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In this paper, we used an interpolation function with strong trigonometric components to derive a numerical integrator that can be used for solving first order initial value problems in ordinary differential equation. This numerical integrator has been tested for desirable qualities like stability, convergence and consistency. The discrete models have been used for a numerical experiment which makes us conclude that the schemes are suitable for the solution of first order ordinary differential equation.

Finite difference schemes have been in the forefront of the methods of using discrete models to approximate the solution of ordinary differential equations. Among the techniques used in building finite difference, scheme is the use of interpolation which requires the design of a basis function that is adequately differentiable in the domain of the numerical integration. Such basis function is then used to create a discrete version of the differential equation involved. This method has been used in the works of [

[

[

The need for the nonstandard method came up due to some shortcomings of the standard method, in which the qualitative properties of the exact solutions are not usually transferred to the numerical solution. These shortcomings may create a lot of problems, which may affect the stability properties of the standard approach [

The concept of numerical instability and its proof by [

A finite difference scheme is called nonstandard finite difference method, if at least one of the following conditions is met [

a) In the discrete derivative, the traditional denominator is replaced by a non-negative function φ such that, φ ( h ) = h + 0 ( h 2 ) as h → 0 ;

b) Non-linear terms that occur in the differential equation are approximated in a non-local way, i.e. by a suitable function of several points of the mesh. The concept of nonstandard finite difference schemes was proposed by [

Since the discovery of this method, researchers like [

Let assume that a solution of a differential equation can be represented by a function

F ( x ) = A cos x + B e − ∝ x − Q x (1)

where − ∝ and Q are simulation parameters and A and B are arbitrary constants.

Let a first order ordinary differential equation possess a real valued solution and be differentiable in its domain several times, then from (1) we can write:

y = A cos x + B e − ∝ x − Q x

y ′ = − A sin x − ∝ B e − ∝ x − Q (2)

y ″ = − A cos x + ∝ 2 B e − ∝ x (3)

y ‴ = + A sin x − ∝ 3 B e − ∝ x (4)

y i v = A cos x + ∝ 4 B e − ∝ x (5)

From (3) and (5) we have

y ″ + y i v = B ( ∝ 4 + ∝ 2 ) e − ∝ x

B = ( y ″ + y i v ) ( ∝ 4 + ∝ 2 ) e − ∝ x (6)

From (2) and (4) we have

y ′ + y ‴ = − B ( ∝ 3 + ∝ ) e − ∝ x

y ‴ − y ′ = 2 A sin x + B ( ∝ − ∝ 3 ) e − ∝ x

A = ( y ‴ − y ′ ) − B ( ∝ − ∝ 3 ) e − ∝ x 2 sin x (7)

Also from (3) and (5) we have

y i v − y ″ = 2 A cos x + ( ∝ 4 − ∝ 2 ) B e − ∝ x

A = ( y i v − y ″ ) − B ( ∝ 4 − ∝ 2 ) e − ∝ x 2 cos x (8)

For the discrete representation

y = A cos x + B e − ∝ x − Q x

y ( x ) = A cos x + B e − ∝ x − Q x

y ( x n ) = A cos x n + B e − ∝ x n − Q x n (9)

y ( x n + 1 ) = A cos x n + 1 + B e − ∝ x n + 1 − Q x n + 1 (10)

y ( x n + 1 ) ≡ y n + 1 and y ( x n ) ≡ y n

y ( x n + 1 ) − y ( x n ) ≡ y n + 1 − y n

e ∝ x n + 1 − e ∝ x n = e ∝ x n ( e ∝ h − 1 )

M = ( e ∝ h − 1 )

y n + 1 = y n − 2 A sin ( x n + h 2 ) sin h 2 + B M e − ∝ x n − Q h (11)

From (7), (8) and (11)

y n + 1 = y n − 2 { ( y i v − y ″ ) − B ( ∝ 4 − ∝ 2 ) e − ∝ x n 2 cos x } sin ( x n + h 2 ) sin ( h 2 ) + { ( y ″ + y i v ) ( ∝ 4 + ∝ 2 ) e − ∝ x n } ( e ∝ h − 1 ) e − ∝ x n − Q h (12)

Or

y n + 1 = y n − 2 { ( y ‴ − y ′ ) − B ( ∝ − ∝ 3 ) e − ∝ x 2 sin x n } sin ( x n + h 2 ) sin ( h 2 ) + { ( y ″ + y i v ) ( ∝ 4 + ∝ 2 ) e − ∝ x n } ( e ∝ h − 1 ) e − ∝ x n − Q h (13)

