Abstract
The performance of the numerical computation based on the diagonally implicit multistep block method for solving Volterra integrodifferential equations (VIDE) of the second kind has been analyzed. The numerical solutions of VIDE will be computed at two points concurrently using the proposed numerical method and executed in the predictor-corrector (PECE) mode. The strategy to obtain the numerical solution of an integral part is discussed and the stability analysis of the diagonally implicit multistep block method was investigated. Numerical results showed the competence of diagonally implicit multistep block method when solving Volterra integrodifferential equations compared to the existing methods.
1. Introduction
Consider the Volterra integrodifferential equation of the second kindwhereThe numerical methods are generated to solve (1) which is a standard algorithm for ordinary differential equations and Newton-Cote integration formulae are required for solving the integral part since it cannot be solved explicitly. These equations usually appeared in physics, biology, and engineering applications such as biological models, neutron diffusion, wind ripple in the desert, heat transfer, and many more.
For many years, several methods had been applied to solve first-order problem of VIDE. Day [1] proposed Newton-Cotes integration formula of the trapezoidal rule for the solutions of outer and inner integral to obtain approximate solutions of integrodifferential equations. A comparison between the variational iteration method and trapezoidal rule revealing that the variational iteration method is more efficient and convenient to solve linear VIDE was discussed by Saadati et al. [2].
In [3], finite difference method is used for solving linear VIDE by Raftari. He transforms the Volterra integrodifferential equation in a matrix form and solved it by using finite difference method based on Simpson’s rule and trapezoidal rule. A fourth-order robust numerical method was presented by Filiz [4] with a combination of the trapezoidal rule and Simpson’s rule to evaluate the solution of VIDE for kernel equal to one. Then, he extended his work with a Runge-Kutta-Verner method in [5] and used higher rules of numerical integration method for solving the integral part.
The extended trapezoidal method [6] was proposed for the numerical solution of VIDE of the second kind and implemented the method in scheme. Mohamed and Majid [7] had solved the second kind of VIDE using one-step block method and the Newton-Cotes quadrature formula was employed for finding the solution of the integral part. The multistep block method in [8] had implemented two approaches for solving VIDE for and .
2. Numerical Method
The proposed numerical method is in the form of block method and it will generate two or more solutions at the same time. The proposed method is a two-point block method; hence it will generate two solutions in one block.
In Figure 1, the two approximate values of and will be computed simultaneously in a block. The approximate values of can be developed by integrating (1) over the interval while the interval for values is . Hence, the formulae of and can be obtained asTherefore,where . Then, function in (4) will be approximated using Lagrange interpolating polynomial, and the interpolation points involved in obtaining the corrector formula of are . Taking , , and replacing into (4), the limit of integration in (4) will be to .
The formulation of can be obtained when three points are involved in the interpolation polynomial, that is, . Considering , in (4) and the limit of integration will be changed from to . The corrector formulae of and will be obtained using MAPLE software. The predictor formulae are one order less than the corrector formulae and the same process of derivation is applied.
Diagonally Implicit Multistep Block Method
Predictor
CorrectorThe matrix form of the corrector formulae iswhich is equivalent to the difference equationswhere , , , and are the coefficients with -vectors , , , and defined as
3. Analysis of Diagonally Implicit Multistep Block Method
3.1. Order and Convergence of the Method
The order of the method can be obtained by referring [9]where is the order of the linear multistep method, is the local truncation error, and is defined as
Definition 1. The numerical method is said to be in order if the linear operator of numerical method iswhere is called as an error constant of the method.
The order of diagonally implicit multistep block method in (7) can be determined by applying the formula in (11); hence the values of and are obtained as follows:Substitute the values of and into (11) and obtainTherefore, the diagonally implicit multistep block method is third-order where the coefficient of error constant is
Definition 2. The local truncation error at of the method is defined to be expression , when is the theoretical solution of the initial value problem
For the formula ,Since , Taylor expansion will be applied to the derivatives and where ,Then, since we haveSo, the local truncation error for is . For the formula ,The Taylor expansion for is given asThen, we will haveThis shows that the local truncation error for is .
Definition 3. The numerical method is said to be consistent if the order of method is and the method is consistent if and only if
Proof. (i) ,(ii) ,Therefore,By Definitions 1 and 3, the diagonally implicit multistep block method is consistent.
Definition 4. A block method is said to be zero-stable if and only if providing the roots of , of the first characteristic polynomial, , specified assatisfies and those roots with .
Proof. The values of can be obtained in (7):The diagonally implicit multistep block method is zero stable since .
Theorem 5. The method is said to be convergent if and only if the method is consistent and zero-stable.
Proof. By Definitions 1, 3, and 4, the diagonally implicit multistep block method is convergent.
3.2. Stability Region of the Method
In this section, the stability region of the diagonally implicit multistep block method of order three is discussed for the numerical solution of VIDE. The test equation for first-order VIDE of the second kind [10] iswhere and are real constants, and . Therefore,
Definition 6. The method is said to be -stable if and only if the region of absolute stability contains at the quarter plane , .
