Open access peer-reviewed chapter

Continuous One Step Linear Multi-Step Hybrid Block Method for the Solution of First Order Linear and Nonlinear Initial Value Problem of Ordinary Differential Equations

Written By

Kamoh Nathaniel, Kumleng Geoffrey and Sunday Joshua

Submitted: September 30th, 2020 Reviewed: December 22nd, 2020 Published: September 8th, 2021

DOI: 10.5772/intechopen.95619

Chapter metrics overview

264 Chapter Downloads

View Full Metrics


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.


  • collocation
  • hybrid block method
  • consistent
  • zero stable
  • convergent

1. Introduction

The development of mathematics parallels the human endeavor to understand our physical environment. Differential equations were discovered when the need to understand the behavior of nearly all systems undergoing change became more demanding. They are found in science and engineering as well as economics, social science, biology, business and health care. Many systems described by differential equations are so large and complex that a purely analytic solution is sometimes not traceable [1, 2, 3, 4, 5]. However mathematicians have studied the nature of these equations for decades of years and there are many well-developed numerical methods for the solution of different order initial value problems of ordinary differential equations. Unfortunately, in trying to achieve efficient and accurate solution, the choice of the numerical method to be adopted becomes very essential [4, 6, 7, 8].

The main goal of this paper is to derive a one step continuous hybrid block method using shifted Legendre polynomials basis function with the expectation that the numerical (proposed) method will give a solution that is close to the close form solution of the initial value problems of first order nonlinear ordinary differential equations. The paper is structured as follows. In Section 2, we derived and analyze the obtained schemes for consistency, zero stability and convergence. Some first order nonlinear problems of ordinary differential equations were solved using the derived schemes and the main results are presented in Section 3. Finally, we end with some concluding remarks in Section 4, where we compared our results with some earlier results contained in the literature.

1.1 Linear multistep methods (LMMs)

Linear Multi-step Methods (LMMs) are very popular for solving Initial Value Problems (IVPs) of Ordinary Differential Equations (ODEs). They are also applied in solving higher order ODEs. LMMs are not self-starting and therefore, need starting values from single-step methods like Euler’s method and Runge–Kutta family of methods [1, 9, 10].

The general kstep LMM of the discrete form as given in [11, 12, 13, 14] is;


where αj and βj are uniquely determined and αj+βj0. The LMM (1) generates discrete schemes which are used to solve first order ODEs. However, the continuous Linear Multi-step Methods (CLMMs) which is now being used was introduced by [15] and used by so many researchers such as [6, 7, 9, 16, 17] leading to the development of what is now called continuous Linear Multi-step Methods (CLMMs) given by;


where αjx and βjx are expressed as continuous functions of x and are continuously differentiable at least m times m1. According to [1, 2, 11, 12, 18], the existing methods of deriving the LMMs in discrete form include the interpolation approach, numerical integration, Taylor series expansion. While the collocation and interpolation technique is widely used for the derivation of CLMMs, this method is derived using different techniques and approaches.

The introduction of CLMMs have numerous advantages which is of great importance; better global error is estimated, it can be used to recover some standard schemes, it provides a simplified form of coefficients for further analytical work at different points and guarantee easy approximation of solutions at all interior points of the integration interval [1, 7, 16, 19, 20].

In this work, the CLMM is developed for the solution of (linear and nonlinear) first-order initial value problems of ordinary differential equations using the shifted Legendre polynomials basis function. The corresponding discrete schemes are obtained from the evaluation of the continuous scheme at some selected grid points.

1.2 Shifted Legendre polynomials

The shifted Legendre polynomials are well known family of orthogonal polynomials on the interval0A and are denoted byPit, thePit can be obtained by the recurrence formula:


where P0t=1 andP1t=2t1

The first few terms of the shifted Legendre polynomials on the interval [0, A] with A=1 are:


1.3 Collocation method

A collocation is a method which involves the determination of an approximate solution of a functional equation in a suitable set of functions called trial or basis functions. The approximate solution is required to satisfy the equation and its supplementary conditions at certain points in the range of interest called collocation points.


2. Derivation of one step hybrid block methods with shifted Legendre polynomials

We consider the first order ordinary differential equation of the form


whereyx is the unknown function to be determined. The idea here is to approximate the exact solution yx of (3) in the partition In=a=x0<x1<x2<<xn=b of the integration interval ab with a constant step sizeh=xixi1,i=1,,n by a shifted Legendre polynomial basis function of degree s+r1 of the form;


where ciR, C1ab and t=xxn. The first derivative of (4), is substituted into (3), to obtain a differential system of the form


Now interpolating (4) at xn+s,s=12,34 and collocating (5) atxn+r,r=0,14,12,34,k where s and r represents the interpolation and collocation points respectively, and k is the step number, after some substitutions and manipulations the continuous scheme of the form;


is obtained with the following continuous coefficients


In order to obtain the block discrete scheme for (K=1), Eq. (7) is evaluated atx=xn,xn+18,xn+14,xn+1 and its first derivative at xn+18 to give the following discrete schemes;


Eq. (7) is the continuous scheme while (8) is the block discrete schemes for step numberK=1.

