Transformed Generate Approximation method for generalized boundary value problems using first-kind Chebychev polynomials

In this paper, a new numerical method, the Transformed Generate Approximation Method (TGAM) is proposed for generalized boundary value problems with first-kind Chebychev polynomials as trial functions. In this method, the trial functions are substituted into the transformed system of ordinary differential equations in order to generate systems of linear algebraic equations satisfying the boundary conditions, which on solving yield the required approximate solution. The method is structurally simple as it requires no perturbation or discretization. The method is reliable in seeking the solution of boundary value problems as numerical illustrations reveal. Results obtained were compared with the exact solution and other methods available in literature. Also, convergence analysis of the method is presented. All computations are carried out with Maple 18 software.Keywords: Boundary value problem, Chebychev polynomials, Trial functions, Approximate solutio

Quadratic spline quasi-interpolants and collocation methods

International audienceUnivariate and multivariate quadratic spline quasi-interpolants provide interesting approximation formulas for derivatives of approximated functions that can be very accurate at some points thanks to the superconvergence properties of these operators. Moreover, they also give rise to good global approximations of derivatives on the whole domain of definition. From these results, some collocation methods are deduced for the solution of ordinary or partial differential equations with boundary conditions. Their convergence properties are illustrated on some numerical examples of elliptic boundary value problems

Analytical approximation schemes for solving exact renormalization group equations in the local potential approximation

The relation between the Wilson-Polchinski and the Litim optimized ERGEs in the local potential approximation is studied with high accuracy using two different analytical approaches based on a field expansion: a recently proposed genuine analytical approximation scheme to two-point boundary value problems of ordinary differential equations, and a new one based on approximating the solution by generalized hypergeometric functions. A comparison with the numerical results obtained with the shooting method is made. A similar accuracy is reached in each case. Both two methods appear to be more efficient than the usual field expansions frequently used in the current studies of ERGEs (in particular for the Wilson-Polchinski case in the study of which they fail).Comment: Final version to appear in Nucl. Phys. B. Some references added correctl

Numerical solutions of matrix differential models using higher-order matrix splines

The final publication is available at http://link.springer.com/article/10.1007%2Fs00009-011-0159-zThis paper deals with the construction of approximate solution of first-order matrix linear differential equations using higher-order matrix splines. An estimation of the approximation error, an algorithm for its implementation and some illustrative examples are included. © 2011 Springer Basel AG.Defez Candel, E.; Hervás Jorge, A.; Ibáñez González, JJ.; Tung, MM. (2012). Numerical solutions of matrix differential models using higher-order matrix splines. Mediterranean Journal of Mathematics. 9(4):865-882. doi:10.1007/s00009-011-0159-zS86588294Al-Said E.A., Noor M.A.: Cubic splines method for a system of third-order boundary value problems. Appl. Math. Comput. 142, 195–204 (2003)Ascher U., Mattheij R., Russell R.: Numerical solutions of boundary value problems for ordinary differential equations. Prentice Hall, New Jersey, USA (1988)Barnett S.: Matrices in Control Theory. Van Nostrand, Reinhold (1971)Blanes S., Casas F., Oteo J.A., Ros J.: Magnus and Fer expansion for matrix differential equations: the convergence problem. J. Phys. Appl. 31, 259–268 (1998)Boggs P.T.: The solution of nonlinear systems of equations by a-stable integration techniques. SIAM J. Numer. Anal. 8(4), 767–785 (1971)Defez E., Hervás A., Law A., Villanueva-Oller J., Villanueva R.: Matrixcubic splines for progressive transmission of images. J. Math. Imaging Vision 17(1), 41–53 (2002)Defez E., Soler L., Hervás A., Santamaría C.: Numerical solutions of matrix differential models using cubic matrix splines. Comput. Math. Appl. 50, 693–699 (2005)Defez E., Soler L., Hervás A., Tung M.M.: Numerical solutions of matrix differential models using cubic matrix splines II. Mathematical and Computer Modelling 46, 657–669 (2007)Mazzia F., Trigiante A.S., Trigiante A.S.: B-spline linear multistep methods and their conitinuous extensions. SIAM J. Numer. Anal. 44(5), 1954–1973 (2006)Faddeyev L.D.: The inverse problem in the quantum theory of scattering. J. Math. Physics 4(1), 72–104 (1963)Flett, T.M.: Differential Analysis. Cambridge University Press (1980)Golub G.H., Loan C.F.V.: Matrix Computations, second edn. The Johns Hopkins University Press, Baltimore, MD, USA (1989)Graham A.: Kronecker products and matrix calculus with applications. John Wiley & Sons, New York, USA (1981)Jódar L., Cortés J.C.: Rational matrix approximation with a priori error bounds for non-symmetric matrix riccati equations with analytic coefficients. IMA J. Numer. Anal. 18(4), 545–561 (1998)Jódar L., Cortés J.C., Morera J.L.: Construction and computation of variable coefficient sylvester differential problems. Computers Maths. Appl. 32(8), 41–50 (1996)Jódar, L., Ponsoda, E.: Continuous numerical solutions and error bounds for matrix differential equations. In: Int. Proc. First Int. Colloq. Num. Anal., pp. 73–88. VSP, Utrecht, The Netherlands (1993)Jódar L., Ponsoda E.: Non-autonomous riccati-type matrix differential equations: Existence interval, construction of continuous numerical solutions and error bounds. IMA J. Numer. Anal. 15(1), 61–74 (1995)Loscalzo F.R., Talbot T.D.: Spline function approximations for solutions of ordinary differential equations. SIAM J. Numer. Anal. 4(3), 433–445 (1967)Marzulli P.: Global error estimates for the standard parallel shooting method. J. Comput. Appl. Math. 34, 233–241 (1991)Micula G., Revnic A.: An implicit numerical spline method for systems for ode’s. Appl. Math. Comput. 111, 121–132 (2000)Reid, W.T.: Riccati Differential Equations. Academic Press (1972)Rektorys, K.: The method of discretization in time and partial differential equations. D. Reidel Pub. Co., Dordrecht (1982)Scott, M.: Invariant imbedding and its Applications to Ordinary Differential Equations. Addison-Wesley (1973

