217 research outputs found

    RKN-type parallel block PC methods with Lagrange-type predictors

    Get PDF
    AbstractThis paper describes the construction of block predictor-corrector methods based on Runge-Kutta-Nyström correctors. Our approach is to apply the predictor-corrector method not only at step points, but also at off-step points (block points), so that in each step, a whole block of approximations to the exact solution at off-step points is computed. In the next step, these approximations are used to obtain a high-order predictor formula using Lagrange interpolation. By suitable choice of the abscissas of the off-step points, a much more accurately predicted value is obtained than by predictor formulas based on last step values. Since the block of approximations at the off-step points can be computed in parallel, the sequential costs of these block predictor-corrector methods are comparable with those of a conventional predictor-corrector method. Furthermore, by using Runge-Kutta-Nyström corrector methods, the computation of the approximation at each off-step point is also highly parallel. Application of the resulting block predictor-corrector methods to a few widely-used test problems reveals that the sequential costs are reduced by a factor ranging from 4 to 50 when compared with the best sequential methods from the literature

    Numerical solution of Black–Scholes Partial Differential Equation using Direct solution of second-order Ordinary Differential Equation with two-step hybrid Block Method of Order seven

    Get PDF
    This paper proposes a new numerical solution of Black-Scholes Partial Differential Equation using Direct solution of second-order Ordinary Differential Equation ODE with two-step hybrid Block Method of Order seven directly. The method is developed using interpolation and collocation techniques. The use of the power series approximate solution as an interpolation polynomial and its second derivative as a collocation equation is considered in deriving the method. Properties of the method such as zero stability, order, consistency, convergence and region of absolute stability are investigated The new method is then applied to solve Black–Scholes equation after converting it to the system of second-order ordinary differential equations and the accuracy is better when compared with the existing methods in terms of error

    Formulation of a new implicit method for group implicit BBDF in solving related stiff ordinary differential equations

    Get PDF
    This paper proposed a new alternative approach of the implicit diagonal block backward differentiation formula (BBDF) to solve linear and nonlinear first-order stiff ordinary differential equations (ODEs). We generate the solver by manipulating the numbers of back values to achieve a higher-order possible using the interpolation procedure. The algorithm is developed and implemented in C ++ medium. The numerical integrator approximates few solution points concurrently with off-step points in a block scheme over a non-overlapping solution interval at a single iteration. The lower triangular matrix form of the implicit diagonal causes fewer differentiation coefficients and ultimately reduces the execution time during running codes. We choose two intermediate points as off-step points appropriately, which are proven to guarantee the method's zero stability. The off-step points help to increase the accuracy by optimizing the local truncation error. The proposed solver satisfied theoretical consistency and zero-stable requirements, leading to a convergent multistep method with third algebraic order. We used the well-known and standard linear and nonlinear stiff IVP problems used in literature for validation to measure the algorithm's accuracy and processor time efficiency. The performance metrics are validated by comparing them with a proven solver, and the output shows that the alternative method is better than the existing one

    One step hybrid block methods with generalised off-step points for solving directly higher order ordinary differential equations

    Get PDF
    Real life problems particularly in sciences and engineering can be expressed in differential equations in order to analyse and understand the physical phenomena. These differential equations involve rates of change of one or more independent variables. Initial value problems of higher order ordinary differential equations are conventionally solved by first converting them into their equivalent systems of first order ordinary differential equations. Appropriate existing numerical methods will then be employed to solve the resulting equations. However, this approach will enlarge the number of equations. Consequently, the computational complexity will increase and thus may jeopardise the accuracy of the solution. In order to overcome these setbacks, direct methods were employed. Nevertheless, most of these methods approximate numerical solutions at one point at a time. Therefore, block methods were then introduced with the aim of approximating numerical solutions at many points simultaneously. Subsequently, hybrid block methods were introduced to overcome the zero-stability barrier occurred in the block methods. However, the existing one step hybrid block methods only focus on the specific off-step point(s). Hence, this study proposed new one step hybrid block methods with generalised off-step point(s) for solving higher order ordinary differential equations. In developing these methods, a power series was used as an approximate solution to the problems of ordinary differential equations of order g. The power series was interpolated at g points while its highest derivative was collocated at all points in the selected interval. The properties of the new methods such as order, error constant, zero-stability, consistency, convergence and region of absolute stability were also investigated. Several initial value problems of higher order ordinary differential equations were then solved using the new developed methods. The numerical results revealed that the new methods produced more accurate solutions than the existing methods when solving the same problems. Hence, the new methods are viable alternatives for solving initial value problems of higher order ordinary differential equations directly