2.1 Order and error constant

Expanding (8), in Taylor’s series gives;


and collecting like terms in powers of h, gives c0=c1=c2==c6=00000T and c7=17851558403111981808640177414401123863041352583691264T. Hence, the method has order p=66666T and with error constants ofc7=17851558403111981808640177414401123863041352583691264T.

2.2 Consistency

The linear multi-step method (8) is said to be consistent if the following conditions hold:

  1. it has order pˇ1,

  2. j=0kαˇj=0,

  3. j=0kjαˇj=j=0kβˇj,

  4. ρ1=0 and ρ1=σ1,

whereρr andσr are the first and the second characteristic polynomials of (8) respectively, [21]. Following [8, 14], (i) is sufficient condition for the block method (8) to be consistent sincep=66666T>1. Hence, the method is consistent.

2.3 Zero stability

The block solution (8), is said to be zero stable if the roots zr;r=1,,n of the first characteristic polynomialpz, defined by


satisfies zr1 and every root with zr=1 has multiplicity not exceeding the order of the differential equation in the limit ash0.

Calculations from all available information revealed that the block method (8) has the following roots


Hence the block method is zero stable, since all roots with modulus one do not have multiplicity exceeding the order of the differential equation in the limit ash0.

2.4 Convergence

According to [8, 14, 22], we can safely assert the convergence of the block method (8) since the method is consistent and zero stable.

2.5 Region of absolute stability of the block method

Reformulating the block (8) as a General Linear Method (GLM) containing a partition of matrices A and B using the stability polynomialArB, where




we obtain the region of absolute stability shown in Figure 1 below

Figure 1.

Region of absolute stability.


3. Numerical experiments

This section discusses the implementation of the derived method by solving some first order nonlinear initial value problems of ordinary differential equations.

Figure 2.

Solution curve for problem 1.

Figure 3.

Solution curve for problem 2.

Figure 4.

Solution curve for problem 3.

Problem 1

We consider a nonlinear first order initial value problem of ordinary differential problem which was solved by [23]. yx=10y12; y0=2,h=0.01 With exact solution given asyx=1+11+10x, the result is shown in Table 1, while the theoretical and numerical results are presented graphically in Figure 2.

xExact solutionResult of Proposed MethodError in Proposed MethodError in [23]

Table 1.

(Problem 1): Comparing results of proposed method with [23].

Problem 2

Given a nonlinear first order ordinary differential problem solved by [24] (Table 2). yx=2xy, y0=1,h=0.1x01 with exact solution given byyx=ex2, the result is shown in Table 2, Figure 3 shows the solution curve for problem 2.

xExact solutionResult of Proposed MethodError in Proposed MethodError in [24]

Table 2.

(Problem 2): Comparing results of proposed method with [24].

Problem 3

Considering the first order initial value problem of ordinary differential problem solved by [25] (Table 3). yx=y2, y0=1,h=0.01x01 with exact solution given byyx=11+x with the result is shown in Table 3, Figure 4 compare the two results (theoretical and numerical graphically).

xExact solutionResult of Proposed MethodError in Proposed MethodError in [25]

Table 3.

(Problem 3): Comparing results of proposed method with [25].


4. Conclusion

In this paper, we derived one step block hybrid continuous linear multi-step method for solving first order initial value problems of ordinary differential equations, the method was found to be consistent, zero stable and convergent. The method was implemented on some nonlinear initial value problems of ordinary differential equations and the numerical results were found to be accurate when compared with the exact solutions and other numerical methods as contained in Tables 13 and their respective solution curves. The new hybrid block method can be suitable candidate for all forms (linear and nonlinear) of first order initial value problems of ordinary differential equations.



The authors express their sincere thanks to the referees for the careful and details reading of their earlier version of the paper and for the very helpful suggestions.


Authors contributions

This work was carried out in collaboration among the authors. Author Kamoh, N.M. proposed, derived and implemented the method. Author Kumleng, G.M. analyzed the method while Author Sunday; J. presented the numerical results graphically. All the authors managed the literature searches, read and approved the final manuscript.


