Bernstein Multiscaling polynomials and application by solving Volterra integral equations

In this paper, we present a direct computational method to solve Volterra integral equations. The proposed method is a direct method based on approximate functions with the Bernstein Multiscaling polynomials. In this method, using operational matrices, the integral equation turns into a system of equations. Our approach can solve nonlinear integral equations of the first kind and the second kind with piecewise solution. The computed operational matrices in this article are exact and new. The comparison of obtained solutions with the exact solutions shows that this method is acceptable. We also compared our approach with two direct and expansion–iterative methods based on the block-pulse functions. Our method produces a system, which is more economical, and the solutions are more accurate. Moreover, the stability of the proposed method is studied and analyzed by examining the noise effect on the data function. The appropriateness of noisy solutions with the amount of noise approves that the method is stable.

BMSPs are more general forms of BPs. One of the advantages of BMSPs compared to BPs is that they can approximate piecewise functions. In addition, using BMSP basis, we will have two degrees of freedom which increase accuracy of the method. One of these parameters is m, the degree of polynomials, and the other one k, which corresponds to the number of partitions in the interval ½a; bÞ.
In this paper, we review BP properties and preliminary theorems in Subsect. 1 of Sect. ''Review of Bernstein polynomials''. In Subsects. ''Tranformation matrices'' and ''Operational matrices'', transformation matrices and operational matrices for BPs are computed. In Sect. ''Bernstein Multiscaling polynomials'', BMSPs are defined. Transformational matrices and operational matrices for BMSPs are obtained in Sects. ''Transformation matrices'' and ''BMSPs operational matrices'', respectively. In Sect. ''Solution of Volterra integral equation'', by applying obtained matrices and functions approximation, the integral equations are turned into a system of equations. We present some numerical examples to illustrate the accuracy and ability of this method in Sect. ''Numerical examples''. In next section, we compare our approach with two direct and expansion-iterative methods based on block-pulse functions. In Sect. ''Stability of method'', stability is shown and we end this paper with a short conclusion in Sect. ''Conclusion''. For convenience, we consider a; b

Review of Bernstein polynomials
We denote U m , an m þ 1 À column vector, as follows: The BPs have many interesting properties [24][25][26][27][28][29][30]. However, here, some of them that are useful in our work are stated: The following theorems are a fundamental tool that justifies the use of polynomials. Theorem Next theorem indicates that dual matrix is symmetric and invertible.

Transformation matrices
Transformation matrix is used to change the dimension of the problem. In other words, this matrix can convert U m to U n and vice versa.
Suppose m is less than n, T n m is an ðm þ 1Þ Â ðn þ 1Þ matrix, called increasing transformation matrix, that converts U m to U n . In other words, U m ¼ T n m :U n : The increasing transformation matrix can be computed as follows: ; otherwise: It is sufficient to use p3, k times, where k ¼ n À m.
In addition, decreasing transformation matrix is an ðn þ 1Þ Â ðm þ 1Þ matrix, which is shown by T m n , and converts U n to U m , where n is greater than m. In other words, U n ¼ T m n U m : The ith row of decreasing transformation matrix can be calculated as follows:

Operational matrices
Operational matrix is a matrix that works on basis, such as an operator; in other words, if K is an operator, an operational matrix is a matrix, such as P, such that KðUÞ ' PU.

Operational matrix of integration
Operational matrix of product Using p1 gives: Lemma 5. Let u be an arbitrary ðm þ 1Þ vector, then whereũ is an ðm þ 1Þ Â ð2m þ 1Þ matrix with elements Proof. Property p1 implies Now, the ith entry of the above matrix can be rewritten as follows: Bernstein Multiscaling polynomials Definition 3. Suppose B i;m ðxÞ is the ith BPs of degree m on interval ½0; 1; Bernstein Multiscaling polynomials on ½0; 1Þ define as follows: where k ! 1 is the number of partitions on ½0; 1 and i ¼ 0; . . .; m, and in addition, j ¼ 0; . . .; k À 1: Now, every function f 2 L 2 ð 0; 1 ½ ÞÞ has the unique best approximation with respect to span space by BMSPs as follows: Equation (4) implies that w 0 i s are disjoint. In other words when In addition, Eq. (4) and p3 imply Lemma 6. Suppose Q is the dual operational matrix of BMSPs, then where 0 is an ðm þ 1Þ Â ðm þ 1Þ zero matrix, and Q is the dual operational matrix of BPs.