SIMULASI NUMERIK METODE BEDA HINGGA NON UNIFORM GIRD DALAM MENYELESAIKAN MASALAH NILAI BATAS PERSAMAAN DIFERENSIAL BIASA

Pada penelitian ini disimulasikan implementasi dari metode beda hingga nonuniform grid untuk menyelesaikan masalah nilai batas persamaan differensial biasa. Penelitian ini bertujuan mendapatkan pemahaman yang baik mengenai implementasi metode beda hingga nonuniform gird, dan mendapatkan solusi dengan akurasi tinggi untuk penyelesaian numerik persamaan diferensial biasa. Dibentuk aproksimasi turunan berdasarkan deret Taylor yang kemudian di implementasikan pada persamaan diferensial biasa yang telah didiskritisasi dengan gird yang berbeda. Penyelesaian persamaan diferensial tersebut disimulasikan menggunakan program MATLAB sehingga dapat ditarik beberapa kesimpulan tentang karakteristik, efisiensi, serta, akurasi dari metode beda hingga non uniform gird.In this paper, we simulate the implementation of finite difference method with uniform gird to solve boundary value problem of ordinary differential equation. This study aims to understand well the implementation of finite difference method and also to get numerical solution with high accuracy for ordinary differential equation. The derivative approximation is based on the Taylor series which is then implemented in ordinary differential equations that have been discretized  with nonuniform gird. Completion of differential equations is simulated using the MATLAB program. So, conclusions can be drawn about the characteristics, efficiency, and accuracy of finite differece methods

Modifications to a Runge-Kutta type software package for the numerical solution of boundary value ordinary differential equations

ix, 90 leaves : ill. ; 28 cm.Includes abstract.Includes bibliographical references (leaves 86-90).MIRKDC [Enright, Muir, 1996] is a software package for the numerical solution of systems of first order, nonlinear, boundary value ordinary differential equations (ODEs), with separated boundary conditions. It employs mono-implicit Runge-Kutta methods for the discretization of the ODEs and monitors the quality of the numerical solution using defect control. The discrete systems are solved by modified Newton iterations and extensive use of adaptive mesh refinement is employed. This thesis describes modifications to the MIRKDC software package in order to incorporate a number of performance enhancements including computational derivative approximation, analytic derivative assessment, problem sensitivity (conditioning) assessment, introduction of new optimized Runge-Kutta formulas, improvement of the defect control strategy and the introduction of an auxiliary global error indicator. Numerical results to demonstrate the impact of these enhancements are presented

Iterative method for solving a nonlinear fourth order boundary value problem

AbstractIn the study of transverse vibrations of a hinged beam there arises a boundary value problem for fourth order ordinary differential equation, where a significant difficulty lies in a nonlinear term under integral sign. In recent years several authors considered finite approximation of the problem and proposed an iterative method for solving the system of nonlinear equations obtained. The essence of the iteration is the simple iteration method for a nonlinear equation, although this is not shown in the papers of the authors.In this paper we propose a new approach to the solution of the problem, which is based on the reduction of it to finding a root of a nonlinear equation. In both cases, when the explicit form of this equation is found or not, the use of the Newton or Newton-type methods generate fast convergent iterative process for the original problem. The results of many numerical experiments confirm the efficiency of the proposed approach

Numerical approximations of second-order matrix differential equations using higher-degree splines

