Optimization of Adams-type difference formulas in Hilbert space $W_{2}^{(2,1)}(0,1)$

It is known that the solutions of many practical problems lead to solutions of differential equations or their systems. Although differential equations have so many applications and only a small number of them can be solved exactly using elementary functions and their combinations. Even in the analytical analysis of differential equations, their application can be inconvenient due to the complexity of the obtained solution. If it is very difficult to obtain or impossible to find an analytic solution to a differential equation, one can find an approximate solution.

In the present paper we consider the problem of approximate solution to the first order linear ordinary differential equation

with the initial condition

We assume that $f(x,y)$ is a suitable function and the differential equation (1.1) with the initial condition (1.2) has a unique solution on the interval $[0,1]$.

For approximate solution of problem (1.1)-(1.2) we divide the interval $[0,1]$ into $N$ pieces of the length $h=\frac{1}{N}$ and find approximate values $y_{n}$ of the function $y(x)$ for $n=0,1,...,N$ at nodes $x_{n}=nh$.

A classic method of approximate solution of the initial-value problem (1.1)-(1.2) is the Euler method. Using this method, the approximate solution of the differential equation is calculated as follows: to find an approximate value $y_{n+1}$ of the function at the node $x_{n+1}$, it is used the approximate value $y_{n}$ at the node $x_{n}$:

where $y_{n}^{\prime}=f(x_{n},y_{n})$, so that $y_{n+1}$ is a linear combination of the values of the unknown function $y(x)$ and its first-order derivative at the node $x_{n}.$

Everyone are known that there are many methods for solving the initial-value problem for ordinary differential equation (1.1). For example, the initial-value problem can be solved using the Euler, Runge-Kutta, Adams-Bashforth and Adams-Moulton formulas of varying degrees [1]. In [2] by Ahmad Fadly Nurullah Rasedee, et al., research they discussed the order and stepsize strategies of the variable order stepsize algorithm. The stability and convergence estimations of the method are also established. In the work [3] by Adekoya Odunayo M. and Z.O.Ogunwobi, it was shown that the Adam-Bashforth-Moulton method is better than the Milne Simpson method in solving a second-order differential equation. Some studies have raised the question of whether Nordsieck’s technique for changing the step size in the Adams-Bashforth method is equivalent to the explicit continuous Adams-Bashforth method. And in N.S.Hoang and R.B.Sidje’s work [4] they provided a complete proof that the two approaches are indeed equivalent. In the works [5] and [6] there were shown the potential superiority of semi-explicit and semi-implicit methods over conventional linear multi-step algorithms.

However, it is very important to choose the right one among these formulas to solve the Initial-value problem and it is not always possible to do this. Also, in this work, in contrast to the above-mentioned works, exact estimates of the error of the formula is obtained.

Our aim, in this paper, is to construct new difference formulas that are exact for $e^{-x}$ and optimal in the Hilbert space $W_{2}^{(2,1)}(0,1)$. Also these formulas can be used to solve certain classes of problems with great accuracy.

The rest of the work is organized as follows. In the first paragraph, an algorithm for constructing an explicit difference formula in the space is given. The above algorithm is used to obtain an analytical formula for the optimal coefficients of an explicit difference formula. In the second section, the same algorithm is used to obtain an analytical formula for the optimal coefficients of the implicit difference formula. In the third and fourth sections, respectively, exact formulas are given for the square of the norm of the error functionals of explicit and implicit difference formulas. Numerical experiments are presented at the end of the work.

We consider a difference formula of the following form for the approximate solution of the problem (1.1)-(1.2) [7, 8]

where $h=\frac{1}{N},$ $N$ is a natural number, $C[\beta]$ and ${C}_{1}[\beta]$ are the coefficients, functions $\varphi$ belong to the Hilbert space $W_{2}^{(2,1)}(0,1)$. The space $W_{2}^{(2,1)}(0,1)$ is defined as follows

equipped with the norm [9, 10]

The following difference between the sums given in the formula (2.1) is called the error of the formula (2.1) [11]

To this error corresponds the error functional [12]

where $\delta(x)$ is Dirac’s delta-function. We note that $(\ell,\varphi)$ is the value of the error functional $\ell$ at a function $\varphi$ and it is defined as [13, 14]

