As is known, very often we are faced with solving the initial-value problem for the Ordinary differential Equation, which can be presented as the follows:

(r) (j) (j)

F(t, x, δ1x, δ ˙ 2x, ..., δ ¨ rx ) = 0, x (t0) = x (j = 0, 1, 2, .., r − 1). (1.1)

The numerical solution of this problem has been well studied for the case j ≤ 2. For the finding the numerical solution of this problem constructed and investigated some class methods of one and multistep types. Among of them one can be recommending some of classes multistep methods with the constant coefficients. Noted that some specialists usually recommended to reducing of the problem (1.1) to the system of differential equation of the first order.

However, adaptation methods for solving ordinary differential equation of higher orders can be more accurate. By taking into account of this, let us to consider investigation of the numerical solution of the following initial-value problem:

′′′ ′ ′′ (j) (j)

y (x) = f(x, y(x), δ1y (x), δ2y (x)), y = y (j = 0, 1, 2), x0 ≤ x ≤ X. (1.2)

(x0) 0

For the finding the numerical solution of this problem assume that continues of totality of ar-guments function by, its f(t, z) is defined in some closed set in some which has the continuous partial derivatives up to P, inclusively. To construct numerical methods for solving problem (1.1), let us denote by y(xi) the exact value of the solution to problem (1.1) at the mesh-point xi (i = 0, 1, ..., N). And by the yi denote approximately values of the numerical solution of the problem (1.1) at the corresponding mesh-point. Here xi+1 = xi + h (i = 1, 2, ..., N − k) and h > 0.

In usually for the finding numerical solution of this problem are used the following multistep methods with constant coefficients (see for example [3], [15], [29], [34] and [37]):

k m t s

X X X X

′ 2 ′′ 3

αiyn+i = h βiyn+i + h γiyn+i + h lifn+i, n = 0, 1, ..., . (1.3)

i=0 i=0 i=0 i=0

It is clear that by choosing the constant k, m, s, t one can be construct the numerical methods with the different properties. In the future, will be assume that the amount of the points one and the same; k = m = t = s.

It is not difficult to prove that, this method can be applied with equal success to solve the problems below:

′′ ′ ′

y (x) = φ(x, y, δ1y ), y(x0) = y0, y (x0) = y0, x0 ≤ x ≤ X, (1.4)

y (x) = F(x, y), y(x0) = y0, x0 ≤ x ≤ X. (1.5)

Noted that method (1.3) in the application to solve problem (1.4), usually used for the values li = 0 (i = 0, 1, ..., s), but for the problem (1.5), usually it’s coefficients choosing as the γi = 0 (i = 0, 1, ..., t)and li = 0 (i = 0, 1, ..., s). Noted that method (1.3) is more accurate, than the others. It is known that all the methods has own advantages and disadvantages. Methods receiving from the formula (1.3) is more accurate, but in the application that to solve any problems arises some

′ ′′

question about the calculation values yi and yi therefor the next paragraph is devoted to the study of this issue.

It is obvious that one of the main issues in the study of numerical methods is their convergence. It is known that the convergence of the multistep methods determined by using the values of the coefficients, which are usually called as stability of the multistep methods. Therefor let us to consider defined the stability of the Multistep Methods with constant coefficients. The definition of the conception of stability for the method (1.3), can it be formulated in the following form:

Definition 2.1. The method (1.3) is called as the stable, if the roots of the following polynomial

k k−1

ρ(λ) ≡ αkλ + αk−1λ + α1λ + α0

located in the unit circle, on the boundary of which there is not multiple root.

Have been proven, that the conception of stability is the necessary and sufficient for the convergence of the multistep method of type (1.3), if the condition |βm| + |βm+1| + ... + |β1| + |β0| ̸= 0 is holds (see for example [30], [35]). Noted that if the specified condition does not hold, then the method (1.3) does not converge. Consequently, if βi = 0 (i = 0, 1, ..., m), then the structure of method (1.3) is changes.

Let us noted that, in solving some problems arises for the study of methods, when βi = 0, (i = 0, 1, ..., m). For example, let us to consider solving problem (1.3) in the case δ1 = 0. As is known one of the effective methods for solving the problem (1.4) is the Stermer method, which can be presented as the following form (see for example [16]-[23] and [39]-[40]):

2 ′′

αiyn+i = h γiyn+i, n = 0, 1, 2, ..., . (2.1)

i=0 i=0

The concept of the stability for the method (2.1) can be formulated in the following form: Definition 2.2. Method (2.1) is called as the stable, if the roots of the polynomial

k k−1

ρ(λ) = αkλ + αk−1λ + ... + α1λ + α0

are located in the unite circle on the boundary of which there is not multiple roots except for double root λ = 1 . Noted that the maximal value for the degree of the method (2.1) and of the following:

αiyn+i = h βiyn+i, n = 0, 1, 2, ..., (2.2)

i=0 i=0

are the same. This method investigated by many authors (see for example [5]-[13], [40]-[41] and [44]).

