# Block Third Derivative Trigonometrically-Fitted Methods for Stiff and Periodic Problems

### Abstract

This article constructed and implemented a family of a third derivative trigonometric fitted method of order k+3 whose coefficients are functions of frequency and step size for the integration of systems of first-order stiff and periodic Initial Value Problems. The Block Third Derivative Trigonometric Fitted methods (BTDTFMs) are constructed via multistep collocation technique and applied in block form as simultaneous numerical integrators which make them self-starting. The convergence, accuracy, and efficiency of the methods are established through some standard numerical examples.

### References

G. G. Dahlquist, A special stability problem for linear multistep methods, BIT. 3 (1963) 27-43.

T. A. Biala S. N. Jator, R. B. Adeniyi, P. L. Ndukum, Block Hybrid Simpson’s Method with Two Offgrid Points for Stiff Systems. International Journal of Nonlinear Science, 20 (2015) 3-10.

C. W. Gear, Hybrid methods for initial value problems in ordinary differential equations, SIAM J. Numer. Anal. 2 (1965) 69-86.

W. H. Enright, Second Derivative Multistep Methods for Stiff ordinary differential equations, SIAM J. Numer. Anal. 11 (1974) 321-331.

J. R. Cash, On the Integration of Stiff Systems of ODEs Using Extended Backward Differentiation Formulae, Numerische Mathematik, 34 (1980) 235-246.

X. U. Wu. A sixth-order A-stable explicit one-step method for stiff systems, Comput math. Applic. 35(1998) 59-64.

G. Hojjati, M. Rahimi, S. M. Hosseini. New second derivative multistep methods for stiff systems, Appl. Math. Model., 30 (2006) 466 - 476.

S. N. Jator. Solving second order initial value problems by a hybrid multistep method without predictors. Applied Mathematics and Computation, 277(2010), 4036-4046.

R. K Sahi, S. N. Jator, N. A. Khan, A Simpson’s-Type Second Derivative Method for Stiff System. International Journal of Pure and Applied Mathematics, 81(4), 619-633.

F. F. Ngwane, S. N. Jator, Block hybrid-second derivative method for stiff systems, Intern. J. Pure Appl. Math. 4(2012): 543-559.

O. A. Akinfenwa, S. N. Jator, , N. Yao, Continuous block backward differentiation formula for stiff ordinary differential equations. Computer and Mathematics with Applications, 65(2013), 996-1005.

O. A. Akinfenwa, S. N. Jator, Extended continuous block backward differentiation formula for stiff systems. Fasciculi Mathematici. https://doi.org/10.1515/fascmath-2015-0010.

O. A. Akinfenwa, S. A. Okunuga, B. I. Akinnukawe, U. O. Rufai, R. I. Abdulganiy,

Multiderivative hybrid implicit Runge-Kutta method for solving Stiff system of a first order Differential equation. Far East Journal of Mathematical Sciences, 106 (2018), 543-562.

M. Mehdizadeh, M. Molayi, A new class of L-stable hybrid one-step methods for the numerical solution of ordinary differential equations. Journal of Computer Science and Applied Mathematics 1(2015)39-44.

R. I. Abdulganiy, O. A. Akinfenwa, S. A. Okunuga, A family of L_0 stable Third derivative Block Methods for solving systems of First Order Initial Value Problems. J. of NAMP, 36(2016), 47-54.

O. A. Akinfenwa, R. I. Abdulganiy S. A. Okunuga, V. Irechukwu, Simpson’s 3⁄8Type Block Method For Stiff Systems of Ordinary Differential Equations. Journal of the Nigerian Mathematical Society, 36(2017), 503-514.

S. A. Okunuga, A fourth order composite two step method for stiff problems. International Journal of Computer Mathematics, 72 (1999), 39-47.

J. M. Vaquero and J. Vigo-Aguiar, Exponential fitted Runge-Kutta methods of collocation type based on Gauss, Radau, and Labatto traditional methods, in Proceedings of the International Conference on Computational and Mathematical Methods in Science and Engineering (CMMSE ’07), 289–303.

C. E. Abhulimen, O. F. Otunta, A family of two step Exponentially fitted Multiderivative methods for the numerical integration of stiff IVPs on ODEs. International Journal of Numerical Mathematics, 13(2007),1-21.

C. E. Abhulimen, G. E. Omeike, A Six-Order Exponentially Fitted Scheme for The Numerical Solution of Systems of Ordinary Differential Equations. Journal of Applied Mathematics and Bioinformatics, 1(2011), 175-186.

J. O. Ehigie, S. A. Okunuga, A. B. Sofoluwe, A class of exponentially-Fitted second derivative extended backward differentiation formula for solving stiff problems. Fasciculi Mathematici, Nr51, (2013), 71-84.

A. O. Adesanya, R. O. Onsachi, , M. R. Odekunle, New algorithm for first order stiff initial value problems. Fasciculi Mathematici Nr 58-2017-0002.