  1. 1. Hairer E, Wanner G. Solving Ordinary Differential Equations II, Stiff and Differential-Algebraic Problems. Second Revised Edition. Springer-Series in Computational Mathematics. 1996;14:75-77
  2. 2. Rosser JB. A Runge-Kutta Method for All Seasons. SIAM Review. 1967;9(3): 417-452
  3. 3. Abbas S. Derivations of New Block Method for the Numerical Solution of First Order IVPs. International Journal of Computer Mathematics. 1997;64: 11-25
  4. 4. Areo E. A., Ademiluyi R. A. and Babatola P. O. , Three-Step Hybrid Linear Multistep Method for the Solution of First Order Initial Value Problems in Ordinary Differential Equations, J.N.A.M.P, Vol. 19, (2011), pp. 261–266
  5. 5. Burden, R. L., and J. D. Faires, Numerical Analysis, 4th ed., pp. 373–376, PWS-Kent, Boston, Mass., 1988
  6. 6. Jator, S. N., A sixth order linear multistep method for the direct solution of y″ = (x, y, y0) International journal of pure and applied Mathematics, 40(1), (2007), pp. 457–472
  7. 7. Kamoh, N. M, Gyemang, D. G and Soomiyol, M. C (2017): On One Justification on the Use of Hybrids for the Solution of First Order Initial Value Problems of Ordinary Differential Equations, Pure and Applied Mathematics Journal, 6(5): 137–143
  8. 8. J. D. Lambert, Numerical Methods for Ordinary Differential Equation, John Wiley and sons New York, 1991
  9. 9. Anake, T. A. (2011). Continuous Implicit Hybrid One-Step Methods for the Solution of Initial Value Problems of General Second Order ODEs, PhD Thesis Submitted to School of Postgraduate Studies, Covenant University, Ota, Nigeria 170pp
  10. 10. T. Aboiyar., T. Luga., B.V. Iyorter Derivation of Continuous Linear Multistep Methods Using Hermite Polynomials as Basis Functions American Journal of Applied Mathematics and Statistics 2015; 3(6):220–225. doi: 10.12691/ajams-3-6-2
  11. 11. Dahlquist GG. Numerical integration of ordinary differential equations. Math. Scand. 1956; 4:33-50
  12. 12. Dahlquist G. A Special Stability Problem for Linear Multistep Methods, BIT. Vol. 1963;3:27-43
  13. 13. Areo EA, Adeniyi RB. "A Self-Starting Linear Multistep Method for Direct Solution of Second Order Differential Equations", International Journal of Pure and Applied Mathematics. Bulgaria. 2013;82(3):345-364
  14. 14. Lambert JD. Computational Methods in Ordinary Differential Equations. New York: John Wiley and Sons; 1973
  15. 15. Onumanyi PA,Woyemi DO, Jator SN, Sirisena UW. New Linear Multistep Methods with Continuous Coefficients for First Order Initial Value Problems. Journal of Nigeria Mathematics Society. 1994;13:37-51
  16. 16. GraggWB, Stetter HJ. Generalized Multistep Predictor-Corrector Methods. Journal of Association of Computing Machines. 1964;11(2):188-209
  17. 17. Mohammed U, Yahaya YA. Fully implicit four point block backward difference formula for solving first-order initial value problems. Leonardo Journal of Sciences. 2010;16:21-30
  18. 18. Dahlquist G. Stability and error bounds in the numerical integration ofordinary differential equations. Stockholm: Dissertation; 1958
  19. 19. Fatokun J., Onumanyi P. and Sirisena U. W., Solution of First Order System of Ordering Differential Equation by Finite Difference Methods with Arbitrary, J.N.A.M.P, (2011), pp. 30–40
  20. 20. Adesanya AO, Odekunle MR, James AA. Starting Hybrid Stomer- Cowell More Accurately by Hybrid Adams Method for the Solution of First Order Ordinary Differential Equation. European Journal of Scientific Research. 2012;77(4):580-588
  21. 21. Fatunla, S. O. (1988). Numerical Methods for Initial Value Problems for Ordinary Differential Equations; Academy press San Diego, 3rd edition. 295pp
  22. 22. Fatunla, S. O. (1988). Numerical Methods for Initial Value Problems for Ordinary Differential Equations Academy press San Diego, 3rd edition. 295pp
  23. 23. Fotta, A.U., Alabi, T.J and Abdulqadir, B (2015): Block Method with One Hybrid Point for the Solution of First Order Initial Value Problems of Ordinary Differential Equations International Journal of Pure and Applied Mathematics, Vol. 103 No. 3, 511–521
  24. 24. Ayinde S. O., Ibijola E. A. (2015) A New Numerical Method for Solving First Order Differential Equations. American Journal of Applied Mathematics and Statistics, Vol. 3, No. 4, 156–160. DOI: 10.12691/ajams-3-4-4
  25. 25. Odejide SA, Adeniran AO. A Hybrid Linear Collocation Multistep Scheme for Solving First Order Initial Value Problems Journal of the Nigerian Mathematical Society. Vol. 2012;31:229-241

Written By

Kamoh Nathaniel, Kumleng Geoffrey and Sunday Joshua

Submitted: September 30th, 2020 Reviewed: December 22nd, 2020 Published: September 8th, 2021