By simple comparison methods (2.1) and (2.2) receive that these class of methods are separate class methods. And by the comparison of the classes (1.3) and (2.1), receive that the classes (1.3) and (2.1) are also the separate. Therefor these class of methods are investigated in separate form. Noted that for the construction more exact methods one can used corresponding integral equations or other ways (see for example [24]-[27], [30]-[33], [42], [45] and [54]).

And now let us to consider investigation of numerical problem (1.2). For this aim let us to use the formula (1.3). It is easy to understand that for the application of the method (1.3) it

′ ′′

is necessary to use methods for calculation of the value ym and ym. In this case arises some

′ ′′

difficult related with the calculation of the values ym and ym with the corresponding to accuracy. Noted that the conception of order of accuracy for the Multistep Methods usually is as the degree, which can be determined as the following.

Definition 2.3. The integer values is called as the degree for the method (1.3) (in the case: (s = t = m = k), if the following is holds:

′ 2 ′′ 3 ′′′ p+1

(αiy(x+ih)−hβy (x+ih)−h γiy (x+ih)−h liy (x+ih)) = O(h) , h → 0. (2.3)

As is known, one of the basic question in the application of the method (1.3) is the defined the value of pmax(maximum value for p). For this aim, let us to consider the following expansions, which are satisfies for smooth function:

′ (ih) ′′ (ih) (p) p+1 y(x + ih) = y(x) + ihy (x) + y (x) + ... + y + O(h) , h → 0. (2.4) 2! p!

2 p−1

′ ′ ′′ (ih) ′′′ (ih) (p−1) p y (x + ih) = y (x) + ihy (x) + y (x) + ... + y + O(h) , h → 0. (2.5) 2! (p − 1)!

2 p−2

′′ ′′ ′′′ (ih) IV (ih) (p−2) y (x + ih) = y (x) + ihy (x) + y + ... + y , h → 0. (2.6)

(x) (x)

2! (p − 2)!

2 p−3

′′′ ′′′ IV (ih) V (ih) (p−3) p−2 y (x+ih) = y (x) +ihy (x) + y +...+ y +O(h) , h → 0. (2.7)

(x) (x)

2! (p − 3)!

Let us suppose that the method (1.3) has the degree of p. It follows that the asymptotic equality of (2.3) takes place. Then by using the asymptotic equality (2.4)-(2.7) in the equality of (2.3), receive:

P P P 2

k k ′ 2 k i ′′

αiy(x) − h (iαi − βi)y (x) − h ( αi − iβi − γi)y (x)−

i=0 i=0 i=0 2!

P 3 2

3 k i i ′′′

−h ( αi − βi − iγi − li)y (x) − ...− (2.8)

i=0 3! 2!

P p p−1 p−2 p−3

p k i i i i (p) p+1

−h ( αi − βi − γi − li)y = O(h ), h → 0.

i=0 p! (p−1)! (p−2)! (p−3)! (x) 2 p ′ (p)

By taking into account that the system 1, x, x , ..., x or the system of functions y(x), y (x), .., y (y ̸≡ 0, j = 0, 1, ..., p) is linear independent, one can be write:

P P P P P 2

k k k k k i

αi = 0; βi = iαi; (iβi + γi) = αi; ...;

i=0 i=0 i=0 i=0 i=0 2!

(2.9) P p−1 p−2 p−3 P p

k i i i k i

( βi + γi + li) = αi.

i=0 (p−1)! (p−2)! (p−3)! i=0 p!

The system of (2.9) is the homogenies linear algebraic equation. The amount of the equation in this system equal p+1, but amount of the unknowns equal to 4k+k. It is known that the homoge-nies system always has a trivial (zero) solution. Noted that in the application of the method (1.3), suppose that αk ̸= 0, because the valueyn+k is the solution of the equation (1.3) (usually for the application of the method (3) suppose that the initial values y0, y1, .., yk−1are known and therefor yn+kis the solution of the finite-difference equation (1.3). Without breaking the generality, one can put αk = 1. By taking into the value αk = 1in the system (2.9) receive nonhomogeneous linear algebraic equation. It can be proven that the system (2.9) has the solution different from zero, if p + 1 ≤ 4k + 3 or p ≤ 4k + 2. One can be prove that if p = 4k + 2, then the corre-sponding method will be unique and if p < 4k+2, then the amount of the corresponding method will be more than 1 (one). Both theoretically and practical interest represent a stable methods so the question arises, about how one can determine maximum value of the degree for the stable methods. For this aim let us to consider the following theorems.

Theorem 2.4. Suppose that the method (1.3) stable and has degree of p and αk ̸= 0, for the case (s = t = m = k). Then in the class of the methods (2.8), there are methods with the degree p ≤ 3k + 4 and there are stable method with degree p = 3k + 4 for k = 2j. If k-is odd (k = 2j − 1), then there are stable method with the degree p = 3k + 3.

