Nonlenear Integro-differential Equations and Splines of the Fifth Order

of Approximation

I. G. BUROVA, Yu. K. DEM’YANOVICH

Department of Computational Mathematics,

St. Petersburg State University,

RUSSIA

Abstract: - In this paper, we consider the solution of nonlinear Volterra–Fredholm integro-differential equation,

integro-differential equation, the corresponding types of splines are used: the left, the right or the middle splines

of the fifth order of approximation. When using the spline approximations, we also obtain the corresponding

formulas for numerical differentiation. which we also apply for the solution of integro-differential equations.

The formulas for approximation of the function and its derivative are presented. The results of the numerical

solution of several integro-differential equations are presented. The proposed method is shown that it can be

applied to solve integro-differential equations containing the second derivative of the solution.

Key-Words: - Nonlinear Volterra–Fredholm integro-differential equations, polynomial splines, fifth order of

approximation

Received: August 22, 2021. Revised: July 23, 2022. Accepted: August 24, 2022. Published: September 23, 2022

discussed in the works of Volterra himself [1].

Volterra first began to study integral equations in

1884 (see [2]). This work is devoted to the

distribution of electric charge on a spherical

segment. Volterra showed that this problem leads

(in modern terms) to the solution of an integral

equation of the first kind with a symmetric kernel.

Volterra's first work on integro-differential

equations was a work on the theory of elasticity. As

is known, integro-differential equations connect an

the history of the electromagnetic field of the

substance at all previous moments (hysteresis)[3].

Methods of solving of integral equations is

considered in books [4], [5].

As noted in paper [6 ], “integral equations have been

one of the principal tools in various areas of applied

mathematics, physics and engineering. Scientists

have investigated the topic of integro-differential

equations through their work in many scientific

applications such as heat transfer, the diffusion

through a network of connected excitatory neurons

to a continuous model, defined by a Volterra-

Fredholm integro-differential equation, which

includes memory effects of the past in the

propagation. In paper [8], an effective direct method

to determine the numerical solution of the specific

nonlinear Volterra–Fredholm integro-differential

equations is proposed. The method is based on new

vector forms for the representation of triangular

functions and its operational matrix. In paper [9],

numerical solution of periodic Fredholm–Volterra

integro–differential equations of first-order is

discussed in a reproducing kernel Hilbert space. A

new O(n) time complexity numerical method for

computing the solutions of Basset integro-

differential equations is presented in paper [11]. A

new class of two-step collocation methods for the

Fredholm integro-differential equations is used.

Polynomial local splines of the fifth order of

approximation have proven themselves well in

solving interpolation problems, solving boundary

value problems and solving Fredholm and Volterra

integral equations [15]. In this paper, we will

consider the solution of integro-differential

equations from papers [6] and [9] using polynomial

local splines of the fifth order of approximation.

This method transforms the nonlinear Volterra-

In section 2, we consider the properties of splines

of the fifth order of approximation. In section 3, we

consider the solution of integro-differential

equations using splines of the fifth order of

The general theory of constructing local

interpolation splines is considered in the monograph

by prof. Yu.K. Dem’yanovich and I. G. Burova. Let

the function 󰇛󰇜 be known at the nodes of the grid

󰇝󰇞        . Approximation

with the local splines of the fifth order of

approximation is built separately on each grid

Denote  󰇛󰇜. At the beginning of the interval

󰇟󰇠, we apply the approximation with the right

where

Proof. Differentiating identity (1) we have the

relation:

Since the absolute value of the integral does not

exceed the integral of the absolute value of the

integrand, we obtain the inequality:

Here 󰇝󰇞 means the smallest segment

containing the points . Let us calculate the

󰇛󰇜󰇛󰇜󰇠󰇛󰇜 . (6)

By integrating in parts, we get the equality

 

By relations (4) and (7) we deduce the formula

  

Let us estimate the first term of relation (8). We take

out the maximum of the absolute value of the

function 󰇛󰇜󰇛󰇜 from the expression under the

integrals and then use equality (10). As a result, we

get the inequality:

A similar estimate is obtained for the second term in

(8).

