Journal of the Nigerian Society of Physical Sciences
It is a known fact that in most cases, to integrate an oscillatory problem, higher order A-stable... more It is a known fact that in most cases, to integrate an oscillatory problem, higher order A-stable methods are often needed. This is because such problems are characterized by stiffness, chaos and damping, thus making them tedious to solve. However, in this research, an accuracy-preserving relatively lower order Block Hybrid Algorithm (BHA) is proposed for solution of second-order physical systems with oscillatory solutions. The sixth order algorithm was derived using interpolation and collocation of power series within a single step interval [tn; tn+1]. In order to circumvent the Dahlquist-barrier and also obtain an accuracy-preserving algorithm, four o-step points were incorporated within the single step interval. A number of special cases of oscillatory problems were solved using the proposed method and the results obtained clearly showed that it outperformed other existing methods we compared our results with even though the BHA is of lower order relative to such methods. Some of...
Over the years, researches have shown that fixed (constant) step-size methods have been efficient... more Over the years, researches have shown that fixed (constant) step-size methods have been efficient in integrating a stiff differential system. It has however been observed that for some stiff differential systems, non-fixed (variable) step-size methods are required for efficiency and for accuracy to be attained. This is because such systems have solution components that decay rapidly and/or slowly than others over a given integration interval. In order to curb this challenge, there is a need to propose a method that can vary the step size within a defined integration interval. This challenge motivated the development of Non-Fixed Step-Size Algorithm (NFSSA) using the Lagrange interpolation polynomial as a basis function via integration at selected limits. The NFSSA is capable of integrating highly stiff differential systems in both small and large intervals and is also efficient in terms of economy of computer time. The validation of properties of the proposed algorithm which include...
Journal of the Nigerian Society of Physical Sciences
Over the years, the systematic search for stiff model solvers that are near-optimal has attracted... more Over the years, the systematic search for stiff model solvers that are near-optimal has attracted the attention of many researchers. An attempt has been made in this research to formulate an implicit Four-Point Hybrid Block Integrator (FPHBI) for the simulations of some renowned rigid stiff models. The integrator is formulated by using the Lagrange polynomial as basis function. The properties of the integrator which include order, consistency, and convergence were analyzed. Further analysis showed that the proposed integrator has an A-stability region. The A-stability nature of the integrator makes it more robust and fitted for the simulation of stiff models. To test the computational reliability of the new integrator, few well-known technical stiff models such as the pharmacokinetics, Robertson and Van der Pol models were solved. The results generated were then compared with those of some existing methods including the MATLAB solid solvent, ode 15s. From the results generated, the ...
In this paper, an efficient hybrid algorithm shall be formulated for the computation of second-or... more In this paper, an efficient hybrid algorithm shall be formulated for the computation of second-order Fredholm integro-differential equations. In developing the algorithm using the method of interpolation and collocation, power series was adopted as the basis function with the integration carried out within a one-step interval. The algorithm derived was then applied on some modeled second-order Fredholm integrodifferential equations and from the results obtained; it is obvious that the algorithm is computationally reliable. The basic properties of the algorithm derived were also analyzed. (
Recent Developments in the Solution of Nonlinear Differential Equations, 2021
In this paper, a collocation approach for solving initial value problem of ordinary differential ... more In this paper, a collocation approach for solving initial value problem of ordinary differential equations (ODEs) of the first order is presented. This approach consists of reducing the problem to a set of linear multi-step algebraic equations by approximating the ODE with a shifted Legendre polynomial basis function to determine the unknown constants. The proposed method is simple and efficient; it approximates the solutions very closely to the closed form solutions. Some problems were considered using Maple Software to illustrate the simplicity, efficiency and accuracy of the method. The results obtained revealed that the hybrid method can be suitable candidate for all forms of first order initial value problems of ordinary differential equations.
Journal of the Nigerian Society of Physical Sciences
It is a known fact that in most cases, to integrate an oscillatory problem, higher order A-stable... more It is a known fact that in most cases, to integrate an oscillatory problem, higher order A-stable methods are often needed. This is because such problems are characterized by stiffness, chaos and damping, thus making them tedious to solve. However, in this research, an accuracy-preserving relatively lower order Block Hybrid Algorithm (BHA) is proposed for solution of second-order physical systems with oscillatory solutions. The sixth order algorithm was derived using interpolation and collocation of power series within a single step interval [tn; tn+1]. In order to circumvent the Dahlquist-barrier and also obtain an accuracy-preserving algorithm, four o-step points were incorporated within the single step interval. A number of special cases of oscillatory problems were solved using the proposed method and the results obtained clearly showed that it outperformed other existing methods we compared our results with even though the BHA is of lower order relative to such methods. Some of...
Over the years, researches have shown that fixed (constant) step-size methods have been efficient... more Over the years, researches have shown that fixed (constant) step-size methods have been efficient in integrating a stiff differential system. It has however been observed that for some stiff differential systems, non-fixed (variable) step-size methods are required for efficiency and for accuracy to be attained. This is because such systems have solution components that decay rapidly and/or slowly than others over a given integration interval. In order to curb this challenge, there is a need to propose a method that can vary the step size within a defined integration interval. This challenge motivated the development of Non-Fixed Step-Size Algorithm (NFSSA) using the Lagrange interpolation polynomial as a basis function via integration at selected limits. The NFSSA is capable of integrating highly stiff differential systems in both small and large intervals and is also efficient in terms of economy of computer time. The validation of properties of the proposed algorithm which include...
Journal of the Nigerian Society of Physical Sciences
Over the years, the systematic search for stiff model solvers that are near-optimal has attracted... more Over the years, the systematic search for stiff model solvers that are near-optimal has attracted the attention of many researchers. An attempt has been made in this research to formulate an implicit Four-Point Hybrid Block Integrator (FPHBI) for the simulations of some renowned rigid stiff models. The integrator is formulated by using the Lagrange polynomial as basis function. The properties of the integrator which include order, consistency, and convergence were analyzed. Further analysis showed that the proposed integrator has an A-stability region. The A-stability nature of the integrator makes it more robust and fitted for the simulation of stiff models. To test the computational reliability of the new integrator, few well-known technical stiff models such as the pharmacokinetics, Robertson and Van der Pol models were solved. The results generated were then compared with those of some existing methods including the MATLAB solid solvent, ode 15s. From the results generated, the ...
In this paper, an efficient hybrid algorithm shall be formulated for the computation of second-or... more In this paper, an efficient hybrid algorithm shall be formulated for the computation of second-order Fredholm integro-differential equations. In developing the algorithm using the method of interpolation and collocation, power series was adopted as the basis function with the integration carried out within a one-step interval. The algorithm derived was then applied on some modeled second-order Fredholm integrodifferential equations and from the results obtained; it is obvious that the algorithm is computationally reliable. The basic properties of the algorithm derived were also analyzed. (
Recent Developments in the Solution of Nonlinear Differential Equations, 2021
In this paper, a collocation approach for solving initial value problem of ordinary differential ... more In this paper, a collocation approach for solving initial value problem of ordinary differential equations (ODEs) of the first order is presented. This approach consists of reducing the problem to a set of linear multi-step algebraic equations by approximating the ODE with a shifted Legendre polynomial basis function to determine the unknown constants. The proposed method is simple and efficient; it approximates the solutions very closely to the closed form solutions. Some problems were considered using Maple Software to illustrate the simplicity, efficiency and accuracy of the method. The results obtained revealed that the hybrid method can be suitable candidate for all forms of first order initial value problems of ordinary differential equations.
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