Two derivative Runge-Kutta method with FSAL property for the solution of first order initial value problems

A new Two Derivative Runge-Kutta method (TDRK) based on First Same as Last (FSAL) technique for the numerical solution of first order Initial Value Problems (IVPs) is derived. We present a fourth order three stages TDRK method designed using the FSAL property. The stability of the new method is analyzed. The numerical experiments are carried out to show the efficiency of our methods in comparison with other existing Runge-Kutta methods (RK)

Pembelajaran Analisis Berangka dalam Talian

Kemajuan dalam teknologi komputer dan antara muka manusia-komputer telah banyak membuka peluang barn kepada perkembangan pembelajaran berbantukan komputer dan sistem kecerdasan penunjukajar. Malangnya, kebanyakan bahan tersebut hanya dalam bentuk "buku elektronik", di mana pembelajaran masih lagi melalui pembacaan teks dan pemaparan imej dua dimensi. Projek ini telah menghasilkan Makmal Pengiraan Serangka (MPS) yang bersifat interaktif dan sistem pembelajaran yang berasaskan web. Sistem MPS terdiri daripada kalkulator anal isis berangka, nota, tutorial &n beberapa pautan yang membolehkan pelajar mendapatkan maklumat yang berguna. Melalui MPS, pelajar boleh mencapai nota kuliah dalam masa 24 jam sehari, menyelesaikan tutorial bagi menguji kefahaman, dan pada masa yang sama mereka boleh menggunakan kalkulator yang mempunyai ciri penyelesai persamaan, animasi, dan grafik sebagai bahan sokongan pembelajaran serta kemudahan e-mel untuk menerima maklumbalas berkenaan bahan kursus. Kalkulator analisis berangka telah dibangunkan menggunakan Tool Command Language/ Tool Kit (Telffk). Telffk adalah bahasa skrip yang menyokong banyak ciri seperti pembinaan Antara Muka Grafik Pengguna (GUI) yang canggih, aplikasi cross-platform, integrasi fleksibel, aplikasi rangkaian dan aplikasi terbenam (embedded). Pemalam Tel merupakan kemudahan yang membolehkan aplikasi dilarikan atas penyemak seimbas (browser). Melalui kalkulator, pelajar boleh memaparkan dan menambah kefahaman tentang konsep kaedah berangka seperti menyelesaikan Persamaan Linear Serentak, Pembezaan dan Pengamiran Berangka, Interpolasi, Persamaan Tak Linear, Penghampiran Fungsi dan Pemadanan Data dan Persamaan Pembezaan Biasa. Nota analisis berangka dan tutorial disediakan menggunakan LaTeX yang kemudiannya ditukar kepada fail PDF

Runge-Kutta-Nystrom Methods For Solving Oscillatory Problems

New Runge-Kutta-Nyström (RKN) methods are derived for solving system of second-order Ordinary Differential Equations (ODEs) in which the solutions are in the oscillatory form. The dispersion and dissipation relations are imposed to get methods with the highest possible order of dispersion and dissipation. The derivation of Embedded Explicit RKN (ERKN) methods for variable step size codes are also given. The strategies in choosing the free parameters are also discussed. We analyze the numerical behavior of the RKN and ERKN methods both theoretically and experimentally and comparisons are made over the existing methods. In the second part of this thesis, a Block Embedded Explicit RKN (BERKN) method are developed. The implementation of BERKN method is discussed. The numerical results are compared with non block method. We find that the new code on Block Embedded Explicit RKN (BERKN) method is more efficient for solving system of second-order ODEs directly. Next, we discussed the derivation of Diagonally Implicit RKN (DIRKN) methods for solving stiff second order ODEs in which the solutions are oscillating functions. The dispersion and dissipation relations are developed and again are imposed in the derivation of the methods. For solving oscillatory problems with high frequency, method with P-stability property is discussed. We also derive the Embedded Diagonally Implicit RKN (EDIRKN) methods for variable step size codes. To see the preciseness and effectiveness of the methods, the constant and variable step size codes are developed and numerical results are compared with current methods given in the literature. Finally, the Parallel Embedded Explicit RKN (PERKN) method is developed. The parallel implementation of PERKN on the parallel machine is discussed. The performance of the PERKN algorithm for solving large system of ODEs are presented. We observe that the PERKN gives the better performance when solving large system of ODEs. In conclusion, the new codes developed in this thesis are suitable for solving system of second-order ODEs in which the solutions are in the oscillatory form

