## Development of Three-Step Implicit Second Derivative Hybrid Block Method for the Solution of the Second Kind Volterra Integral Equations

Published: 2023-09-09

Page: 193-204

Raymond Dominic

Department of Mathematics and Statistics, Federal University Wukari, Taraba State, Nigeria.

Ajia Rita *

Department of Mathematics and Statistics, College of Agriculture, Science and Technology, Jalingo, Nigeria.

Navokhi Joseph

Department of Mathematics and Statistics, Taraba State Polytechnic, Suntai, Jalingo Campus, Nigeria.

Department of Mathematics and Statistics, Federal University Wukari, Taraba State, Nigeria.

Kwetishe Nehemiah Daniel

Department of Mathematics and Statistics, College of Agriculture, Science and Technology, Jalingo, Nigeria.

*Author to whom correspondence should be addressed.

### Abstract

This paper employed the collocation and interpolation approach to construct three-step implicit second derivative hybrid block method to solve Volterra Integral Equations of the second kind using power series and exponentially-fitted function as the basis functions. When the continuous block methods were evaluated at each point, this led to the development of the discrete block methods. Such that, each discrete scheme obtained from the simultaneous solution of the block using the block techniques guaranteed the main method to have the same level of accuracy as the main continuous method. Therefore, a new class of k-step procedures was developed, and each of which provided stable region of absolute stability and a high order of accuracy with a very low error constant. Meanwhile, the fundamental properties of the methods were examined and found to be convergent, zero-stable, and consistent. Similarly, on a few Volterra Integral Equations of the second kind problems, the effectiveness of the techniques was evaluated and result from the numerical experiments demonstrates that our novel approach converge more faster compared to other methods used in existing literatures.

Keywords: Three-step, power series, exponentially-fitted, hybrid point, second kind, volterra integral equations.

#### How to Cite

Raymond Dominic, Ajia Rita, Navokhi Joseph, Adu Agyemang, & Kwetishe Nehemiah Daniel. (2023). Development of Three-Step Implicit Second Derivative Hybrid Block Method for the Solution of the Second Kind Volterra Integral Equations. Asian Research Journal of Current Science, 5(1), 193–204. Retrieved from https://globalpresshub.com/index.php/ARJOCS/article/view/1856

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