Analysis of the Use of Block Methods in VIDEs Research

Abstract

This segment of the block method essay describes a review of some related previous studies of Volterra Integro-Differential Equations. Here is discussed the Block Backward Differentiation Formula method that is proposed in this research.

Introduction

Many researchers use block methods to solve ordinary differential equations to find numerical solutions. For example, Mohamed and Majid presented a one-step block method for solving VIDEs of the second kind. The proposed block method will solve the differentiation part and quadrature rules were applied to calculate an integral part of VIDEs. Later in 2016, they have used a multistep block method to solve linear and nonlinear Volterra integrodifferential equations (VIDEs) of the second kind. Moreover, followed by Baharum, Majid and Senu, their research is about diagonally implicit multistep block method for solving VIDEs. The proposed method will be executed by using a predictor-corrector (PECE) scheme. In a recent study by Majid and Mohamed have solved the fifth-order multistep block method for solving the linear and non-linear of VIDEs. In this research, they have developed the combination of a multistep method, in particular, simple form of Adams Moulton type and Boole’s quadrature rule to solve VIDEs.

Block Backward Differentiation Formula Method

Ordinary differential equations appear in various mathematics fields such as physics, biology, chemistry, ecology, growth, applied mathematics, engineering and many more. ODEs can be classified into two elements that are stiff and non-stiff ODEs. An implicit method usually used to solve stiff equations while non-stiff equations is an explicit method. Some ODEs can be determined explicitly in terms of known functions and integrals. Thus, numerical methods which also known as numerical integration can be used for solving ODEs to find better and more accurate approximation solution.

Multistep methods, Euler’s methods and Runge–Kutta methods are the numerical solutions of ODEs that have been introduced in many studies. According to Zill, Euler’s methods are the easiest numerical methods for approximate solutions of first-order ODEs while Runge-Kutta methods are one of the most accurate numerical procedures used in obtaining approximate solutions to solve ODEs. Both of these methods are examples of single-step methods. Furthermore, the multistep methods or continuing methods use the values from computed steps to obtain the value of . This method is divided into implicit and explicit approaches. The implicit multistep methods are Adams-Moulton methods and backward differentiation formula while explicit multistep methods are Adams–Bashforth methods. The used of numerical method to solve stiff equation have first been studied by Curtiss and for solving ODEs. They have solved the first order ODEs to obtain the numerical solution of stiff equations using forward interpolation, in particular, BDF methods.

There are various existing methods for solving ODEs. However, these methods only give the numerical approximation solutions at one step at a time. Therefore, block methods have been created to develop faster ways of getting solutions simultaneously at each point of the algorithm. Block methods can be formed with single-step or multistep methods. As stated by Ibrahim, in single-step block methods, the approximations values created in each new block depends only on the last point of the previous block. However, in multistep block methods, every point of the previous block are used in producing the approximation values for the new block.

Ibrahim, Othman and Suleiman has derived an implicit two-point block method based on Backward Differentiation Formula of variable step size for solving first order stiff initial value problems for ODEs. This proposed method was compared with non-block BDF of variable step order. Numerical results show that the resulting of this study outperform the non-block BDF method in both execution time and accuracy. Implicit block backward differentiation formula methods to solve stiff equations of ODEs are studied by various researchers such as Yatim et al., Zawawi et al. and Nasir et al. These methods were developed to solve the first-order of ODEs.

Abasi et al. presented a two-point block method based on backward differentiation formula to solve a stiff equation of ODEs with off-step points. The method was estimated by two numerical values and two off-step points simultaneously at each point. Convergence and Stability region of the method is also shown. Recently, Waeleh and Majid have proposed new alternative algorithm for solving higher order of ODEs using variable step size. The strategy of this method is using block multistep method with variable step size to compute the solutions at four points simultaneously and the derivation was based on numerical methods also by using the interpolation approach. The convergence of this study is proved under suitable conditions of consistency and stability. The outcomes were compared with the existing methods for confirming the validity and reliability of the proposed method.

Conclusion

In this segment of the block method essay one of the previous studies of Volterra Integro-Differential Equations was reviewed and analysed through the using of Block backward differentiation formula method. To sum up, The Block Backward Differentiation Formula (BBDF) method is a numerical method used to solve initial value problems (IVPs) for ordinary differential equations (ODEs). It is an extension of the Backward Differentiation Formula (BDF) method, which is a widely used family of implicit numerical methods for stiff ODEs. The BBDF method involves approximating the derivative of a function at a particular time using a backward difference formula, which uses information from previous time steps.