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Interest in the construction of efficient methods for solving initial value problems that have some peculiar properties with it or its solution is recently gaining wide popularity. Based on the assumption that the solution is representable by nonlinear trigonometric expressions, this work presents an explicit single-step nonlinear method for solving first order initial value problems whose solution possesses singularity. The stability and convergence properties of the constructed scheme are also presented. Implementation of the new method on some standard test problems compared with those discussed in the literature proved its accuracy and efficiency.

Many of the numerical methods for obtaining the solution of the first order ordinary differential equation

y ′ = f ( x , y ( x ) ) , x ∈ [ x 0 , X ] , y ( x 0 ) = η (1)

are based on the assumption that the solution is locally representable by a polynomial. However, when a given initial value problem or its theoretical solution u ( t ) is known to posse a singularity, then it is particularly inappropriate to represent y ( x ) , in the neighbourhood of the singularity by a polynomial [

In this work, we assumed that the theoretical solution y ( x ) of (1) can locally be represented by a rational interpolant r ( x ) , of the form

r ( x ) = a 0 + a 1 x + a 2 x 2 + a 3 x 3 + a 4 x 4 b 0 + x . (2)

To construct an explicit single-step method with (2) for (1), it requires that r ( x ) satisfies the following:

r ( x n + j ) = y n + j , j = 0 , 1 , r ( i ) ( x n + j ) = y n + j ( i ) , j = 0 , i = 0 , 1 , 2 , 3 , 4 , 5. } (3)

Substituting for expressions and simplifying (3) yields

y n = a 4 x n 4 + a 3 x n 3 + a 2 x n 2 + a 1 x n + a 0 b 0 + x n y n + 1 = a 4 ( h + x n ) 4 + a 3 ( h + x n ) 3 + a 2 ( h + x n ) 2 + a 1 ( h + x n ) + a 0 b 0 + h + x n y ′ n = 4 a 4 b 0 x n 3 + 3 a 3 b 0 x n 2 + 2 a 2 b 0 x n + a 1 b 0 + 3 a 4 x n 4 + 2 a 3 x n 3 + a 2 x n 2 − a 0 ( b 0 + x n ) 2 y n ( 2 ) = 2 ( b 0 ( b 0 ( a 4 b 0 2 − a 3 b 0 + a 2 ) − a 1 ) + a 0 ( b 0 + x n ) 3 − a 4 b 0 + 3 a 4 x n + a 3 ) y n ( 3 ) = 6 a 4 − 6 ( b 0 ( b 0 ( a 4 b 0 2 − a 3 b 0 + a 2 ) − a 1 ) + a 0 ) ( b 0 + x n ) 4 y n ( 4 ) = 24 ( a 4 b 0 4 − a 3 b 0 3 + a 2 b 0 2 − a 1 b 0 + a 0 ) ( b 0 + x n ) 5 y n ( 5 ) = 120 ( a 4 b 0 4 − a 3 b 0 3 + a 2 b 0 2 − a 1 b 0 + a 0 ) ( b 0 + x n ) 6 } (4)

Eliminating the undetermined coefficients a 0 , a 1 , a 2 , a 3 , a 4 and b 0 in (4) results in

y n + 1 = y n + h y ′ n + 1 2 h 2 y ″ n + 1 6 h 3 y n ( 3 ) − 5 h 4 ( y n ( 4 ) ) 2 24 ( h y n ( 5 ) − 5 y n ( 4 ) ) . (5)

The resulting method (5) is explicit, self-starting and nonlinear. We shall refer to (5) as NLM4 which is the method proposed in this work. The new method NLM4 is suitable for solving initial value problems whose solution possesses singularities.

In this section, the local truncation error (lte) and the absolute stability properties of the new method proposed in this work are considered.

Local Truncation Error: The local truncation error T n + 1 at x n + 1 of the general explicit one step method

y n + 1 = y n − h ϕ ( x n , y n , h ) (6)

is given as

T n + 1 = y ( x n + 1 ) − y ( x n ) − h ϕ ( x n , y ( x n ) , h ) (7)

where, y ( x n ) is the theoretical solution. Using the above definition, it follows that the local truncation error of the constructed one step method can be written as

T n + 1 = y ( x n + 1 ) − y n + 1 (8)

Order of a Ordinary Differential Equation: A numerical method is said to be of order p if p is the largest integer for which T n + 1 = O ( h p + 1 ) for every n and p ≥ 1 . Following the above definition, the local truncation error of the method constructed in this work is obtained as the residual when y n + 1 is replaced by y ( x n + 1 ) . Below is the local truncation error for the method constructed in this work.

T n + 1 = 1 600 y ( 4 ) ( h 6 ( y ( 5 ) ) 2 ) (9)

A scheme is said to be consistent if the difference equation of the integrating formula exactly approximates the differential equation it intends to solve as the step size approaches zero. In order to establish the consistency property of the constructed method, it is sufficient to show that

lim h → 0 y n + 1 − y n h = 0 (10)

Now,

lim h → 0 y n + 1 − y n h = lim h → 0 ( h y ′ n + 1 2 h 2 y ″ n + 1 6 h 3 y n ( 3 ) − 5 h 4 ( y n ( 4 ) ) 2 24 ( h y n ( 5 ) − 5 y n ( 4 ) ) ) = 0 (11)

the above indicates that the constructed schemes satisfy the consistency property.

To get the stability behaviour of the constructed scheme, the scheme is implemented on the standard test problem

y ′ = λ y , R e ( λ ) < 0 (12)

and the stability polynomial R ( z ) = y n + 1 y n , where z = λ h is obtained. The stability function of (5) is obtained as

R ( z ) = y n + 1 y n = − z 4 − 8 z 3 − 36 z 2 − 96 z − 120 24 ( z − 5 ) (13)

and the region of absolute stability is seen in

The first problem considered in this work is the nonlinear initial value problem

y ′ = 1 + y 2 ; y ( 0 ) = 1 (14)

whose theoretical solution is given as

y ( x ) = tan ( x + π 4 ) . (15)

For this problem, the absolute errors of the results obtained by the method proposed in this work are first compared with those of Non-linear One-Step methods for initial value problems of [

The second test problem considered is given as

y ′ = y 2 ; y ( 0 ) = 1. (16)

The exact Solution is

y ( x ) = 1 1 − x . (17)

The logarithm of absolute errors for the solutions obtained is compared with other methods discussed in [

The explicit single-step nonlinear method constructed in this work is consistent and absolutely stable. Its region of absolute stability is larger than those of the methods discussed in the literature. The method gave more accurate result on the standard test problems compared with other methods discussed. Hence, the method is suitable for solving problems whose solution possesses singularity.

The authors declare no conflicts of interest regarding the publication of this paper.

Bakre, O.F., Wusu, A.S. and Akanbi, M.A. (2020) An Explicit Single-Step Nonlinear Numerical Method for First Order Initial Value Problems (IVPs). Journal of Applied Mathematics and Physics, 8, 1729-1735. https://doi.org/10.4236/jamp.2020.89130