Let f n = y ′ , f n 1 − y ″ , f n 2 = y ‴ , f n 3 = y i v

Then

y n + 1 = y n − 2 { sin ( x n + h 2 ) sin ( h 2 ) 2 cos x n ( f n 3 − f n 1 ) − { ∝ 4 − ∝ 2 ∝ 4 + ∝ 2 } ( f n 3 + f n 1 ) } + { ( e ∝ h − 1 ) ( f n 3 + f n 1 ) ( ∝ 4 + ∝ 2 ) } − Q h (14)

Or

y n + 1 = y n − 2 { sin ( x n + h 2 ) sin ( h 2 ) 2 sin x n ( f n 2 − f n ) − { ∝ − ∝ 3 ∝ 4 + ∝ 2 } ( f n 3 + f n 1 ) } + { ( e ∝ h − 1 ) ( f n 3 + f n 1 ) ( ∝ 4 + ∝ 2 ) } − Q h (15)

E = sin ( x n + h 2 ) sin ( h 2 ) 2 cos x n , F = ∝ 4 − ∝ 2 ∝ 4 + ∝ 2 ,

G = e ∝ h − 1 ∝ 4 + ∝ 2 , H = ∝ − ∝ 3 ∝ 4 + ∝ 2 and I = sin ( x n + h 2 ) sin ( h 2 ) 2 sin x n (16)

y n + 1 = y n − 2 { E ( f n 3 − f n 1 ) − F ( f n 3 + f n 1 ) } + { G ( f n 3 + f n 1 ) } − Q h , cos x n ≠ 0

y n + 1 = y n − 2 { I ( f n 2 − f n ) − H ( f n 3 + f n 1 ) } + { G ( f n 3 + f n 1 ) } − Q h , sin x n ≠ 0

y n + 1 = y n + ( 2 F − 2 E + G ) f n 3 + ( 2 E + 2 F + G ) f n 1 − Q h , cos x n ≠ 0 (17)

y n + 1 = y n + ( 2 H + G ) f n 3 − ( 2 I ) f n 2 + ( 2 H + G ) f n 1 + ( 2 I ) f n − Q h , sin x n ≠ 0 (18)

Let

R = ( 2 F − 2 E + G ) , S = ( 2 E + 2 F + G ) ,

U = ( 2 H + G ) , T = ( − 2 I ) , M = ( 2 I ) (19)

Substitute (19) into (17) and (18), we have the integrator in the form (20) and (21) respectively

y n + 1 = y n + R f n 3 + S f n 1 − Q h (20)

y n + 1 = y n + U f n 3 + T f n 2 + U f n 1 + M f n − Q h (21)

Any algorithm for solving a differential equation in which the approximation y n + 1 to the solution at x n + 1 can be calculated iff x n , y n and h are known is called a one step method. It is a common practice to write the functional dependence y n + 1 on the quantities x n , y n and h in the form

y n + 1 = y n + ϕ ( x n , y n , h )

where ϕ ( x n , y n , h ) is the incremental function.

A numerical scheme with an incremental ϕ ( x n , y n , h ) is said to be consistent with the initial value problem y ′ = f ( x , y ) , y ( x 0 ) = y 0 if the incremental function is identically zero at t 0 when h = 0 .

Let the incremental function of the scheme defined in the one step scheme above be continuous and jointly as a function of its arguments in the region defined by

x ∈ [ a , b ] and y ∈ ( − ∞ , ∞ ) , 0 ≤ h ≤ h 0

where h 0 > 0 and let there exists a constant L such that ϕ ( x n , y n , h ) − ϕ ( x n , y n * , h ) ≤ L | y n − y n * | for all ( x n , y n , h ) and ( x n , y n * , h ) in the region just defined then the relation ( x n , y n , 0 ) = ( x n , y n * ) is a necessary condition for the convergence of the new scheme.

Let y n = y ( x n ) and p n = p ( x n ) denote two different numerical solution of the differential equation with the initial condition specified a y 0 = y ( x 0 ) = ξ and p 0 = p ( x 0 ) = ξ * respectively such that | ξ − ξ * | < ε , ε > 0 .

