without requiring time-consuming computations.

## References

Aboodh, K.S. (2013) ‘The new integral transform “Aboodh transform”’, Global Journal of Pure and Applied Mathematics, 9(1), pp. 35–43.

Aggarwal, S., Sharma, S.D. and Gupta, A.R. (2019) ‘A New Application of Mohand Transform for Handling Abel ’ s Integral Equation’, Journal of Emerging Technologies and Innovative Research, 6(3), pp. 600–608.

Aggarwal, S. and Singh, G.P. (2019) ‘Sawi Transform for Population Growth and Decay Problems’, International Journal of Latest Technology in Engineering, Management & Applied Science (IJLTEMAS), VIII(VIII), pp. 157–162.

Al-rikabi, Y.A. (2022) ‘Using the Elzaki Transform to solve systems of partial differential equations’, Eurasian Journal of Physics, Chemistry and Mathematics, 5, pp. 1–11.

Attaweel, M.E.H., Almassry, H.A.A. and Khyar, S.F. (2019) ‘A New Application of Sawi Transform for Solving Volterra Integral Equations and Volterra Integro-differential Equations’, 2, pp. 65–77.

El-Mesady, A.I., Hamed, Y.S. and Alsharif, A.M. (2021) ‘Jafari transformation for solving a system of ordinary differential equations with medical application’, Fractal and Fractional, 5(3), pp. 1–12.

Higazy, M. and Aggarwal, S. (2021) ‘Sawi transformation for system of ordinary differential equations with application’, Ain Shams Engineering Journal, 12(3), pp. 3173–3182.

Hilmi, H., MohammedFaeq, S.J. and Fatah, S.S. (2024) ‘Exact and Approximate Solution of Multi-Higher Order Fractional Differential Equations Via Sawi Transform and Sequential Approximation Method’, JOURNAL OF UNIVERSITY OF BABYLON for Pure and Applied Sciences, 32(1), pp. 311–334.

Joel L. Schiff (1975) The Laplace Transform Theorey and Applications, Pure and Applied Mathematics. New York: Springer.

Kamal Sedeeg, A.H. (2016) ‘The New Integral Transform “Kamal Transform”’, Advances in Theoretical and Applied Mathematics, 11(4), pp. 451–458. Available at: http://www.ripublication.com.

Kharrat, B.N. and Toma, G.A. (2020) ‘A new integral transform: Kharrat - Toma transform and its properties’, World Applied Science, 38(5), pp. 436–443.

Kiliçman, A., Eltayeb, H. and Agarwal, R.P. (2010) ‘On sumudu transform and system of differential equations’, Abstract and Applied Analysis, 2010.

Kumar, A., Bansal Shikha and Aggarwal, S. (2022) ‘A New Novel Integral Transform “Anuj Transform” with Application’, Journal of Scientific Research, 14(2), pp. 521–532.

Latif Shaikh, S. (2018) ‘Introducing a New Integral Transform: Sadik Transform’, American International Journal of Research in Science, pp. 100–102.

Murray R.Spiegel (1965) Schaums’s -Laplace Transforms. United states: Mcgraw Hill.

Patil, D.P., Borse, S.R. and Kapadi, D.P. (2023) ‘Kushare Transform for the Solution of Models in Health Sciences’, international journal of novel research and development, 8(1).

Patil, D.P., Shweta, L. and Gatkal, N.D. (2022) ‘Applications of “ KUSHARE Transform ” in the System of Differential Equations’, International Advanced Research Journal in Science, Engineering and Technology, 9(7).

Patil, D.P., Suryawanshi, Y.S. and Nehete, M.D. (2022) ‘Application of Soham Transform for Solving Mathematical Models Occurring in Health Science and Biotechnology’, International Journal of Mathematics, Statistics and Operations Research, 2(2), pp. 273–288.

R.kumar, J.Chandel and S.Aggarwal (2022) ‘A New Integral Transform “Rishi Transform” with Application’, Journal of Scientific Research, 14(2), pp. 521–532.

Turab, A. et al. (2024) ‘The Rishi Transform method for solving multi-high order fractional differential equations with constant coefficients’, AIMS Mathematics, 9(2), pp. 3798–3809.

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*Zanco Journal of Pure and Applied Sciences*,

*36*(5), 37–46. https://doi.org/10.21271/ZJPAS.36.5.4

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Copyright (c) 2024 Hozan Hilmi, Karwan HamaFaraj Jwamer

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