It should be also noted that since the error functional $\ell$ is defined on the space $W_{2}^{(2,1)}(0,1)$ it satisfies the following conditions

These give us the following equations with respect to coefficients $C[\beta]$ and $C_{1}[\beta]$:

Based on the Cauchy-Schwartz inequality for the absolute value of the error of the formula (2.1) we have the estimation

Hence, the absolute error of the difference formula (2.1) in the space $W_{2}^{(2,1)}$ is estimated by the norm of the error functional $\ell$ on the conjugate space $W_{2}^{(2,1)*}$. From this we get the following[15].

Problem 1. Calculate the norm $\|\ell|W_{2}^{(2,1)*}\|$ of the error functional $\ell$.

From the formula (2.3) one can see that the norm $\|\ell|W_{2}^{(2,1)*}\|$ depends on the coefficients $C[\beta]$ and ${C}_{1}[\beta]$.

Problem 2. Find such coefficients ${C}_{1}[\beta]=\mathring{C}_{1}[\beta]$ that satisfy the equality

In this case $\mathring{C}_{1}[\beta]$ are called the optimal coefficients and the corresponding difference formula (2.1) is called the optimal difference formula.

A function $\psi_{\ell}$ satisfying the following equation is called the extremal function of the difference formula (2.1) [13]

Since the space $W_{2}^{(2,1)}(0,1)$ is a Hilbert space, then from the Riesz theorem on the general form of a linear continuous functional on a Hilbert space there is a function $\psi_{\ell}$ (which is the extremal function) that satisfies the following equation [16, 17]

and the equality $\|\ell|W_{2}^{(2,1)*}\|=\|\psi_{\ell}|W_{2}^{(2,1)}\|$ holds, here $\langle\varphi,\psi_{\ell}\rangle_{W_{2}^{(2,1)}}$ is the inner product in the space $W_{2}^{(2,1)}(0,1)$ and is defined as follows [18]

According to the above mentioned Riesz’s theorem, the following equalities is fulfilled

By direct calculation from the last equality for the norm of the error functional for the difference formula (2.1) we have the following result [18].

It is known that stability in the Dahlquist sense, just like strong stability, is determined only by the coefficients $C\left[\beta\right]$, $\beta=\overline{0,k}.$ For this reason, our search for the optimal formula is only related to finding $C_{1}\left[\beta\right]$. Therefore, in this subsection we consider difference formulas of the Adams-Bashforth type, i.e. $C\left[k\right]=-C\left[k-1\right]=1$ and $C\left[k-i\right]=0$, $i=\overline{2,k}$, [19, 20]. Then is easy to check, that the coefficients satisfy the condition (2.4).

In this work, we find the minimum of the norm (2.9) by the coefficients $C_{1}\left[\beta\right]$ under the condition (2.5) in the space $W_{2}^{(2,1)}(0,1)$ [21]. Then using Lagrange method of undetermined multipliers we get the following system of linear equations with respect to the coefficients $C_{1}[\beta]$:

$\beta=\overline{0,k-1},$

It is easy to prove that the solution of this system gives the minimum value to the expression (2.9) under the condition (2.5). Here $d$ is an unknown constant, ${\mathop{C}\limits^{\circ}}_{1}\left[\beta\right]$ are optimal coefficient. Given that $C\left[k\right]=1$, $C\left[k-1\right]=-1$, $C\left[k-i\right]=0$, $i=\overline{2,k}$ the system (2.10),(2.11) is reduced to the form,

where

Assuming that $C_{1}\left[\beta\right]=0,$ for $\beta<0$ and $\beta>k-1$, we rewrite the system (2.12), (2.13) in the convolution form

thus, ${\mathop{C}\limits^{\circ}}_{1}\left[k-1\right]=\frac{e^{h}-1}{he^{h}}$ for $\beta=k-1.$

Finally, we have proved the following theorem.

Thus, the optimal explicit difference formula in $W_{2}^{(2,1)}(0,1)$ has the form

where $n=0,1,...,N-k,\,\,k\geq 1.$