The Class of Atomic Exponential Basis Functions EFupn(x,ω)-Development and Application

Nives Braji Kurbaa,Bla Gotovac and Vedrana Kozuli

University of Split,Faculty of Civil Engineering,Architecture and Geodesy,Split,21000,Croatia

ABSTRACT The purpose of this paper is to present the class of atomic basis functions(ABFs)which are of exponential type and are denoted by EFupn(x,ω).While ABFs of the algebraic type are already represented in the numerical modeling of various problems in mathematical physics and computational mechanics,ABFs of the exponential type have not yet been sufficiently researched.These functions,unlike the ABFs of the algebraic type Fupn(x),contain the tension parameter ω,which gives them additional approximation properties.Exponential monomials up to the nth degree can be described exactly by the linear combination of the functions EFupn(x,ω).The function EFupn for n=0 is called the“mother”ABF of the exponential type,i.e.,EFup0(x,ω)≡Eup(x,ω).In other words,the functions EFupn(x,ω)are elements of the linear vector space EUPn and retain all the properties of their“mother”function Eup(x,ω).Thus,this paper,in terms of its content and purpose,can be understood as a sequel of the article by Braji′c Kurbaa et al.,which shows the basic properties and application of the basis function Eup(x,ω).This paper presents,in an analogous way,the development and application of the exponential basis functions EFupn(x,ω).Here,for the first time,expressions for calculating the values of the functions EFupn(x,ω)and their derivatives are given in a form suitable for application in numerical analyses,which is shown in the verification examples of the approximations of known functions.

KEYWORDS Exponential atomic basis functions;Fourier transform;compact support;tension parameter

1 Introduction

Numerical methods are indispensable for the successful simulation of physical and engineering problems.Many different numerical approaches and methods have been proposed in recent decades.The classical methods are the finite element method(FEM),the finite difference method(FDM),the finite volume method(FVM),the boundary element method(BEM),and the discrete element method(DEM)[1–3].In addition to traditional mesh-based methods,there are many others,such as various meshless methods[4–6].

The choice of the basis functions plays a key role in all numerical methods.The idea of choosing basis functions that correspond to the class of solutions of the problems we are solving has long been accepted, but, in practice, rarely implemented.Polynomials are fundamental to modeling and numerical methods.They provide canonical local approximations to smooth functions and are used extensively in geometric design.Polynomials not only provide very accurate approximations of smooth functions but also guarantee convergence for any continuous function on a compact interval.

Whereas classical polynomials have dominated in the field of numerical analysis, spline-based basis functions [7] play a crucial role in the field of computational geometry.The true popularity of spline functions for numerical analysis was achieved by the introduction of the concept of isogeometric analysis(Hughes et al.[8]and Cottrell et al.[9]).B-splines play an important role in many areas of applied mathematics, computer science, and engineering.Typical applications arise in the approximation of functions and data,automated design and manufacturing,computer graphics,and numerical simulations.This diversity of areas and techniques involved makes B-splines an extremely interesting research topic, which has attracted a growing number of scientists in universities and industry.

In addition to spline functions, relatively lesser-known atomic basis functions have been used in recent times [10–13].Atomic basis functions can be placed between classical polynomials and spline functions.However,in practice,their use as basis functions is closer to splines or wavelets(see Beylkin et al.[14]).Rvachev et al.[10],in their pioneering work,called these basis functions“atomic”because they span the vector spaces of all three fundamental functions in mathematics: algebraic,exponential,and trigonometric polynomials.The authors of this article have worked intensively on the development and application of ABFs of algebraic type in solving problems of structural mechanics and have therefore demonstrated their significant potential compared to conventional procedures with finite elements.Gotovac [12] systematized the existing knowledge regarding atomic basis functions of algebraic type and transformed them into a numerically appropriate form, especially Fup basis functions as a typical member of the atomic class of basis functions.Gotovac et al.[15] showed the basic possibilities of using atomic functions in structural mechanics and numerical analysis.The work in [16] gives a generalization of atomic functions to the multivariable case.The use of Fup basis functions, which are atomic functions of the algebraic type, has been shown to solve the problem of signal processing [17], the initial value problem [18], the boundary value problems using the Fup Collocation Method[19],the boundary-initial value problems[20],elasto-plastic analysis of prismatic bars subjected to torsion [21], and modeling of groundwater flow and transport problems [22].Gotovac et al.[23]presented a true multiresolution approach based on the Adaptive Fup Collocation Method(AFCM).Kamber et al.[24]set the foundation for an efficient adaptive spatial procedure by developing a one-dimensional hierarchical Fup(HF)basis functions.The works in[25,26]gave a brief analysis of the current publications regarding ABFs,from the first publications to current ones.