Transformation matrices
Let W m and W n be two different BMSPs and m n. There are two matrices, s n m and s m n ; such that W m ¼ s n m :W n and W n ¼ s m n :W m . These transformation matrices have dimensions kðm þ 1Þ Â kðn þ 1Þ and kðn þ 1Þ Â kðm þ 1Þ, respectively: where T n m and T m n are increasing transformation matrix and decreasing transformation matrix, respectively.

BMSP operational matrices
Operational matrices for BMSPs are obtained by BPs operational matrices and results are similar.

Solution of Volterra integral equation
In this section, we are going to convert an integral equation to a system.

Linear Volterra integral equation of the first kind
Consider the following Volterra integral equation of the first kind: where f and k are known, but u is not. Moreover, kðx; tÞ 2 l 2 ð½0; 1Þ Â ½0; 1ÞÞ and f ðtÞ 2 l 2 ð½0; 1ÞÞ: Approximating functions f ; u; and k with respect to BMSPs gives: where the vectors F; U, and matrix K are BMSP coefficients of f ðxÞ; uðtÞ; and kðx; tÞ, respectively. Now, replacing (15) into (14) gives: Using (12) follows: Using operational matrix of integration M; in Eq. (16) gives Let Equation (17) changes to: Using Eq. (11) in (18) gives: : Using decreasing transformation matrix s m 2m , gives the final system: where U T ¼Û ÃT s m 2m :

Nonlinear Volterra integral equation of the first kind
Consider the following nonlinear Volterra integral equation: Put wðxÞ ¼ gðuðxÞÞ and subsequently wðxÞ ¼ W T W m ðxÞ: ð21Þ where W is an unknown kðm þ 1Þ vector. Following the same procedure, the final system is as follows: Finally, uðxÞ ¼ g À1 ðwðxÞÞ is the desire solution.
One advantage of this method is solving linear or nonlinear Volterra integral equation of the second kind with piecewise functions. In these equations, solution, kernel, or data function can be piecewise. It is essential k, the number of partitions, be chosen, such that discontinuity points lie on boundary point of partitions.

Linear Volterra integral equation of the second kind
Consider linear Volterra integral equations as the following form: Substituting (15) into (22) and a process similar to the previous state final linear system is U À U ¼ F:

Numerical examples
Now, we test our method on some numerical examples; in every example, we use a table to show approximations, exact solution, and absolute errors in some points.
Example 1 Suppose uðxÞ ¼ e Àx be the exact solution of the following Volterra integral equation of the first kind: Table 1 shows results of Example 1.
Example 2 Consider the following integral equation: with the exact solution uðxÞ ¼ 2sinx. Table 2 shows approximated solutions, absolute errors, and exact solution in some points. Table 3 shows results of Example 3.

Comparison
Block-pulse functions are a special case of BMSPs. However, our method is different from the methods, as presented in [32] and [33]. Consider Example 1, in Table 6, the expansion-iterative method and direct method are compared with our method. In Table 6, we presented mean-absolute errors for the expansion-iterative method and direct method and absolute error of BMSPs for two different values of m and k. With respect to dimensions of the final system, our method is more accurate than the expansion-iterative method and direct method. Consider Example 2, Table 7 shows a comparison between BMSPs method and the expansion-iterative method and direct method with block-pulse functions. Mean-absolute errors for methods are presented in Table 7.

Stability of method
To demonstrate the stability of the method, we review effect of noise on data function. In other words, we replace f ðxÞ by 1 þ ep ð Þf ðxÞ into integral equation. Where p is a real random number between -1 and 1, and e is percent of noise. Now, we want to show that our method is stable and noise is proportional to the variations of solutions.

Conclusion
BMSPs that we use to solve Volterra integral equations have acceptable accuracy. Operational matrices, which we have computed, are exact. These exact matrices lead to fewer errors in our computations. In addition, BMSPs can solve piecewise Volterra integral equations of the second kind. Effect of noise on data function shows that our method is reliable and ill-posedness does not occur. This method with respect to complexity of computations and desirable accuracy is recommended. Furthermore, this method can be used to solve optimal control equations, differential equations, and systems of integral or differential equations.
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