Theorem 2.5. Let us suppose that the method (1.3) has the degree p, stable, αk ̸= 0 and k ≥ max(m, s, t). Then p ≤ m + t + s + 4 and there are methods with degree p = m + t + s + 4 for the case t + s = −1 and m = k = 2j also for the m = t = s = k = 2j.

In the case 1)k = 2j, m = 2ν + 1, t = 2j, s = 2j + 1; 2)k = 2m + 1, t = 2j + 1; s = 2ν, m = 2j; 3)k = 2j, m = 2j, t = 2j + 1, s = 2ν; 4)k = 2j + 1, m = 2ν + 1, t = 2j, s = 2i + 1 there are method with degree p ≤ m + t + s + 2, but in other case there are methods with the degree p ≤ m + t + s + 3.

Noted that for the construction of more exact stable methods, specialists are constructed hybrid methods (see for example [14], [24]-[27], [38]-[39] and [46]-[55]).

As is known, the method (1.3) is called as the explicit, if αk ̸= 0, max(m, s, t) < k and if αk ̸= 0, max(m, s, t) = k, then method (1.3) called as the implicit. If αk ̸= 0 and max(m, s, t) > k, then method (1.3) called as the advanced or forward-jumping. Thus have determine the status of receiving methods from the method (1.3) as a special case. And now let us to consider the construction some specific methods.

Noted that from the method (1.3) one can receive the following explicit and implicit methods the special case:

′ ′ 2 ′′ ′′ 3 ′′′ ′′′

h(y + y ) h (y − y ) h (y + y )

n+1 n+2 n+1 n+2 n+1 n+2

yn+2 = yn + + + , (3.1) 2 10 120

′ ′ 2 ′′ ′′ 3 ′′′ ′′′

h(15y − 13yn) h (31y + 29yn) h (111y − 49yn )

n+1 n+1 n+1

yn+2 = − + , (3.2) 2 10 120

′ ′ 2 ′′ ′′ 3 ′′′ ′′′

(yn+1 + yn) h(31y − 25yn) h (63y + 57yn) h (233y − 97yn )

n+1 n+1 n+1

yn+2 = + − + . (3.3) 2 4 20 240 All the methods are stable, method (3.1) is the implicit one step method and has the degree p = 6, but methods (3.2) and (3.3) are explicit two step methods and has the degree p = 6.

If method (1.3) is stable and has the maximum degree, then the signs of the senior members alternate, that is βkγk < 0 and γklk < 0. If γk = 0, then the βkγk < 0is hold. The specified property is mainly used when constructing bilateral methods.

Methods (3.2) and (3.3) are the stable and explicit therefor these methods can be used as the predictor methods, if their accuracy are satisfactory, then methods (3.1) and (3.2) can be used as the main method for solving the given problem. However, often the explicitly methods are used as the predictor formula in the predictor-corrector methods. How does it follow from here, by using methods (3.1)-(3.3) one can construct two different predictor-corrector methods. Noted that, methods (3.1)-(3.3) can be applied to solve some problems, if are known the values of the solution of investigated problem at the points of xn and xn+1.

It is not difficult to understand that methods (3.1)-(3.3), can be applied to solve the initial-value problem for ODEs of the first and second order with the some success. For example let us to consider the following problem:

′′ ′ ′ ′

y (x) = φ(x, y, y ), y(x0) = y0, y (x0) = y (x0). (3.4)

The method (3.2) for the application to solve problem (3.4) can be presented as the following form:

′ ′ 2 3

h(15y − 13yn) h (31φn+1 + 29φn) h (119gn+1 − 49gn) yn+2 = yn+1 + − + , 2 10 120

here g(x, y, y ) = dφ(x, y, y )/dx.

As is known, among of the Numerical Methods, which have applied to solve initial-value prob-lems for ODEs, more popular as the Multistep Multiderivative Methods with the constant coeffi-cients. Noted that Multistep Methods and Multistep Second Derivative Methods fundamentally investigated by Dahlquist. But Multistep Multiderivative Methods in some general cases were investigated by Ibrahimov. Here have considering some simple form of the initial-value problem for ODEs of third order. For this have used some concrete methods with the second or third derivative, for the application of which to solve some initial-value problem, one can have con-structed predictor-corrector methods in different forms. It is known that in the construction of the predictor-corrector methods are arises some difficulties arise related with properties of the predictor-corrector methods. For example, in some cases as a predictor method one can use the unstable methods. And in some cases one can use the corrector method as the predictor. Note that the receiving results by these predictor-corrector methods are not the same. It is obvious that if the order of the ODEs increases, then the number of methods is also increasing. There-fore, the options for selecting methods are also increasing. And are tried here, describing some ways for a suitable method to obtain better results. Considering the difficulty of constructing algorithms for solving some problems, we tried to construct explicit methods. We hope that the results receiving here will be found over its following.

The authors express their sincere gratitude to the reviewers and editors for their useful comments.