D. G. Yakubu, M. Aminu, P. Tumba, M. Abdulhameed, An efficient family of second derivative Runge-Kutta collocation methods for oscillatory systems. Journal of the Mathematical Society, 37(2018)111-138.

G. Vanden Berghe, H. De Meyer, M. Van Daele, T. V. Hecke, Exponentially-Fitted explicit Runge-Kutta methods. Computer Physics Communications, 123(1999), 7-15.

H. Van de Vyver, Frequency evaluation for exponentially fitted Runge-Kutta methods. Journal of Computational and Applied Mathematics 184(2005)442-463.

W. Gautschi, Numerical Integration of Ordinary Differential Equations Based on Trigonometric Polynomials, Numerische Mathematik, 3(1961), 381-397.

T. Lyche, Chebyshevian Multistep Methods for Ordinary Differential Equations, Numerische Math., 19(1972), 65-75.

J. P. Coleman, S. C. Duxbury, Mixed collocation methods for y''=f(x,y). Journal of computational and Applied Mathematics, 126(2000), 47-75.

L. Gr. Ixaru , G. Vanden Berghe, H. De Meyer, Frequency evaluation in exponentially-fitted algorithms for ODEs. Journal of Computational and Applied Mathematics, 140(2002), 423-434.

G. Vanden Berghe, Ixaru L. Gr., M. Van Daele, Optimal implicit exponentially fitted Runge-Kutta methods. Computer Physics Communications, 140(2001), 346-357.

G. Vanden Berghe, M. Van Daele, Exponentially-fitted Numerov methods, J. Comp. Appl. Math., 200(2007), 140-153.

T. E. Simos, An Exponentially-Fitted Runge-Kutta Method for the Numerical Integration of Initial Value Problems with Periodic or Oscillating Solutions. Computer Physics Communications, 115(1998), 1-8.

T. E. Simos, Exponentially-Fitted Runge-Kutta-Nyström Method for the Numerical Solution of Initial Value-Problems with Oscillating Solutions. Applied Mathematics Letters, 15(2002), 217-225.

Ch. Tsitouras, T. E. Simos, Optimized Runge-Kutta pairs for problem with oscillatory solutions, Journal of computational and Applied Mathematics 147(2002), 397-409

H. S. Nguyen, R. B. Sidje, N. H. Cong, Analysis of trigonometric implicit Runge-Kutta methods. Journal of computational and Applied Mathematics, 198(2007), 187-207.

N. Senu, M. Suleimon, F. Ismail, M. Othman, A New Diagonally Implicit Runge-Kutta-Nystrom Method for Periodic IVPs. WSEAS Transactions on Mathematics, 9(2010), 679-688.

S. N. Jator, Trigonometric symmetric boundary value method for oscillating solutions including the sine-Gordon and Poisson equations. Applied & Interdisciplinary Mathematics, 3(2016), 1-16.

F. F. Ngwane, S. N. Jator, Solving Oscillatory Problems Using a Block Hybrid Trigonometrically Fitted Method with Two Off-Step Points. Texas State University. San Marcos, Electronic Journal of Differential Equation, 20(2013),119-132.

F. F. Ngwane, S. N. Jator, Block hybrid method using trigonometric basis for initial problems with oscillating solutions. Numerical Algorithm, 63(2013), 713-725.

S. N. Jator, S. Swindell, R. D. French, Trigonometrically Fitted Block Numerov Type Method for y^''=f(x,y,y^''). Numer Algor, 62(2013), 13-26.

F. F. Ngwane, S. N. Jator, Trigonometrically-Fitted Second Derivative Method for Oscillatory Problems. Springer Plus, 3(2014),304.

F. F. Ngwane, S. N. Jator, A Family of Trigonometrically Fitted Enright Second Derivative Methods for Stiff and Oscillatory Initial Value problems. Journal of Applied Mathematics, 2015, 1-17.

R. I. Abdulganiy, O. A. Akinfenwa, S. A. Okunuga, Maximal Order Block Trigonometrically Fitted Scheme for the Numerical Treatment of Second Order Initial Value Problem with Oscillating Solutions. International Journal of Mathematical Analysis and Optimization, 2017, 168 – 186.

R. I. Abdulganiy, O. A. Akinfenwa, S. A. Okunuga, G. O. Oladimeji, A robust block hybrid trigonometry method for the numerical integration of oscillatory second order nonlinear initial value problem. AMSE journals-AMSE IIETA publication 2017 series, 54(2017), 497-518.

R. I. Abdulganiy, O.A. Akinfenwa, S. A. Okunuga, Construction of L stable second derivative trigonometrically fitted block backward differentiation formula for the solution of oscillatory initial value problems, African Journal of Science, Technology, Innovation and Development, 10(2018), 411-419.

S. N. Jator and E. Agyingi, Block Hybrid

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