In the mentioned works, the advantage of atomic basis functions of algebraic type, which significantly improve the quality of numerical solutions in relation to classical basis functions, for example,splines and wavelets,is confirmed.The numerical results thus obtained were the motivation for the development of ABF of the exponential type which are wider than algebraic space,moreover algebraic ABFs space is contained in exponential ABFs space.

The numerical modeling of different physical and engineering problems characterized by large local gradients and singularities often presents a challenge in terms of choosing a numerical approach and basis functions.Classic examples of such are the advection–dispersion equation and the heat conduction equation,which describe the transfer of mass and energy,respectively;beams and plates on a flexible foundation;and special problems of loss of stability.For the simulation of such physical problems, exponential basis functions would be a good choice.Improving the quality of numerical analyzes of problems whose solutions have an exponential form is the main motivation of this paper.

The atomic functions of the exponential type have been developed only at the basic level.In[12], the previous knowledge about ABF of the exponential type was presented, which was later expanded and upgraded in[27].Reference[28],partly resulted from[27],showed the basic properties and application of the maternal basis functionEup(x,ω),by which the whole class of atomic functions of the exponential typeEFupn(x,ω)is generated and given in this article as natural sequel of the[28]to complete“the story”of the ABFs of the exponential type.

The content of this work is focused on the mathematical background,approximation properties,and applications of exponential basis functionsEFupn(x,ω).There are no articles in the literature that deal with these basis functions.So, this paper is intended to provide novel information for scientists and engineers who are interested in applying the state-of-the-art atomic exponential basis functions to solve real-life problems.The paper presents expressions for the necessary mathematical operations of the ABFsEFupn(x,ω)in a simpler, more understandable and more user-friendly way.New expressions have been derived,especially the expression for calculating the value of the function and the desired number of derivatives at an arbitrary point of the basis function support,which is the original contribution of this paper and,most importantly,the rules(elements)for their practical use.

The following section of the article refers to the description of the ABF class of the algebraic type.The procedure used to generate the class of functionsFupn(x)and the determination of their derivatives are presented,and the basic properties are given in a new and original way,and that is starting from the well-known Fourier transform and the convolution theorem in a way suitable for defining and deriving the ABFs of the exponential typeEFupn(x,ω), shown in Section 3.The implementation of ABFEFupn(x,ω)in the numerical approximations of the given functions is shown in Section 4.Finally,the conclusions are given in Section 5.

2 ABFs of the Algebraic Type:Fupn(x)

Atomic Basis Functions(ABFs)are infinitely derivable finite solutions of functional differential equations of the type:

whereLis a linear differential operator with constant coefficients,λis a scalar quantity other than zero,Ckare the solution coefficients,a ＞0 is the support length parameter of the finite function,andbkare the coefficients that determine the displacements of the finite basis functions[10–12,15].

The type of finite functiony(x)from the class of atomic basis functions is determined by choosing the operatorLin Eq.(1).Thus,we distinguish the atomic basis functions of the algebraic,exponential,and trigonometric types.

The functionsFupn(x)are finite ABFs of the algebraic type from the classC∞with a compact support,and they are also elements of the universal vector spaceUPn.The indexndenotes the highest degree of a polynomial that can be accurately represented in the form of a linear combination of basis functions obtained by moving the functionFupn(x)for the characteristic segmentΔxn= 2－n.Forn=0,it holds that:

The functionsFupn(x)retain all the good properties of the“maternal”functionup(x)[10–12,15],while for the development of a given function,a much smaller number than that of the basis functions obtained by moving the functionup(x)is required.For a sufficiently highnfunctionFupn(x)has a very small support length,so any functionFupk(x),k ＜n,including the functionup(x),can be expressed using the functionFupn(x).

