## THERMAL SCIENCE

International Scientific Journal

### THE HAAR WAVELETS BASED NUMERICAL SOLUTION OF RECCATI EQUATION WITH INTEGRAL BOUNDARY CONDITION

**ABSTRACT**

A Haar wavelet collocation method (HWCM) is presented for the solution of Riccati equation subject to the two-point and integral boundary condition. The quasilinearization technique is applied to linearized the Riccati equation and then the linearized equation with boundary condition is solved by converting into system of algebraic equation with the help of Haar wavelets. We have considered three different form of Reccati equation, two having integral boundary condition and one with two-point boundary condition. The numerical results obtained by HWCM are stable, efficient and convergent.

**KEYWORDS**

PAPER SUBMITTED: 2022-06-01

PAPER REVISED: 2022-06-14

PAPER ACCEPTED: 2022-06-24

PUBLISHED ONLINE: 2023-04-08

- Fakhrodin, M., Hosseini, M. M., A Comparative Study of Numerical Methods for Solving Quadratic Riccati Differential Equations, Journal of the Franklin Institute, 348 (2011), 2, 156-164
- Yuanlu, Li., et al., Wavelet Operational Matrix Method for Solving the Riccati Differential Equation, Communications in Non-Linear Science and Numerical Simulation, 19 (2014), 3, pp. 483-493
- El-Tawil, M. A., et al., Solving Riccati Differential Equation Using Adomian's Decomposition Method, Applied Mathematics and Computation, 157 (2004), 2, pp. 503-514
- Abbasbandy, S., A New Application of He's Variational Iteration Method for Quadratic Riccati Differential Equation by Using Adomian's Polynomials, Journal of Computational and Applied Mathematics, 207 (2007), 1, pp. 59-63
- Abbasbandy, S., Homotopy Perturbation Method for Quadratic Riccati Differential Equation and Comparison with Adomian's Decomposition Method, Applied Mathematics and Computation, 172 (2006), 1, pp. 485-490
- Momani, S., Shawagfeh, N., Decomposition Method for Solving Fractional Riccati Differential Equations, Applied Mathematics and Computation, 182 (2006), 2, pp. 1083-1092
- Geng, F., et al., A Piecewise Variational Iteration Method for Riccati Differential Equations, Computers and Mathematics with Applications, 58 (2009), 11-12, pp. 2518-2522
- Carinena, J., et al., Related Operators and Exact Solutions of Schrodinger Equations, International Journal of Modern Physics A, 13 (1998), 28, pp. 4913-4929
- Diaz, L., et al., Daubechies Wavelet Beam and Plate Finite Elements, Finite Elements in Analysis and Design, 45 (2009), 3, pp. 200-209
- Siraj-ul-Islam, et al., An Improved Method Based on Haar Wavelets for Numerical Solution of Non-Linear Integral and Integro-Differential Equations of First and Higher Orders, Journal of Computational and Applied Mathematics, 260 (2014), Apr., pp. 449-469
- Lepik, U., Hein, H., Solving PDE with the Aid of 2-D Haar Wavelets, in: Haar Wavelets, Springer, Berlin, Germany, 2014, pp. 97-105
- Liu,Y., et al., Daubechies wavelet meshless method for 2-D elastic problems, Tsinghua Science and Technology, 13 (2008), 5, pp. 605-608
- Jang, Q. W., et al., Remesh-Free Shape Optimization Using the Wavelet-Galerkin Method, International Journal of Solids and Structures, 41 (2004), 22, pp. 6465-6483
- Elden, L., The Numerical Solution of a Non-Characteristic Cauchy Problem for a Parabolic Equation, in: Numerical Treatment of Inverse Problems in Differential and Integral Equations, Springer, Berlin, Germany, 1983, pp. 246-268
- Yeih, W., Liu, C. S., A Three-Point BVP of Time Dependent Inverse Heat Source Problems and Solving by a TSLGSM, Computer Modelling in Engineering and Sciences (CMES), 46 (2009), 2, pp. 107-127
- Siraj-ul-Islam, et al. A Multi-Resolution Collocation Procedure for Time-Dependent Inverse Heat Problems, International Journal of Thermal Sciences, 128 (2018), June, pp. 160-174
- Ahsan, M., A Numerical Haar Wavelet-Finite Difference Hybrid Method for Linear and Non-Linear Schrodinger equation, Mathematics and Computers in Simulation, 165 (2019), Nov., pp. 13-25
- Liu, X., Haar Wavelets Multi-Resolution Collocation Procedures for 2-D Non-Linear Schrodinger Equation, Alexandria Engineering Journal, 60 (2021), 3, pp. 3057-3071
- Ahsan, M., Haar Wavelets Multi-Resolution Collocation Analysis of Unsteady Inverse Heat Problems, Inverse Problems in Science and Engineering, 27 (2018), 1, pp. 1-23
- Liu, X., Applications of Haar Wavelet-Finite Difference Hybrid Method and Its Convergence for Hyperbolic Non-Linear Schrodinger Equation with Energy and Mass Conversion, Energies, 14 (2021), 23, 7831
- Ahsan, M., A Haar Wavelet-Based Scheme for Finding the Control Parameter in Non-Linear Inverse Heat Conduction Equation, Open Physics, 19 (2021), 1, pp. 722-734
- Nazir, S., Birthmark Based Identification of Software Piracy Using Haar Wavelet, Mathematics and Computers in Simulation, 166 (2019), Dec., pp. 144-154
- Shiralashetti, S., Deshi, A., Haar Wavelet Collocation Method for Solving Riccati and Fractional Riccati Differential Equations, Bulletin of Mathematical Sciences and Applications, 17 (2016), Nov., pp. 46-56
- Rubin, S. G., Graves Jr, R. A., A Cubic Spline Approximation for Problems in Fluid Mechanics, NASA Tech. Rep., NASA-TR-R-436, 1975