Study of One Dimensional Hyperbolic Telegraph Equation Via a Hybrid Cubic B-Spline Differential Quadrature Method

Brajesh Kumar Singh1, J. P. Shukla1, Mukesh Gupta1
1School of Physical and Decision Sciences, Department of Mathematics, Babasaheb Bhimrao Ambedkar University, Lucknow, India

Tóm tắt

Từ khóa


Tài liệu tham khảo

El-Azab, M.S., El-Ghamel, M.: A numerical algorithm for the solution of telegraph equations. Appl. Math. Comput. 190, 757–764 (2007)

Mohanty, R.K., Jain, M.K., George, K.: On the use of high order difference methods for the system of one space second order non-linear hyperbolic equations with variable coefficients. J. Comput. Appl. Math. 72, 421–431 (1996)

Twizell, E.H.: An explicit difference method for the wave equation with extended stability range. BIT Numer. Math. 19(3), 378–383 (1979)

Mohebbi, A., Dehghan, M.: High order compact solution of the one-space-dimensional linear hyperbolic equation. Numer. Methods Partial Differ. Equ. 24, 1222–1235 (2008)

Gao, F., Chi, C.M.: Unconditionally stable difference schemes for a one space- dimensional linear hyperbolic equation. Appl. Math. Comput. 187(2), 1272–1276 (2007)

Mohanty, R.K.: New unconditionally stable difference schemes for the solution of multidimensional telegraphic equations. Int. J. Comput. Math. 86(12), 2061–2071 (2009)

Srivastava, V.K., Singh, B.K.: A Robust finite difference scheme for the numerical solutions of two dimensional time dependent coupled nonlinear Burgers’ equations. Int. J. Appl. Math. Mech. 10(7), 28–39 (2014)

Singh, B.K., Arora, G., Kumar, P.: A note on solving the fourth-order Kuramoto–Sivashinsky equation by the compact finite difference scheme. Ain Shams Eng. J. 9(4), 1581–1589 (2018)

Lakestani, M., Saray, B.N.: Numerical solution of telegraph equation using interpolating scaling functions. Comput. Math. Appli. 60(7), 1964–1972 (2010)

Saadatmandi, A., Dehghan, M.: Numerical solution of hyperbolic telegraph equation using the Chebyshev Tau method. Numer. Methods Partial Differ. Equ. 26(1), 239–252 (2010)

Jafari, H., Tajadodi, H., Baleanu, D.: A numerical approach for fractional order Riccati differential equation using B-spline operational matrix. Fract. Calc. Appl. Anal. 18(2), 387 (2015)

Rashidinia, J., Jamalzadeh, S., Esfahani, F.: Numerical solution of one dimensonal telegraph equation using cubic B-spline collocation method. J. Interpol. Approx. Sci. Comput. 2014, 1–14 (2014)

Dosti, M., Nazemi, A.: Quartic B-Spline collocation method for solving one-dimensional hyperbolic telegraph equation. J. Inform. Comput. Sci. 7(2), 83–090 (2012)

Mittal, R.C., Bhatia, R.: Numerical solutions of second order one dimensonal hyperbolic telegraph equation by cubic Bspline collocation method. Appl. Math. Comput. 222, 496–506 (2013)

Singh, S., Singh, S., Arora, R.: Numerical solution of second-order one-dimensional hyperbolic equation by exponential B-spline collocation method. Numer. Anal. Appl. 10(2), 164–176 (2017)

Wasim, I., Abbas, M., Amin, M.: Hybrid B-spline collocation method for solving the generalized Burgers–Fisher and Burgers–Huxley equations. Math. Probl. Eng. 18 (2018), Article ID 6143934. https://doi.org/10.1155/2018/6143934

Arora, G., Mittal, R.C., Singh, B.K.: Numerical solution of BBM-Burger equation with quartic B-spline collocation method. J. Eng. Sci. Technol. 9(1), 104–116 (2014)

Ersoy, O., Dag, I.: Numerical solutions of the reaction diffusion system by using exponential cubic B-spline collocation algorithms. Open Phys. 13, 414–427 (2015)

Ramezani, M., Jafari, H., Johnston, S.J., Baleanu, D.: Complex B-spline collocation method for solving weakly singular Volterra integral equations of the second kind. Miskolc Math. Notes 16(2), 1091–1103 (2015)

Jafari, H., Khalique, C.M., Ramezani, M., Tajadodi, H.: Numerical solution of fractional differential equations by using fractional B-spline. Cent. Eur. J. Phys. 11(10), 1372–1376 (2013)