If the two numerical estimates are generated by the integration scheme, we have

y n + 1 = y n + h ϕ ( x n , y n , h )

p n + 1 = p n + h ϕ ( x n , p n , h )

The condition that | y n + 1 − p n + 1 | ≤ K | ξ − ξ * | is the necessary and sufficient condition for the stability and convergence of the schemes.

The increment function ∅ ( x n , y n ; h ) can be written in the form

∅ ( x n , y n ; h ) = { M f ( x n , y n ) + U f ( 1 ) ( x n , y n ) + T f ( 2 ) ( x n , y n ) + U f ( 3 ) ( x n , y n ) − Q h } (22)

Consider Equation (19), we can also write

∅ ( x n , y n * ; h ) = { M f ( x n , y n * ) + L f ( 1 ) ( x n , y n * ) + T f ( 2 ) ( x n , y n * ) + U f ( 3 ) ( x n , y n * ) − Q h }

∅ ( x n , y n * ; h ) − ∅ ( x n , y n ; h ) = M [ f ( x n , y n * ) − f ( x n , y n ) ] + U [ f ( 1 ) ( x n , y n * ) − f ( 1 ) ( x n , y n ) ] + T [ f ( 2 ) ( x n , y n * ) − f ( 2 ) ( x n , y n ) ] + U [ f ( 3 ) ( x n , y n * ) − f ( 3 ) ( x n , y n ) ] − Q h + Q h (23)

Let y ¯ be defined as a point in the interior of the interval whose points are y and y * , applying mean value theorem, we have

f ( x n , y n * ) − f ( x n , y n ) = ∂ f ( x n , y ¯ ) ∂ y n ( y n * − y n )

f ( 1 ) ( x n , y n * ) − f ( 1 ) ( x n , y n ) = ∂ f ( 1 ) ( x n , y ¯ ) ∂ y n ( y n * − y n )

f ( 2 ) ( x n , y n * ) − f ( 2 ) ( x n , y n ) = ∂ f ( 2 ) ( x n , y ¯ ) ∂ y n ( y n * − y n )

f ( 3 ) ( x n , y n * ) − f ( 3 ) ( x n , y n ) = ∂ f ( 3 ) ( x n , y ¯ ) ∂ y n ( y n * − y n )

We define

L = sup ( x n , y n ) ∈ D ∂ f ( x n , y n ) ∂ y n , L 1 = sup ( x n , y n ) ∈ D ∂ f ( 1 ) ( x n , y n ) ∂ y n

L 2 = sup ( x n , y n ) ∈ D ∂ f ( 2 ) ( x n , y n ) ∂ y n , L 3 = sup ( x n , y n ) ∈ D ∂ f 0 ( x n , y n ) ∂ y n

Therefore

∅ ( x n , y n * ; h ) − ∅ ( x n , y n ; h ) = M [ f ( x n , y n * ) − f ( x n , y n ) ] + U [ f ( 1 ) ( x n , y n * ) − f ( 1 ) ( x n , y n ) ] + T [ f ( 2 ) ( x n , y n * ) − f ( 2 ) ( x n , y n ) ] + U [ f ( 3 ) ( x n , y n * ) − f ( 3 ) ( x n , y n ) ] = M L ( y n * − y n ) + U L 1 ( y n * − y n ) + T L 2 ( y n * − y n ) + U L 3 ( y n * − y n ) (24)

Taking the absolute value of both sides

| ∅ ( x n , y n * ; h ) − ∅ ( x n , y n ; h ) | ≤ | M L ( y n * − y n ) + U L 1 ( y n * − y n ) + T L 2 ( y n * − y n ) + U L 3 ( y n * − y n ) | ≤ | M L + U L 1 + T L 2 + U L 3 | | y * − y | (25)

If we let K = | M L + U L 1 + T L 2 + U L 3 |

then our Equation (23) turns to

| ∅ ( x n , y n * ; h ) − ∅ ( x n , y n ; h ) | ≤ K | y * − y | (26)

which is the condition for convergence.

Consider an initial value problem of the form

y ′ = f ( x , y ) , y ( x o ) = y o (27)

Having an integrator of the form

y n + 1 = y n + ϕ ( x n , y n , h )

which can be obtained using (17) and (18) above and applying the rule in Section 1.2 a).