Unlike in references[10,11]which define ABFs from Eq.(1),the authors of this paper determine ABFs from their known Fourier transform(FT),and then from their known FT determine everything necessary for their use (e.g., derivatives, integrals, moments, etc.).Namely, we can say that in the“frequency domain”the construction of ABFs becomes more transparent.

The FT of the functionFup0(x),according to Eq.(2),corresponds exactly to the FT of the functionup(x)from[15,28],i.e.,

The Fourier transform of the functionFupn(x)is given by the expression[15,28]:

Thus,according to Eq.(4),the functionsFupn(x)can be written in integral form:

From the known FT,as is shown similarly for the functionup(x)in[28],the functionsFupn(x)can also be generated using the convolution theorem.In Eq.(4),it is seen that the FTsFn(t)of the basis functionsFupn(x)are equal to the product of thenth degree B-spline FT compressed on the support of length(n+1)2－nand the functionup(x)FT from Eq.(3)compressed on a support of length 2－n.Thus,the functionsFupn(x)can be written using the convolution theorem in the form:

According to Eq.(6),the support of the functionFupn(x)is an interval composed ofn+2 segments of length 2－n,which are called characteristic segments,that is,

The functional differential equations of the basis functionsFupn(x)are of the following form[12,15]:

whereare binomial coefficients.

Solving the functional differential Eq.(8), or Eqs.(4) and(5), is not numerically convenient for calculating the values of the functionFupn(x).Practically,the most convenient possibility to construct the functionsFupn(x)is in the form of a linear combination of functionsup(x)mutually shifted for the characteristic segment 2－n,i.e.,

The“zeroth”coefficient follows from Eq.(9)and is

The other coefficients are obtained in the formCk(n)=C0(n)·,, where the auxiliary coefficientsare calculated by the recursive formula[15]:

The coefficients from Eq.(11)forn≤6 andk≤9 are given in Table 1.

Table 1: Coefficients n)for n ≤6 and k ≤9

Table 1: Coefficients n)for n ≤6 and k ≤9

C′kn C′0 C′1 C′2 C′3 C′4 C′5 C′6 C′7 C′8 C′9 n 0 1 0 0 0 0 0 0 0 0 0 1 1 －1 1 －1 1 －1 1 －1 1 －1 2 1 －2 2 －2 3 －4 4 －4 5 －6 3 1 －3 4 －4 5 －7 8 －8 10 －14 4 1 －4 7 －8 9 －12 15 －16 18 －24 5 1 －5 11 －15 17 －21 27 －31 34 －42 6 1 －6 16 －26 32 －38 48 －58 65 －76

The derivatives of the functionFupn(x)are obtained by a linear combination of the derivatives of the shifted functionsup(x)using the coefficients from Eq.(11),i.e.,

wheremis the order of derivation,andnis the order of the basis function.Fig.1 shows the functionFup2(x)and its first three derivatives.The third, and all further derivatives, of the functionFup2(x)correspond in parts to the compressed functionup(x).

The integrals of the functionFupn(x)are also obtained by a linear combination of the integrals of the shifted functionsup(x)using the coefficients from Eq.(11):

Figure 1:Function Fup2(x)and it is first three derivatives

3 ABFs of the Exponential Type:EFupn(x,ω)

The functionsEFupn(x,ω)are finite functions of classC∞with compact support, and are the elements of linear vector spaceEUPn[12,27,28],and retain all the properties of their“maternal”basis functionEup(x,ω).The index‘n’denotes the largest degree of an exponential monomial that can be represented exactly in the form of a linear combination of mutually shifted functionsEFupn(x,ω)on a characteristic segment of lengthΔxn=2－n.

3.1 Generating the Fourier Transform of the Function EFupn(x,ω)

The Fourier transform of the atomic basis functionEFupn(x,ω)is constructed by a similar procedure applied to the functionFupn(x)using the so-called “fragmentation process”of the FT as shown below.