Finally, we deduce the next inequality from formula

(8) using condition (9):

󰆒󰇟󰇠

relation. Namely, for some point ξ  󰇟󰇠, the

next equality is valid:

Since by assumption  󰇟󰇠, this implies the

inequality:

From formula (12) with the help of relations (11)

and (13) we obtain the estimate:

From inequality (14) we have relation (3).

The proof of Theorem 2 is complete.

Remark. Theorem 2 is proved when there is a non-

uniform grid of nodes.

On a more detailed note, we have

Differentiating this equality, we get

a uniform grid of knots. In the case of a uniform

grid, we have   Replacing   

,   󰇟󰇠, in the expression  we have:

Now it is easy to obtain error estimates for the right

 

 

Note that the inequalities turn into equalities on

Table 1 shows the actual errors of approximation

of functions and the first derivative of the functions

the first derivative of the functions when   ,

The data presented in the Tables are consistent with

the theoretical results formulated in the Theorems.

formulas of numerical differentiation obtained with

coefficients, we have to solve a system of nonlinear

equations. Then we can visualize the solution by

connecting the obtained points with splines of the

fifth order of approximation. In addition, it is

possible to obtain a piecewise given expression not

only for the desired function, but also for the first

derivative of the desired function. In the figures, the

Figure 1 shows the errors in the solution of problem

1, found using splines of the fifth order of

approximation. Figure 2 shows the errors of

problem 1 found in paper [6]. Figure 3 shows the

solutions to problem 1 found using paper [8]’s .

Fig. 1: The plot of the errors in the solution of

problem 1, found using splines of the fifth order of

approximation

Fig. 3: The plot of the errors in the solution of

problem 1, found in paper [8]

the graph of the first derivative of the solution

restored using splines of the fifth order of

approximation.

Fig. 4: The plot of the solution of problem 1, found

using splines of the fifth order of approximation

Fig. 5: The plot of the errors of the first derivative of

the solution of problem 1, found using splines of the

fifth order of approximation

Example 2 (Example 4.2. from paper [6]). Consider

the nonlinear Volterra integro-differential equation,

as follows:

solution at the grid nodes, obtained using splines of

the fifth order of approximation. The third column

gives the solution presented in paper [6] (at  

Table 4. The results of calculations (Example 2)

Figure 6 shows the errors in absolute values of the

solution of problem 2, found using splines of the

fifth order of approximation. Figure 7 shows the

Fig. 7: The plot of the errors in the solution of

problem 2, found in paper [6]

Figures 8 and 9 show the graphs of the solution and

the graph of the first derivative of the solution,

restored using splines of the fifth order of

approximation.

Example 3 (Example 4.3. from paper [6]). Consider

the nonlinear Volterra–Fredholm integro-differential

equation, as follows:

the fifth order of approximation. The third column

gives the solution presented in paper [6] (at  

using splines of the fifth order of approximation,

abscissa axis, grid nodes from the interval [0,1] are

marked.

Fig. 10: The plot of the errors in the solution of

problem 3, found using splines of the fifth order of

approximation

Fig. 11: The plot of the errors in the solution of

problem 3, found in paper [6]

Example 4. Finally, consider an integro-differential

The exact solution to this problem is the next:

For the approximation of the second derivative, we

obtain the formula in the same way, namely by

twice differentiating the spline approximation of the

function.

In paper [9] with the number of nodes 32, the error

Problem 4 obtained with splines of the 5th order of

approximation.

Such a high accuracy of the solution is explained by

the fact that spline approximations are exact on

the approximation error is zero for polynomials up

to the fourth degree.

4 Conclusion

This paper considers the solution of nonlinear

One example of solution of the integro-differential

equation with the second derivative of the unknown

is given.

Note that it is assumed that the integral of the

product of the kernel and the basis function is

calculated without error. In this case, to obtain a

solution, it is required that the solution be five times

integro-differential equations containing the second

derivative. In addition, cases of using a non-

uniform grid, as well as non-polynomial

approximations, will be considered.

[1] V. Volterra, Leçons sur les équations intégrales