The renormalized nonstandard form of the schemes will be

y n + 1 = y n + φ { R f n 3 + S f n 1 − Q h } , cos x n ≠ 0 (28)

y n + 1 = y n + φ { U f n 3 + T f n 2 + U f n 1 + M f n − Q h } , sin x n ≠ 0 (29)

where y n + 1 = y n + φ ( x n , y n ; h ) , φ = sin ( ∝ h ) then

If h = 0 , E = 0 , F = ∝ 4 − ∝ 2 ∝ 4 + ∝ 2 , G = 0 and H = ∝ − ∝ 3 ∝ 4 + ∝ 2 , I = 0 and φ = 0 then (28) and (29) reduced to y n + 1 = y n

⇒ φ ( x n , y n ; 0 ) = 0 (30)

It is a known fact that a consistent method has order of at least one. Therefore, the new numerical integrator is consistent since Equations (28) and (29) can be reduced to (30) when h = 0 .

We shall establish the stability analysis of the integrator by considering the theorem established by [

Let y n = y ( x n ) and P n = P ( x n ) denote two different numerical solutions of initial value problem of ordinary differential Equation (25) with the initial conditions specified as y ( x o ) = η and p ( x o ) = η * respectively, such that | η − η * | < ε , ε > 0 . If the two numerical estimates are generated by the integrator (19). From the increment function (26), we have

y n + 1 = y n + φ ∅ ( x n , y n ; h ) (31)

P n + 1 = P n + φ ∅ ( x n , p n ; h ) (32)

The condition that

| y n + 1 − P n + 1 | ≤ K | η − η * | (33)

is the necessary and sufficient condition that our new method (19) be stable and convergent.

Proof

From (29) we have

y n + 1 = y n + φ ( 2 H + G ) f n 3 − φ ( 2 I ) f n 2 + φ ( 2 H + G ) f n 1 + φ ( 2 I ) f n − φ Q h

y n + 1 = y n + φ { U f n 3 + T f n 2 + U f n 1 + M f n − Q h } (34)

Then let

y n + 1 = y n + φ { M f ( x n , y n ) + U f ( 1 ) ( x n , y n ) + T f ( 2 ) ( x n , y n ) + U f ( 3 ) ( x n , y n ) − Q h } (35)

and

p n + 1 = p n + φ { M f ( x n , p n ) + U f ( 1 ) ( x n , p n ) + T f ( 2 ) ( x n , p n ) + U f ( 3 ) ( x n , p n ) − Q h } (36)

Therefore,

y n + 1 − p n + 1 = y n − p n + φ { M [ f ( x n , y n ) − f ( x n , p n ) ] + U [ f 1 ( x n , y n ) − f 1 ( x n , p n ) ] + T [ f 2 ( x n , y n ) − f 2 ( x n , p n ) ] + U [ f 3 ( x n , y n ) − f 3 ( x n , p n ) ] } (37)

Applying the mean value theorem as before, we have

y n + 1 − p n + 1 = y n − p n + φ { M [ ∂ f ( x n , p n ) ∂ y n ( y n − p n ) ] } + φ { U [ ∂ f ( 1 ) ( x n , p n ) ∂ y n ( y n − p n ) ] + T [ ∂ f ( 2 ) ( x n , p n ) ∂ y n ( y n − p n ) ] + U [ ∂ f ( 3 ) ( x n , p n ) ∂ y n ( y n − p n ) ] } (38)

We define

L = sup ( x n , y n ) ∈ D ∂ f ( x n , y ¯ n ) ∂ y n , L 1 = sup ( x n , y n ) ∈ D ∂ f ( 1 ) ( x n , y ¯ ) ∂ y n

L 2 = sup ( x n , y n ) ∈ D ∂ f ( 2 ) ( x n , y ¯ ) ∂ y n , L 3 = sup ( x n , y n ) ∈ D ∂ f 3 ( x n , y ¯ ) ∂ y n

Therefore

y n + 1 − p n + 1 = φ M L ( y n − p n ) + φ U L 1 ( y n − p n ) + φ T L 2 ( y n − p n ) + φ U L 3 ( y n − p n ) (39)

Taking the absolute value of both sides

| ∅ ( x n , y n * ; h ) − ∅ ( x n , y n ; h ) | ≤ | φ M L ( y n − p n ) + φ U L 1 ( y n − p n ) + φ T L 2 ( y n − p n ) + φ U L 3 ( y n − p n ) | ≤ | φ M L + φ U L 1 + φ T L 2 + φ U L 3 | | y n − p n | (40)

If we let K = | φ M L + φ U L 1 + φ T L 2 + φ U L 3 |

then our Equation (34) turns to

| ∅ ( x n , y n ; h ) − ∅ ( x n , p n ; h ) | ≤ K | y n − p n | (41)

which is the condition for convergence.

and y ( x o ) = η , P ( x o ) = η * , given ε > 0 , then

| y n + 1 − p n + 1 | ≤ N | y n − p n | (42)

and

| y n + 1 − p n + 1 | ≤ N | η − η * | < ε , for every ε > 0 (43)

Then we conclude that our method (29) is stable and convergent.