    Generalized one-step third derivative implicit hybrid block method for the direct solution of second order ordinary differential equation

    Get PDF
    In this article, an implicit hybrid method of order six is developed for the direct solution of second order ordinary differential equations using collocation and interpolation approach.To derive this method, the approximate solution power series is interpolated at the first and off-step points and its second and third derivatives are collocated at all points in the given interval.Besides having good numerical method properties, the new developed method is also superior to the existing methods in terms of accuracy when solving the same problems

    New strategy of developing a direct two-step implicit hybrid block method with generalized Two off-step points for third order ordinary differential equations

    Get PDF
    This paper proposes new strategy in developing a direct two-step implicit hybrid block method with generalized two off-step points for initial value problems (IVPs) of third order ordinary differential equations (ODEs). In this strategy, two off-step points are confined in the second step of two-step interval. The main continuous schemes are obtained through interpolating approximate solutions in the form of power series at three the grid points while its second derivatives are collocated at all the grid points in the interval. The analysis of the method such as order, zero stability, consistency and convergence are also discussed. It was found that the proposed method outperforms the existing methods. Hence, it can be considered as a viable alternative method to solve the thirdorder of IVPs directly

    One-Step Implicit Hybrid Block Method for The Direct Solution of General Second Order Ordinary Differential Equations

    Get PDF
    A one-step implicit hybrid block solution method for initial value problems of general second order ordinary differential equations has been studied in this paper. The onestep method is augmented by the inclusion of off step points to enable the multistep procedure. This guaranteed zero stability as well as consistency of the resulting method. The convergence and weak stability properties of the new method have been established. Results from the new method compared with those obtained from existing methods show that the new method gives better accuracy

    Implicit Two Step Adam Moulton Hybrid Block Method with Two Off-Step Points for Solving Stiff Ordinary Differential Equations

    Get PDF
    A two step block hybrid Adam Moulton method of uniform order five is presented for the solution of stiff initial value problems. The individual schemes that made up the block method are obtained from the same continuous scheme which is applied to provide the solutions of stiff initial value problems on non overlapping intervals. The constructed block method is consistent, zero – stable and A – stable. Numerical results obtained using the new block method show that it is superior for stiff systems and competes well with existing ones. Keywords: stiff ODEs, Block Method, Adam Moulton method, Stabilit

    Direct solution of higher order ordinary differential equations using one-step hybrid block methods with generalised off-step points In the presence of higher derivative

    Get PDF
    A great number of physical phenomena can be expressed as initial or boundary value problems of higher order ordinary differential equations (ODEs) which may not have analytical solutions. Thus, there is a need to develop numerical methods for approximating the solution of higher order ODEs. One of the well-known direct methods which frequently employed is block method. Even though this method is capable of finding the approximate solutions at several points simultaneously, it fails to overcome the zero-stability barrier. Thus, a hybrid block method was introduced to tackle this drawback. The main benefit of this method is its ability of using data at off-step points which contribute to better accuracy. Most of the existing hybrid block methods, however, only focus on specific off-step point(s) in deriving the methods with the exception of the method proposed by Abdelrahim in 2016. Although he has successfully developed one-step hybrid block methods with generalised off-step point(s) for solving high order ODEs directly, nevertheless, the methods are only confined to initial value problems. Moreover, he did not consider higher derivative in developing those methods. Thus, this study introduced new one-step hybrid block methods with generalised off-step point(s) in the presence of higher derivative for directly solving higher order ODEs. In developing these methods, a power series was used as an approximate solution to the problems of ODEs of order m. The power series was interpolated at m points, while its mth and (m+1)th derivatives were collocated at all points in the given interval. Investigations on the properties of the new methods such as order, error constant, zero-stability, consistency, convergence and region of absolute stability were also carried out. Several initial and boundary value problems of higher order ODEs considered in literature were then solved by using the newly developed methods in order to investigate the accuracy of the solution in terms of error. The numerical results revealed that, in general, the new methods were able to produce smaller errors compared to the existing methods in solving the same problems. In conclusion, this study has successfully developed viable methods for directly solving both initial and boundary value problems of higher order ODEs

    Direct solution of fourth order ordinary differential equations using a one step hybrid block method of order five

    Get PDF
    In this article, a power series of order eight is adopted as a basis function to develop one step hybrid block method with three off step points for solving general fourth order ordinary differential equations. The strategy is employed for the developing this method are interpolating the power series at xnx_n and all off-step points and collocating its fourth derivative at all points in the selected interval. The method derived is proven to be consistent, zero stable and convergent with order five. Taylor’s series is used to supply the starting values for the implementation of the method while the performance of the method is tasted by solving linear and non-linear problems
    corecore