Solution of the Second Order of the Linear Hyperbolic Equation Using Cubic B-Spline Collocation Numerical Method

Aflakha Kharisa; Sri Maryani; Nunung Nurhayati

doi:10.31764/jtam.v6i2.7496

Solution of the Second Order of the Linear Hyperbolic Equation Using Cubic B-Spline Collocation Numerical Method

Aflakha Kharisa, Sri Maryani, Nunung Nurhayati

Abstract

Wave equation is one of the second order of the linear hyperbolic equation. Telegraph equation as a special case of wave equation has interesting point to investigate in the numerical point of view. In this paper, we consider the numerical methods for one dimensional telegraph equation by using cubic B-spline collocation method. Collocation method is one method to solve the partial differential equation model problem. Cubic spline interpolation is an interpolation to a third order polynomial. This polynomial interpolate four point. B-Spline is one of spline function which related to smoothness of the partition. For every spline function with given order can be written as linear combination of those B-spline. As we known that the result of the numerical technique has difference with the exact result which we called as, so that we have an error. The numerical results are compared with the interpolating scaling function method which investigated by Lakestani and Saray in 2010. This numerical methods compared to exact solution by using RMSE (root mean square error), L2 norm error and L_∞ norm error . The error of the solution showed that with the certain function, the cubic collocation of numerical method can be used as an alternative methods to find the solution of the linear hyperbolic of the PDE. The advantages of this study, we can choose the best model of the numerical method for solving the hyperbolic type of PDE. This cubic B-spline collocation method is more efficiently if the error is relatively small and closes to zero. This accuration verified by test of example 1 and example 2 which applied to the model problem.

Keywords

Cubic B-spline collocation method; Telegraph Equation; Interpolating scaling function; Numerical methods;

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References

Akram, T., Abbas, M., Ismail, A. I., Ali, N. H. M., & Baleanu, D. (2019). Extended cubic B-splines in the numerical solution of time fractional telegraph equation. Advances in Difference Equations, 2019(1), 1–20.

Atangana, A. (2015). On the stability and convergence of the time-fractional variable order telegraph equation. Journal of Computational Physics, 293, 104–114.

Biazar, J., & Eslami, M. (2010). Analytic solution for Telegraph equation by differential transform method. Physics Letters A, 374(29), 2904–2906.

Chen, J., Liu, F., & Anh, V. (2008). Analytical solution for the time-fractional telegraph equation by the method of separating variables. Journal of Mathematical Analysis and Applications, 338(2), 1364–1377.

Das, S., Vishal, K., Gupta, P. K., & Yildirim, A. (2011). An approximate analytical solution of time-fractional telegraph equation. Applied Mathematics and Computation, 217(18), 7405–7411.

Dehghan, M., & Ghesmati, A. (2010). Solution of the second-order one-dimensional hyperbolic telegraph equation by using the dual reciprocity boundary integral equation (DRBIE) method. Engineering Analysis with Boundary Elements, 34(1), 51–59.

Dehghan, M., & Salehi, R. (2012). A method based on meshless approach for the numerical solution of the two‐space dimensional hyperbolic telegraph equation. Mathematical Methods in the Applied Sciences, 35(10), 1220–1233.

Dosti, M., & Nazemi, A. (2012). Quartic B-spline collocation method for solving one-dimensional hyperbolic telegraph equation. Journal of Information and Computing Science, 7(2), 83–90.

Hosseini, V. R., Chen, W., & Avazzadeh, Z. (2014). Numerical solution of fractional telegraph equation by using radial basis functions. Engineering Analysis with Boundary Elements, 38, 31–39.

Javidi, M., & Nyamoradi, N. (2013). Numerical solution of telegraph equation by using LT inversion technique. International Journal of Advanced Mathematical Sciences, 1(2), 64–77.

Jiwari, R., Pandit, S., & Mittal, R. C. (2012). A differential quadrature algorithm for the numerical solution of the second-order one dimensional hyperbolic telegraph equation. International Journal of Nonlinear Science, 13(3), 259–266.

Lakestani, M., & Saray, B. N. (2010). Numerical solution of telegraph equation using interpolating scaling functions. Computers & Mathematics with Applications, 60(7), 1964–1972.

Mittal, R. C., & Jain, R. (2012). Redefined cubic B-splines collocation method for solving convection–diffusion equations. Applied Mathematical Modelling, 36(11), 5555–5573.

Phillips, G. M. (2003). Interpolation and approximation by polynomials (Vol. 14). Springer Science & Business Media.

PM, P. (1975). Spline and variational methods. Wiley, New York.

Rashidinia, J., & Jokar, M. (2016). Application of polynomial scaling functions for numerical solution of telegraph equation. Applicable Analysis, 95(1), 105–123.

Saadatmandi, A., & Dehghan, M. (2010). Numerical solution of hyperbolic telegraph equation using the Chebyshev tau method. Numerical Methods for Partial Differential Equations: An International Journal, 26(1), 239–252.

Sharifi, S., & Rashidinia, J. (2016). Numerical solution of hyperbolic telegraph equation by cubic B-spline collocation method. Applied Mathematics and Computation, 281, 28–38.

Srinivasa, K., & Rezazadeh, H. (2021). Numerical solution for the fractional-order one-dimensional telegraph equation via wavelet technique. International Journal of Nonlinear Sciences and Numerical Simulation, 22(6), 767–780.

Wang, K.-L., Yao, S.-W., Liu, Y.-P., & Zhang, L.-N. (2020). A fractal variational principle for the telegraph equation with fractal derivatives. Fractals, 28(04), 2050058.

DOI: https://doi.org/10.31764/jtam.v6i2.7496