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In this paper, the implementation of one-step hybrid block method with quadrature rules will be proposed for solving linear and non-linear rst order Volterra Integro-Di erential Equations (VIDEs) of the second kind. VIDEs have important applications in many branches of sciences and engineering, such as analysing rhythmic biological data can be conducted by utilizing a curve tting technique based on solutions of the VIDEs. The formulation of the hybrid block method is based on the Lagrange interpolation polynomial. The approximation of the integral part in the VIDEs will be estimated using the quadrature rules. The proposed hybrid block method of order ve will compute the numerical solutions at two points simultaneously at each integration steps. The stability analysis such as order of the method, consistency, zero stable and stability region of the method are deliberated. The xed step size is used to generate the results and the code is written in C language. Numerical simulations are presented to show the e ciency and accuracy of the hybrid method when compared to the Runge-Kutta of order four and ve in terms of accuracy, total steps, and total function calls.

In this paper, the implementation of one-step hybrid block method with quadrature rules will be proposed for solving linear and non-linear rst order Volterra Integro-Di erential Equations (VIDEs) of the second kind. VIDEs have important applications in many branches of sciences and engineering, such as analysing rhythmic biological data can be conducted by utilizing a curve tting technique based on solutions of the VIDEs. The formulation of the hybrid block method is based on the Lagrange interpolation polynomial. The approximation of the integral part in the VIDEs will be estimated using the quadrature rules. The proposed hybrid block method of order ve will compute the numerical solutions at two points simultaneously at each integration steps. The stability analysis such as order of the method, consistency, zero stable and stability region of the method are deliberated. The xed step size is used to generate the results and the code is written in C language. Numerical simulations are presented to show the e ciency and accuracy of the hybrid method when compared to the Runge-Kutta of order four and ve in terms of accuracy, total steps, and total function calls.

This paper focuses on the construction of two-point and three-point implicit block methods for solving general second order Initial Value Problems. The proposed methods are formulated using Hermite Interpolating Polynomial. The block methods approximate the numerical solutions at more than one point at a time directly without reducing the equation into the first order system of ordinary differential equations. In the derivation of the method, the higher derivative of the problem is incorporated into the formula to enhance the efficiency of the proposed methods. The order and zero- stability of the methods are also presented. Numerical results presented show the efficiency of these methods compared to the existing block methods.

This paper focuses on the construction of two-point and three-point implicit block methods for solving general second order Initial Value Problems. The proposed methods are formulated using Hermite Interpolating Polynomial. The block methods approximate the numerical solutions at more than one point at a time directly without reducing the equation into the first order system of ordinary differential equations. In the derivation of the method, the higher derivative of the problem is incorporated into the formula to enhance the efficiency of the proposed methods. The order and zero- stability of the methods are also presented. Numerical results presented show the efficiency of these methods compared to the existing block methods.

In recent time, Runge-Kutta methods that integrate special fourth order ordinary differential equations (ODEs) directly are proposed to address efficiency issues associated with classical Runge-Kutta methods. Although, the methods require approximation of y', y'' and y''' of the solution at every step. In this paper, a hybrid type method is proposed, which can directly integrate special fourth order ODEs. The method does not require the approximation of any derivatives of the solution. Algebraic order conditions of the methods are derived via Taylor series technique. Using the order conditions, eight algebraic order method is presented. Absolute stability of the method is analyzed and the stability region presented. Numerical experiment is conducted on some test problems. Results from the experiment show that the new method is more efficient and accurate than the existing Runge-Kutta and hybrid methods with similar number of function evaluation.

In this paper, we present the absolute stability of the existing 2-point implicit block multistep step methods of step number k = 3 and k = 5 and solving special second order ordinary differential equations (ODEs). The methods are then trigonometrically fitted so that they are suitable for solving highly oscillatory problems arising from the special second order ODEs. Their explicit counterparts are also trigonometrically fitted so that in the implementation the methods can act as a predictor-corrector pairs. The numerical results based on the integration over a large interval are given to show the performance of the proposed methods. From the numerical results we can conclude that the new trigonometrically-fitted methods are superior in terms of accuracy and execution time, compared to the existing methods in the scientific literature when used for solving problems which are oscillatory in nature.

In this paper, we present the absolute stability of the existing 2-point implicit block multistep step methods of step number k = 3 and k = 5 and solving special second order ordinary differential equations (ODEs). The methods are then trigonometrically fitted so that they are suitable for solving highly oscillatory problems arising from the special second order ODEs. Their explicit counterparts are also trigonometrically fitted so that in the implementation the methods can act as a predictor-corrector pairs. The numerical results based on the integration over a large interval are given to show the performance of the proposed methods. From the numerical results we can conclude that the new trigonometrically-fitted methods are superior in terms of accuracy and execution time, compared to the existing methods in the scientific literature when used for solving problems which are oscillatory in nature.