Many studies of mechanical systems in engineering are based on second-order matrix models. This work discusses the second-order generalization of previous research on matrix differential equations dealing with the construction of approximate solutions for non-stiff initial problems Y 00(x) = f(x, Y (x), Y 0 (x)) using higher-degree matrix splines without any dimensional increase. An estimation of the approximation error for some illustrative examples are presented by using Mathematica. Several MatLab functions have also been developed, comparing, under equal conditions, accuracy and execution times with built-in MatLab functions. Experimental results show the advantages of solving the above initial problem by using the implemented MatLab functions.The authors wish to thank for financial support by the Universidad Politecnica de Valencia [grant number PAID-06-11-2020].Defez Candel, E.; Tung ., MM.; Solis Lozano, FJ.; Ibáñez González, JJ. (2015). Numerical approximations of second-order matrix differential equations using higher-degree splines. Linear and Multilinear Algebra. 63(3):472-489. https://doi.org/10.1080/03081087.2013.873427S472489633Loscalzo, F. R., & Talbot, T. D. (1967). Spline Function Approximations for Solutions of Ordinary Differential Equations. SIAM Journal on Numerical Analysis, 4(3), 433-445. doi:10.1137/0704038Al-Said, E. A. (2001). The use of cubic splines in the numerical solution of a system of second-order boundary value problems. Computers & Mathematics with Applications, 42(6-7), 861-869. doi:10.1016/s0898-1221(01)00204-8Al-Said, E. A., & Noor, M. A. (2003). Cubic splines method for a system of third-order boundary value problems. Applied Mathematics and Computation, 142(2-3), 195-204. doi:10.1016/s0096-3003(02)00294-1Kadalbajoo, M. K., & Patidar, K. C. (2002). Numerical solution of singularly perturbed two-point boundary value problems by spline in tension. Applied Mathematics and Computation, 131(2-3), 299-320. doi:10.1016/s0096-3003(01)00146-1Micula, G., & Revnic, A. (2000). An implicit numerical spline method for systems for ODEs. Applied Mathematics and Computation, 111(1), 121-132. doi:10.1016/s0096-3003(98)10111-xDefez, E., Soler, L., Hervás, A., & Santamaría, C. (2005). Numerical solution ofmatrix differential models using cubic matrix splines. Computers & Mathematics with Applications, 50(5-6), 693-699. doi:10.1016/j.camwa.2005.04.012Defez, E., Hervás, A., Soler, L., & Tung, M. M. (2007). Numerical solutions of matrix differential models using cubic matrix splines II. Mathematical and Computer Modelling, 46(5-6), 657-669. doi:10.1016/j.mcm.2006.11.027Ascher, U., Pruess, S., & Russell, R. D. (1983). On Spline Basis Selection for Solving Differential Equations. SIAM Journal on Numerical Analysis, 20(1), 121-142. doi:10.1137/0720009Brunner, H. (2004). On the Divergence of Collocation Solutions in Smooth Piecewise Polynomial Spaces for Volterra Integral Equations. BIT Numerical Mathematics, 44(4), 631-650. doi:10.1007/s10543-004-3828-5Tung, M. M., Defez, E., & Sastre, J. (2008). Numerical solutions of second-order matrix models using cubic-matrix splines. Computers & Mathematics with Applications, 56(10), 2561-2571. doi:10.1016/j.camwa.2008.05.022Defez, E., Tung, M. M., Ibáñez, J. J., & Sastre, J. (2012). Approximating and computing nonlinear matrix differential models. Mathematical and Computer Modelling, 55(7-8), 2012-2022. doi:10.1016/j.mcm.2011.11.060Claeyssen, J. R., Canahualpa, G., & Jung, C. (1999). A direct approach to second-order matrix non-classical vibrating equations. Applied Numerical Mathematics, 30(1), 65-78. doi:10.1016/s0168-9274(98)00085-3Froese, C. (1963). NUMERICAL SOLUTION OF THE HARTREE–FOCK EQUATIONS. Canadian Journal of Physics, 41(11), 1895-1910. doi:10.1139/p63-189Marzulli, P. (1991). Global error estimates for the standard parallel shooting method. Journal of Computational and Applied Mathematics, 34(2), 233-241. doi:10.1016/0377-0427(91)90045-lShore, B. W. (1973). Comparison of matrix methods applied to the radial Schrödinger eigenvalue equation: The Morse potential. The Journal of Chemical Physics, 59(12), 6450-6463. doi:10.1063/1.1680025ZHANG, J. F. (2002). OPTIMAL CONTROL FOR MECHANICAL VIBRATION SYSTEMS BASED ON SECOND-ORDER MATRIX EQUATIONS. Mechanical Systems and Signal Processing, 16(1), 61-67. doi:10.1006/mssp.2001.1441Flett, T. M. (1980). Differential Analysis. doi:10.1017/cbo978051189719

The Construction of Finite Difference Approximations to Ordinary Differential Equations

Finite difference approximations of the form Σ^(si)_(i=-rj)d_(j,i)u_(j+i)=Σ^(mj)_(i=1) e_(j,if)(z_(j,i)) for the numerical solution of linear nth order ordinary differential equations are analyzed. The order of these approximations is shown to be at least r_j + s_j + m_j - n, and higher for certain special choices of the points Z_(j,i). Similar approximations to initial or boundary conditions are also considered and the stability of the resulting schemes is investigated