Mittal, R.C., Arora, G.: Numerical solution of the coupled viscous Burgers’ equation. Commun. Nonlinear Sci. Numer. Simulat. 16(2011), 1304–1313 (2010)

Abbas, M., Majid, A.A., Ismail, A.I., Rashid, A.: Numerical method using cubic B-spline for a strongly coupled reaction-diffusion system. PLoS ONE 9(1), e83265 (2014)

Hashmi, M.S., Awais, M., Waheed, A., Ali, Q.: Numerical treatment of Hunter Saxton equation using cubic trigonometric B-spline collocation method. AIP Adv. 7, 095124 (2017)

Mat Zin, S., Abbas, M., Abd Majid, A., Md Ismail, A.I.: A new trigonometric spline approach to numerical solution of generalized nonlinear Klien–Gordon equation. PLoS ONE 9(5), e95774 (2014). https://doi.org/10.1371/journal.pone.0095774

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

Bellman, R., Kashef, B.G., Casti, J.: Differential quadrature: a technique for the rapid solution of nonlinear differential equations. J. Comput. Phy. 10, 40–52 (1972)

Shu, C., Richards, B.E.: Application of generalized differential quadrature to solve two dimensional incompressible Navier–Stokes equations. Int. J. Numer. Methods Fluids 15, 791–798 (1992)

Korkmaz, A., Dag, I.: Cubic B-spline differential quadrature methods and stability for Burgers’ equation. Eng. Comput. Int. J. Comput. Aided Eng. Softw. 30(3), 320–344 (2013)

Arora, G., Singh, B.K.: Numerical solution of Burgers’ equation with modified cubic B-spline differential quadrature method. Appl. Math. Comput. 224, 166–177 (2013)

Singh, B.K., Kumar, P.: A novel approach for numerical computation of Burgers equation (1+1) and (2+1) dimension. Alex. Eng. J. 55(4), 3331–3344 (2016)

Singh, B.K.: A novel approach for numeric study of 2D biological population model. Cogent Math. 3(1), 1261527 (2016). https://doi.org/10.1080/23311835.2016.1261527

Singh, B.K., Arora, G.: A numerical scheme to solve Fisher-type reaction-diffusion equations. Nonlinear Stud. Mesa-Math. Eng. Sci. Aerosp. 5(2), 153–164 (2014)

Singh, B.K., Arora, G., Singh, M.K.: A numerical scheme for the generalized Burgers–Huxley equation. J. Egypt. Math. Soc. (2016). https://doi.org/10.1016/j.joems.2015.11.003

Singh, B.K., Bianca, C.: A new numerical approach for the solutions of partial differential equations in three-dimensional space. Appl. Math. Inf. Sci. 10(5), 1–10 (2016)

Singh, B.K., Kumar, P.: A novel approach for numerical study of two dimensional hyperbolic telegraph equation. Alex. Eng. J. (2016). https://doi.org/10.1016/j.aej.2016.11.009

Singh, B.K., Kumar, P.: An algorithm based on a new DQM with modified extended cubic B-splines for numerical study of two dimensional hyperbolic telegraph equation. Alex. Eng. J. 57(1), 175–191 (2018)

Singh, B.K., Kumar, P.: An algorithm based on DQM with modified trigonometric cubic B-splines for solving coupled viscus Burger’s equations. Commun. Numer. Anal. 2018(1), 21–41 (2018)

Mittal, R.C., Rohila, R.: Numerical simulation of reaction-diffusion systems by modified cubic B-spline differential quadrature method. Chaos Solitons Fractals 92, 9–19 (2016)

Dehghan, M.: On the solution of an initial-boundary value problem that combines Neumann and integral condition for the wave equation. Numer. Methods Partial Differ. Equ. 21, 24–40 (2005)

Dehghan, M., Shokri, A.: A numerical method for solving the hyperbolic telegraph equation. Numer. Methods Partial Differ. Equ. 24, 1080–1093 (2008)

Dehghan, M., Ghesmati, A.: Solution of the second-order one-dimensional hyperbolic telegraph equation by using the dual reciprocity boundary integral equation (DRBIE) method. Eng. Anal. Bound. Elem 34, 51–59 (2010)

Spiteri, J.R., Ruuth, S.J.: A new class of optimal high-order strongstability-preserving time-stepping schemes. SIAM J. Numer. Anal. 40(2), 469–491 (2002)

Dosti, M., Nazemi, A.: Solving one-dimensional hyperbolic telegraph equation using cubic B-spline quasi-interpolation. World Acad. Sci. Eng. Technol. 52, 935–940 (2011)

Jain, M.K.: Numerical Solution of Differential Equations, 2nd edn. Wiley, New York, NY (1983)