We present a new third order convergence iterative method for solving multiple roots of nonlinear equation, which requires one function evaluation and two evaluation of first derivative of function per step. Our present method free from second derivative function. Error term is proved to possess a third order method. Numerical experiments exhibit that our method gives the smallest error of bound per iteration and it is highly accurate as compared to other existing iterative methods.

This numerical study emphasizes on fifth order multistep block method for solving second order boundary value problems (BVPs) imposing Robin boundary conditions. The shooting technique will be utilized to compute the approximate solutions at two point simultaneously. The implementation of predictor-corrector scheme follows the P E(CE) r mode. Numerical results are presented to give a clear view of the performances for the propose method. The order and stability of the method are also discussed.

This numerical study emphasizes on fifth order multistep block method for solving second order boundary value problems (BVPs) imposing Robin boundary conditions. The shooting technique will be utilized to compute the approximate solutions at two point simultaneously. The implementation of predictor-corrector scheme follows the P E(CE) r mode. Numerical results are presented to give a clear view of the performances for the propose method. The order and stability of the method are also discussed.

We present a new third order convergence iterative method for solving multiple roots of nonlinear equation, which requires one function evaluation and two evaluation of first derivative of function per step. Our present method free from second derivative function. Error term is proved to possess a third order method. Numerical experiments exhibit that our method gives the smallest error of bound per iteration and it is highly accurate as compared to other existing iterative methods.

In recent time, Runge-Kutta methods that integrate special fourth order ordinary differential equations (ODEs) directly are proposed to address efficiency issues associated with classical Runge-Kutta methods. Although, the methods require approximation of y', y'' and y''' of the solution at every step. In this paper, a hybrid type method is proposed, which can directly integrate special fourth order ODEs. The method does not require the approximation of any derivatives of the solution. Algebraic order conditions of the methods are derived via Taylor series technique. Using the order conditions, eight algebraic order method is presented. Absolute stability of the method is analyzed and the stability region presented. Numerical experiment is conducted on some test problems. Results from the experiment show that the new method is more efficient and accurate than the existing Runge-Kutta and hybrid methods with similar number of function evaluation.

Numerical analysis is a way to do higher mathematical problems on a computer, a technique widely used by scientists and engineers to solve their problems. A major advantage of numerical analysis is that a numerical answer can be obtained even when a problem has no “analytical” solution. Results from numerical analysis are an approximation, which can be made as accurate as desired. The analysis of errors in numerical methods is a critically important part of the study of numerical analysis. Hence, we will see in this research that computation of the error is a must as it is a way to measure the efficiency of the numerical methods developed. Numerical methods require highly tedious and repetitive computations that can only be done using the computer. Hence in this research, it is shown that computer programs must be written for the implementation of numerical methods. In the early part of related research the computer language used was Fortran. Subsequently more and more computer programs used the C programming language. Additionally, now computations can also be carried out using softwares like MATLAB, MATHEMATICA and MAPLE. Many physical problems that arise from ordinary differential equations (ODEs) have magnitudes of eigenvalues which vary greatly, and such systems are commonly known as stiff systems. Stiff systems usually consist of a transient solution, that is, a solution which varies rapidly at the beginning of the integration. This phase is referred to as the transient phase and during this phase, accuracy rather than stability restricts the stepsize of the numerical methods used. Thus the generally the structure of the solutions suggests application of specific methods for non-stiff equations in the transient phase and specific methods for stiff equations during the steady-state phase in a manner whereby computational costs can be reduced. Consequently, in this research we developed embedded Runge-Kutta methods for solving stiff differential equations so tha

Numerical analysis is a way to do higher mathematical problems on a computer, a technique widely used by scientists and engineers to solve their problems. A major advantage of numerical analysis is that a numerical answer can be obtained even when a problem has no “analytical” solution. Results from numerical analysis are an approximation, which can be made as accurate as desired. The analysis of errors in numerical methods is a critically important part of the study of numerical analysis. Hence, we will see in this research that computation of the error is a must as it is a way to measure the efficiency of the numerical methods developed. Numerical methods require highly tedious and repetitive computations that can only be done using the computer. Hence in this research, it is shown that computer programs must be written for the implementation of numerical methods. In the early part of related research the computer language used was Fortran. Subsequently more and more computer programs used the C programming language. Additionally, now computations can also be carried out using softwares like MATLAB, MATHEMATICA and MAPLE. Many physical problems that arise from ordinary differential equations (ODEs) have magnitudes of eigenvalues which vary greatly, and such systems are commonly known as stiff systems. Stiff systems usually consist of a transient solution, that is, a solution which varies rapidly at the beginning of the integration. This phase is referred to as the transient phase and during this phase, accuracy rather than stability restricts the stepsize of the numerical methods used. Thus the generally the structure of the solutions suggests application of specific methods for non-stiff equations in the transient phase and specific methods for stiff equations during the steady-state phase in a manner whereby computational costs can be reduced. Consequently, in this research we developed embedded Runge-Kutta methods for solving stiff differential equations so tha