A sixth-order RKFD method with four-stage for directly solving special fourth-order ODEs

In this article, the general form of Runge-Kutta method for directly solving a special fourth- order ordinary differential equations denoted as RKFD method is given. The order conditions up to order seven are derived, based on the order conditions, we construct a new explicit four-stage sixth-order RKFD method denoted as RKFD6 method. Zero-stability of the method is proven. Comparisons are made using the existing Runge–Kutta methods after the problems are reduced to a system of first order ordinary differential equations. Numerical results are presented to illustrate the efficiency and competency of the new method

Exponentially-fitted Runge-Kutta Nystrom method of order three for solving oscillatory problems

In this paper the exponentially fitted explicit Runge-Kutta Nystrom method is proposed for solving special second-order ordinary differential equations where the solution is oscillatory. The exponentially fitting is based on given Runge-Kutta Nystrom (RKN) method of order three at a cost of three function evaluations per step. Here, we also developed the trigonometrically-fitted RKN method for solving initial value problems with oscillating solutions. The numerical results compared with the existing explicit RKN method of order three which indicates that the exponentially fitted explicit Runge-Kutta Nystrom method is more efficient than the classical RKN method

Semi implicit hybrid methods with higher order dispersion for solving oscillatory problems

In this paper, two-step fourth order semi implicit hybrid method (SIHM) with dispersion of order six and zero dissipation is constructed for solving second order ordinary differential equations (ODEs). Numerical results show that SIHM is more accurate as compared to the existing hybrid method, Runge-Kutta Nyström (RKN) method, Runge-Kutta (RK) method and Diagonally Implicit Runge-Kutta Nyström (DIRKN) method of the same order. The interval of absolute stability of SIHM for ODE is presented. The comparison of time for solving the test problems for the various methods is also given

Linear 3 and 5-step methods using Taylor series expansion for solving special 3rd order ODEs

Some new linear 3 and 5-step methods for solving special third order ordinary differential equations directly are constructed using Taylor's series expansion. A set of test problems are solved using the new method and the results are compared when the problem is reduced to a system of first order ordinary differential equations and then using the existing Runge-Kutta method. The numerical results have clearly shown the advantage and competency of the new methods

Solving directly special fourth-order ordinary differential equations using Runge-Kutta type method

In this paper, an explicit Runge–Kutta method for solving directly fourth-order ordinary differential equations (ODEs) is constructed and denoted as (RKFD). We present a relevant-colored tree theory and the associated B-series theory for the order conditions. Based on the order conditions a three-stage fourth-order RKFD method and a three-stage fifth-order RKFD method are constructed. Numerical illustrations are presented to show the efficiency of the new RKFD methods by comparing them with other existing Runge–Kutta Nyström (RKN) and Runge–Kutta (RK) methods in the scientific literature after converting the problem into a system of second order ODEs and a system of first order ODEs respectively

A new optimized Runge-Kutta method for solving oscillatory problems

A new explicit Runge-Kutta method of fifth algebraic order is developed in this paper, for solving second-order ordinary differential equations with oscillatory solutions. The new method has zero phase-lag, zero amplification error and zero first derivative of the phase-lag. Numerical results show that the new proposed method is more efficient as compared with other Runge-Kutta methods in the scientific literature, for the numerical integration of oscillatory problems

An accurate block hybrid collocation method for third order ordinary differential equations

The block hybrid collocation method with two off-step points is proposed for the direct solution of general third order ordinary differential equations. Both the main and additional methods are derived via interpolation and collocation of the basic polynomial. These methods are applied in block form to provide the approximation at five points concurrently. The stability properties of the block method are investigated. Some numerical examples are tested to illustrate the efficiency of the method. The block hybrid collocation method is also implemented to solve the nonlinear Genesio equation and the problem in thin film flow