From the proposed method for the numerical solution, the characteristics polynomials , , , and can be developed as follows. Corrector formula for is Corrector formula for is Simpson’s rule isThe stability polynomial of the diagonally implicit multistep block method can be determined by substituting (31), (32), and (33) into this particular formula,where and . Thus, the stability polynomial of the proposed method is obtained:The region of stability polynomial can be illustrated in Figure 2.
Regarding Definition 6, the third-order of diagonally implicit multistep block method in Figure 2 is -stable within the shaded region.
4. Implementation
The one-step methods are required for finding the first starting point at since Volterra integrodifferential equations of the second kind have two types of kernels. For , Runge-Kutta method is involved in solving differential part of VIDE, while, midpoint method is needed to solve VIDE when . Hence, the predictor and the corrector formula can be implemented until the end of the intervalSince in is the integral term in VIDE and cannot be solved explicitly, therefore, Simpson’s rule is adapted for solving the integral part. (i)For , Simpson’s rule is applied to solve the integral term in VIDE:(ii)For , composite Simpson’s rule is employed for solving the integral part:where are Simpson’s rule weights . The unknown value of in (38) can be estimated by using Lagrange interpolation at the point ,
4.1. Algorithm of the Method
The input of the programming is the endpoints of and and the integer, . The developed algorithm for the method is given as follows.
Step 1. Set ; ; ; ; OUTPUT: .
Step 2. For , When , using RK to evaluate the value of . When , applying Midpoint Method.
Step 4. Set .
Step 5. Calculate for and , and .
Step 6. Compute the solution for and , and .
Step 7. Calculate the error.
Step 8. OUTPUT: and the absolute error.
Step 9. STOP.
5. Numerical Results
Four tested problems of first-order Volterra integrodifferential equations were considered in order to study the performance of the diagonally implicit multistep block method.
Example 1 (() linear VIDE). Exact solution is .
Source: [4].
Example 2 (() linear VIDE). Exact solution is .
Source: [11].
Example 3 (() nonlinear VIDE). Exact solution is .
Source: [12].
Example 4 (() nonlinear VIDE). Exact solution is .
Source: [13].
Notations used in Tables 1–4 are as follows: : step size MAXE: maximum error TS: total steps TFC: total functions call Time: the execution time taken RK3: Runge-Kutta method of order 3 with Simpson’s rule by Filiz [4] ABM3: Adam Bashforth Moulton Order 3 with Simpson’s rule. BVMs: Combination of BVMs and third-order Generalized Adams Method by Chen and Zhang [11] 2PMBM: Two points Multistep Block Method of Order 3 with Simpson’s rule by Mohamed and Majid [8] DIMBM: Diagonally implicit multistep block method with Simpson’s rule proposed in this paper
Tables 1–4 display the numerical results for the four tested problems when solved using the proposed block method and the code was written in C language.
The numerical results for Examples 1–4 displayed in Tables 1–4 are solved numerically using the proposed numerical method with Simpson’s rule. In Table 1, the numerical results will be obtained when the step size , , , and for the case when . The maximum error of DIMBM is comparable compared to 2PMBM at all tested but the order of accuracy is the same or one order less compared to RK3 and ABM3. The performance of DIMBM is better in terms of total functions call and total number of steps compared to RK3 and ABM3.
In Tables 2, 3, and 4, the numerical results are solved using the proposed numerical method via composite Simpson’s for the integral part when . In Table 2, the maximum error of DIMBM is comparable compared to 2PMBM and ABM3. The DIMBM manage to obtain less total functions call compared to ABM3 and 2PMBM. For the nonlinear Examples 3 and 4, we could observe that ABM3 and 2PMBM are expensive in terms of total functions call, respectively. Figures 3–6 display the numerical results of maximum error versus total functions call when solving the tested problems. This has shown the advantage of DIMBM in the form of a standard multistep method because the cost per step is cheaper and the numerical results are more accurate when the step size is reduced. In terms of timing, DIMBM gave faster results compared to ABM3 and 2PMBM.
(a) Graph of TFC versus MAXE
(b) Graph of Time versus MAXE
(a) Graph of TFC versus MAXE
(b) Graph of time versus MAXE
(a) Graph of TFC versus MAXE
(b) Graph of Time versus MAXE
(a) Graph of TFC versus MAXE
(b) Graph of time versus MAXE
6. Conclusion
In this research, we proposed the diagonally implicit multistep block method for solving linear and nonlinear Volterra integrodifferential equations and comparisons were made with the existing method. Comparisons with existing method reveal that the diagonally implicit multistep block method is more efficient and cheaper.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
The authors gratefully acknowledged the financial support of Fundamental Research Grant Scheme (FRGS:5524973) and Graduate Research Fund (GRF) from Universiti Putra Malaysia.