The first from the ABF class of the exponential typeEFupn(x,ω)forn= 0 is precisely the“maternal”basis functionEup(x,ω),i.e.,

Thus,according to Eq.(14),the Fourier transform of the functionEFup0(x,ω)is determined by the expression from[28]:

Writing Eq.(15)in an extended form,applying the basic trigonometric relations,and fragmenting the expression thus obtained(omitting the termch(α/2)),after arranging the expression,we obtain the Fourier transform of the finite functionEFup1(x,ω)of the form:

Thus, analogously to the ABFFupn(x), according to Eq.(17), the functionEFupn(x,ω)can be written using the convolution theorem in the following form:

where

When the parameterωweighs zero,the exponential ABF turns into an algebraic ABF,so Eqs.(17)and(20)become Eqs.(4)and(6),respectively.

Figure 2:Generating the exponential functions EFupn(x,ω)for:(a)n=0;(b)n=1;(c)n=2

According to Eq.(18),the support of the functionEFupn(x,ω)is an interval composed of(n+2)segments of length 2－n.The characteristic points are the boundary points of the characteristic segment.

The inverse Fourier transform, i.e., the functionEFupn(x,ω), having satisfied the Paley–Wiener normalization condition,can be expressed in the form:

By developing Eq.(21) in the Fourier series, the “original”of the functionEFupn(x,ω)can be determined at arbitrary points.However, as for the algebraic ABFsFupn(x), the most favorable possibility of constructing theEFupn(x,ω)functions is in the form of a linear combination of shiftedEup(x,ω)functions,as shown below.

Fig.3 shows the functionEFup2(x,ω)for different values of the parameterω.Similar to the“maternal”functionEup(x,ω),the function is tilted to the left for negative values of the parameterω, while for positive ones it is tilted to the right.In the limitary case whenω→0, the exponential functionEFup2(x,ω)is identically equal to the algebraic functionFup2(x).

Figure 3:Function EFup2(x,ω)for different values of the parameter－10 ≤ω ≤10

3.2 Functional Differential Equation of the Function EFupn(x,ω)

Analogous to the algebraic ABFFupn(x), the functional differential equation of the functionEFupn(x,ω)is determined from the Fourier transform(17),which can also be written as follows:

If Eq.(22) is written in the form of exponential functions and multiplied by(ω+it), arranging the members of the left and right sides gives the functional differential equation of the functionEFupn(x,ω)of the form:

where the coefficients are

and

where

In particular,when the value of the parameterω=0,Eqs.(23)–(26)become equivalent to Eq.(8).

3.3 The Values of the Function EFupn(x,ω)in Characteristic Points

The values of the functionsEFupn(x,ω)at arbitrary discrete pointsxcan be determined by Convolution(20)as a solution of the following integral:

wherefn(x,ω)is the corresponding exponential spline defined as the result of the convolution of zerodegree exponential splines,i.e.,

while,ω)are determined by Eq.(19).

However,calculating the integral(27)at arbitrary pointsxis not a simple or numerically favorable procedure, and therefore solving the integral (27) is used only to determine the values of the basis functionsEFupn(xk,ω)at the characteristic pointsxk.

Fig.4 shows the graphical interpretation of the integral(27)for the basis functionEFup2(x,ω)at the characteristic points

For example,the value of the functionEFup2(x,ω)at the pointxk=－1/4,according to Eq.(27),corresponds to the solution of the following integral:

which,when written in exponential form and using the appropriate substitutions after arranging,has the final form:

whereλ0(ω)is the value of the“maternal”functionEup(x,ω)at the local origin,i.e.,the pointx=0,see[28].

Figure 4:The values of the function EFup2(x,ω)at the characteristic points xk = using the convolution theorem

The values at other characteristic points of the functionEFup2(x,ω)are determined by an analogous procedure:

or the values of the basis functionsEFupn(x,ω)at the characteristic points in general.

As seen in Eqs.(30)and(31),the values of the functionsEFupn(x,ω)at the characteristic pointsxkhave a“final”inscription in the form of the product of the corresponding exponential function and the values of the functionEup(x,ω)at the pointx=0,i.e.,λ0(ω)given in[28].

3.4 EFupn(x,ω)as a Linear Combination of Shifted Eup(x,ω)Functions

The values of the basis functionsEFupn(x,ω)at arbitrary points can be determined,among other methods, by developing Eq.(21) in the Fourier series.However, analogous to the algebraic ABF,the most favorable possibility of constructing the functionsEFupn(x,ω)is in the form of a linear combination of mutually shiftedEup(x,ω)basis functions:

whereCk(n)are the coefficients of the linear combination.