Subspace quasi-Newton (SQN) method has been widely used in large scale unconstrained optimization problem. Its popularity is due to the fact that the method can construct subproblems in low dimensions so that storage requirement as well as the computation cost can be minimized. However, the main drawback of the SQN method is that it can be very slow on certain types of non-linear problem such as ill-conditioned problems. Hence, we proposed a preconditioned SQN method, which is generally more effective than the SQN method. In order to achieve this, we proposed that a diagonal updating matrix that was derived based on the weak secant relation be used instead of the identity matrix to approximate the initial inverse Hessian. Our numerical results show that the proposed preconditioned SQN method performs better than the SQN method which is without preconditioning.

Subspace quasi-Newton (SQN) method has been widely used in large scale unconstrained optimization problem. Its popularity is due to the fact that the method can construct subproblems in low dimensions so that storage requirement as well as the computation cost can be minimized. However, the main drawback of the SQN method is that it can be very slow on certain types of non-linear problem such as ill-conditioned problems. Hence, we proposed a preconditioned SQN method, which is generally more effective than the SQN method. In order to achieve this, we proposed that a diagonal updating matrix that was derived based on the weak secant relation be used instead of the identity matrix to approximate the initial inverse Hessian. Our numerical results show that the proposed preconditioned SQN method performs better than the SQN method which is without preconditioning.

In this paper, an implicit 2-point Block Backward Differentiation formula (BBDF) method was considered for solving Delay Differential Equations (DDEs). The method was implemented by using a constant stepsize via Newton Iteration. This implicit block method was expected to produce two points simultaneously. The efficiency of the method was compared with the existing classical 1-point Backward Differentiation Formula (BDF) in terms of execution time and accuracy

In this paper, an implicit 2-point Block Backward Differentiation formula (BBDF) method was considered for solving Delay Differential Equations (DDEs). The method was implemented by using a constant stepsize via Newton Iteration. This implicit block method was expected to produce two points simultaneously. The efficiency of the method was compared with the existing classical 1-point Backward Differentiation Formula (BDF) in terms of execution time and accuracy

In this paper the explicit 2 and 3-point block method for solving large systems of special second order ODEs directly is discussed. Codes based on the methods are executed in sequential and parallel. The numerical results show that parallel to sequential counterpart for solving the large system of special second order ODEs.

In this paper the explicit 2 and 3-point block method for solving large systems of special second order ODEs directly is discussed. Codes based on the methods are executed in sequential and parallel. The numerical results show that parallel to sequential counterpart for solving the large system of special second order ODEs.

A new fourth order four-stage diagonally Implicit Runge-Kutta (DIRK) method which is specially designed for the integration of Linear Ordinary Differential Equations (LODEs) is constructed. In the derivation, Butcher's error equations are used but one of the error equations can be eliminated due to the property of the LODE itself. The stability aspect of the method is investigated and it is found to have a bigger region of stability compared to explicit Runge-Kutta (ERK) method of the same type (designed for the integration of LODE). A set of test problems are used to validate the method and numerical results show that the method produced smaller global error compared to ERK method.

A new fourth order four-stage diagonally Implicit Runge-Kutta (DIRK) method which is specially designed for the integration of Linear Ordinary Differential Equations (LODEs) is constructed. In the derivation, Butcher's error equations are used but one of the error equations can be eliminated due to the property of the LODE itself. The stability aspect of the method is investigated and it is found to have a bigger region of stability compared to explicit Runge-Kutta (ERK) method of the same type (designed for the integration of LODE). A set of test problems are used to validate the method and numerical results show that the method produced smaller global error compared to ERK method.

Interval method is used for the inclusion of a zero of a function. The Ehrman (EHR) method considers the Newton’s iteration in finding the root of a function. This method is modified by using the mid point in the procedure and improved method has a faster convergence rate and less processing time. In this paper, the convergence analysis and the numerical results are shown.

Interval method is used for the inclusion of a zero of a function. The Ehrman (EHR) method considers the Newton’s iteration in finding the root of a function. This method is modified by using the mid point in the procedure and improved method has a faster convergence rate and less processing time. In this paper, the convergence analysis and the numerical results are shown.