A New class of rational multistep methods for solving initial value problem

There exists initial value problem whose solution possesses singularity.Studies show that conventional numerical method such as multistep method fail woefully near the singular point when solving problem whose solution possesses singularity.This is because a multistep method is based on the local representation of polynomial of the theoretical solution of an initial value problem.Therefore, a natural step would appear to be the replacement of the polynomial function for a multistep method, by a rational function due to its smooth behaviour in the neighbourhood of singularity.In this paper, we have developed a new class of two step numerical methods that are based on rational functions in solving general initial value problem and problem whose solution possesses singularity.These new methods are called rational multistep methods.The developments of these rational multistep methods, as well as the local truncation error and stability analysis for each rational multistep method are presented.We have found out that only the second order, third order and fourth order rational multistep methods are A-stable.Numerical experiments have showed that all newly developed rational multistep methods presented in this paper are suitable to solve general initial value problem, stiff problem and problem whose solution possesses singularity

Numerical solution of first order initial value problem using 7-stage tenth order Gauss-Kronrod-Lobatto IIIA method

In this paper, a new implicit Runge-Kutta method which based on a 7-point Gauss-Kronrod-Lobatto quadrature formula is developed.The resulting implicit method is a 7-stage tenth order Gauss-Kronrod-Lobatto IIIA method, or in brief as GKLM(7,10)-IIIA. GKLM(7,10)-IIIA requires seven function of evaluations at each integration step and it gives accuracy of order ten.In addition, GKLM(7,10)-IIIA has stage order seven and being A-stable. Numerical experiments compare the accuracy between GKLM(7,10)-IIIA and the classical 5-stage tenth order Gauss-Legendre method in solving some test problems. Numerical results reveal that GKLM(7,10)-IIIA is more accurate than the 5-stage tenth order Gauss-Legendre method because GKLM(7,10)-IIIA has higher stage orde

New nonlinear four-step method for y"=f(t,y)

In this paper, a study is made on the possibility of developing a nonlinear four-step method based on contraharmonic mean. The study is done since the four-step methods always give higher order than popular methods like Numerov and classical Runge-Kutta methods. A detailed study of consistency, stability, convergence and interval of periodicity has been done to convince ourselves of using this new method. The numerical results shows that the method is more accurate than the existing one

A new fourth-order embedded method based on the harmonic mean

In this paper we formulate an embedded method based on harmonic and arithmetic means of order 4. This method together with RK-Harmonic scheme may be used to estimate the solutions of initial value problems. The absolute stability region of this scheme is also studied and we conclude with a numerical example to justify the effectiveness of the metho

Sin-Cos-Taylor-Like method for solving stiff ordinary diffrential equations

This paper discusses the derivation of an explicit Sin-Cos-Taylor-Like method for solving stiff ordinary differential equations, which is a formulation of the combination of a polynomial and the exponential function. This new method requires extra work to evaluate a number of differentiations of the function involved. However, the result shows smaller errors when compared to the results from the explicit classical fourth-order Runge-Kutta (RK4) and the Adam- Bashforth-Moulton (ABM) methods. Implicit methods could work well for stiff problems but have certain drawbacks especially when discussing about the cost. Although extra work is required, this explicit method has its own advantages. Besides providing excellent results, the cost of computation using this explicit method is much cheaper than the implicit methods. We also considered the stability property for this method since the stability property of the classical explicit fourth order Runge-Kutta method is not adequate for the solution of stiff problems. As a result, we find that this explicit method is of order-6, which has been developed, and proved to be both A-stable and L-stable

A new non-linear multistep method based on centroidal mean in solving initial value problems

A new 2-step fourth order implicit non-linear multistep method based on centroidal mean is considered in this paper. The new method is tested on some test problems; and numerical results show that the new method is able to produce acceptable numerical solutions for these test problems. Comparisons in terms of numerical accuracy between the new method and the classical 2-step Adams-Moulton method are carried out as well. Numerical experiments show that our new method performs better than the classical 2-step Adams-Moulton method in solving these test problem

One step cosine–taylorlike method for solving stiff equations

This paper discusses the derivation of an explicit cosine-Taylorlike method for solving stiff ordinary differential equations. The formulation has resulted in the introduction of a new formula for the numerical solution of stiff ordinary differential equations. This new method needs an extra work in order to solve a number of differentiations of the function involved. however, the result produced is better than the results from the explicit classical fourth-order runge-kutta (rk4) and the implicit adam-bashforth-Moulton (abM) methods. When compared with the previously derived Sine-Taylorlike method, the accuracy for both methods is almost equivalent

Langkah-langkah awal dalam analisis berangka

Walaupun ia bukan merupakan suatu bidang yang baru diusahakan, namun dengan kehadiran alat-alat bantu mengira yang murah sejak kebelakangan ini telah memperluaskan lagi teori dan penggunaan Analisis Berangka. Oleh yang demikian, pelajar-pelajar daripada pelbagai disiplin memperolehi faedah daripada pengetahuan asas dan teknik dalam subjek ini. Langkah-langkah Awal dalam Analisis Berangka merupakan suatu pengenalan terhadap ide-ide asas dan kaedah-kaedah dalam Analisis Berangka. Bahan-bahan subjek ini disusun sebagai topik-topik dasar dan disajikan sebagai suatu siri ‘langkah-langkah’ dengan setiap satunya mengandungi bahan yang mencukupi untuk dua kuliah. Pada akhir setiap langkah diberikan suatu ‘ulang kaji’ agar dapat menguji kefahaman pelajar mengenai konsep-konsep yang telah diperkenalkan. Ini disusuli dengan suatu set latihan-latihan dan jawapannya diberi pada akhir bahagian buku ini. Beberapa ‘langkah’ tambahan (bertanda bintang) disertakan untuk pelajar-pelajar yang mempunyai pengetahuan yang lebih tinggi. Buku ini sesuai sekali dibaca oleh semua peringkat pelajar di maktab-maktab dan sekolah-sekolah yang menawarkan Analisis Berangka sebagai kursus pengenalan

Penganggaran dan pengawalan ralat dalam kaedah HaM-4(4) untuk penyelesaian masalah nilai awal

In this papaer a new fourth order formula based on harmonic and arithmatic means is formulated. The formula when used with the Harmonic-Rungge-Kutta formula is capable of estimating the local truncution error in solving initial value problems. The absolute stability region of the method is also studied. The numerical results presented showed the effectiveness of the metho

One-step exponential-rational methods for the numerical solution of first order initial value problems

In this study, a new class of exponential-rational methods (ERMs) for the numerical solution of first order initial value problems has been developed. Developments of third order and fourth order ERMs, as well as their corresponding local truncation error have been presented. Each ERM was found to be consistent with the differential equation and L-stable. Numerical experiments showed that the third order and fourth order ERMs generates more accurate numerical results compared with the existing rational methods in solving first order initial value problems