Note: Similar arguments can be used to proof the stability, convergence and consistency of the scheme y n + 1 = y n + φ { R f n 3 + S f n 1 − Q h } .

We derive a scheme for the first equation thus

y ′ = 25 + y 2 , y ( 0 ) = 0 , y = 5 tan ( 5 x ) (44)

y ″ = 2 y y ′ = 2 y ( 25 + y 2 ) = 50 y + 2 y 3 (45)

y ‴ = 100 + 50 y 2 + 150 y 2 + 6 y 4 = 100 + 200 y 2 + 6 y 4 (46)

y i v = 5000 + 800 y 3 + 24 y 5 (47)

Since we are integrating from zero and cos x n ≠ 0

y n + 1 = y n + φ { R f n 3 + S f n 1 − Q h }

where f n = y ′ , f n 1 − y ″ , f n 2 = y ‴ , f n 3 = y i v .

We now derive a scheme for the first model

y ′ = y − y 2 , y = y 0 y 0 + ( 1 − y 0 ) e t , y ( 0 ) = 0.5 (48)

y ″ = y ′ − 2 y y ′

y ″ = y − 3 y 2 + 2 y 3 (49)

y ‴ = y − 6 y 2 + 6 y 3 − y 2 + 6 y 3 − 6 y 4 (50)

y i v = y − 15 y 2 + 50 y 3 − 60 y 4 + 24 y 5 (51)

Since we are integrating from zero we will use

y n + 1 = y n + φ { R f n 3 + S f n 1 − Q h } ,

where f n = y ′ , f n 1 = y ″ , f n 2 = y ‴ , f n 3 = y i v .

The new scheme will be obtained by substituting the derivatives and applying it to our scheme

y n + 1 = y n + φ { R y ″ + S y ″ − Q h } (52)

The new standard scheme will be named (New SCH STD) with

φ = 1 , h = h (53)

For each of the examples the hybrid schemes will be obtained by applying the renormalization techniques.

The hybrid scheme (NEW SCH h) is obtained by using φ = sin ( ∝ h ) , step size

h = h (54)

The hybrid scheme (NEW SCH SIN) by using φ = sin ( ∝ h ) , step size

h = sin ( r h ) , r ∈ R (55)

The hybrid scheme (NEW SCH EXP) by using φ = sin ( ∝ h ) , step size

h = ( e λ h − 1 ) λ , λ ∈ R (56)

The choice of this denominator function as step size is informed by the works of [

The following are the 3D graphs obtained from the schemes when applied to the two models (Figures 1-14). We have used same parameters, step size, denominator functions and simulation parameters ∝ and Q to test the two differential equation models.

Graph of the Schemes with parameters: h = 0.001, ∝ = (4.25 - 14), Q = 36, λ = 0.26.

Graph of the Schemes with parameters: h = 0.0001, ∝ = (4.25 - 14), Q = 36, λ = 0.26.

Graph of the Schemes with parameters: h = 0.001, ∝ = (6.39 - 7.3), Q = 36, λ = 0.0026.

Graph of the Schemes of model II with parameters: h = 0.0001, ∝ = (6.39 - 7.3), Q = 36, λ = 0.0016.

The derived simulation models have been tested with the control parameters ∝ and Q. We also applied the Nonstandard method by modifying denominator function φ, which also provides for parameters λ and r that can be chosen to obtain iteratively assigned step size as denominator. The discrete model worked for the tested differential equation. The solution curves of the hybrid schemes follow the analytical solutions of the respective equations monotonically are shown in

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

Obayomi, A.A., Ayinde, S.O. and Ogunmiloro, O.M. (2019) On Trigonometric Numerical Integrator for Solving First Order Ordinary Differential Equation. Journal of Applied Mathematics and Physics, 7, 2564-2578. https://doi.org/10.4236/jamp.2019.711175