The“zeroth”coefficientC0(n)is determined in[27]by the expression

The other coefficients of the linear combinationCk(n),k= 1,...,n+1 are unknown and are determined as described below.

For example, for the basis functionEFup2(x,ω), the linear combination (32) has the following form(hereinafter,the functionsEup(x,ω)will be denoted byyω(x)for transparency):

or written in characteristic points:

where the values of the basis functionEFup2(x,ω)at the characteristic points are known and are calculated as shown in Section 3.3.

The expression for the“zeroth”coefficient follows directly from the first equation in Eq.(35):

By including the values from(30)andyω(－3/4)from[28]in Eq.(36),we obtain

which corresponds to Eq.(33)forn=2.

The“first”coefficient of the linear combination(34)follows from the second equation in Eq.(35)in the form:

By including the coefficientC0(2)and the other required values,we obtain

The expression for the“third”coefficient follows from the third equation in Eq.(35),and so on.By generalizing the presented procedure,a general expression for the coefficientsCk(n)is obtained in the form of a recursive formula:

where the coefficientsC0(n)are determined by(33).

Thus, to determine the coefficientsCk(n),k= 0,...,n+ 1 of the linear combination (32), it is necessary to know the “zeroth”coefficientC0(n)and the values of the functionsEup(x,ω)andEFupn(x,ω)at the characteristic pointsxk.

In the limit whenω→0,the coefficientsCk(n)for the development of the exponential functionsEFupn(x,ω),n= 1,...,6 from Eq.(40) become the coefficientsCk(n)for the development of the algebraic basis functionsFupn(x)in the form of a linear combination of mutually shifted functionsup(x)from Eq.(9).

Fig.5 shows the functionEFup2(x,ω)in the form of a linear combination of mutually shiftedEup(x,ω)basis functions.

Figure 5:Function EFup2(x,ω)as a linear combination of shifted Eup(x,ω)basis functions

3.5 The Derivatives and Integrals of the Function EFupn(x,ω)

The derivatives of the functionEFupn(x,ω)are obtained by a linear combination of the derivatives of the shiftedEup(x,ω)functions using the coefficients specified in the previous section:

Fig.6 shows the basis functionEFup2(x,ω)and its first three derivatives for the value of the parameterω=2.

Figure 6:Function EFup2(x,ω)and the first three derivatives

The integrals of the functionEFupn(x,ω)are also obtained by a linear combination of the integrals of the shiftedEup(x,ω)functions:

3.6 The Connection between the Function EFupn(x,ω)and the Exponential Monomials

Similar to the functionEup(x,ω)[28], which is only a special case of the functionEFupn(x,ω)forn= 0,a connection between the functionsEFupn(x,ω)and exponential monomialse2nω·xcan be established.

For a linear combination of the basis functionsEFupn(x,ω),

(offset from each other for the characteristic sectionΔxn= 2－n) to represent an exponential monomial of degreen, it is necessary and sufficient that by the action of the differential operator from[12,28]for a givenn∈N

on Eq.(43),the linear combination on the right is annuled.

For example, we show the calculation of the coefficients in the case of the basis functionEFup2(x,ω).Fig.7 shows the disposition of the basis functionsEFup2(x,ω).Such an disposition of the basis functions accurately develops the exponential monomials up to and including the second degree, as well as the exponential polynomials formed by their combination.By the action of the operator from Eq.(44)on Eq.(43)forn=2,the following recursion is obtained:

where

or,after reordering:

By introducing the substitution=λkin Eq.(46),we obtain a characteristic equation whose roots are

“Recompositioning”the roots(47)gives the general form of the coefficientsforn=2:

or

The coefficientsfrom Eq.(49)are calculated from the following system of equations:

Fork=0 andΔx=1/4,we obtain

In general, the exponential monomiale2mωx,m= 0,1,...,n,n∈Non a segment of length 2－ncan be accurately represented by the linear combination of the(n+2)·2nbasis functionsEFupn(x,ω)offset from each other by 2－nin the form:

where the coefficientsA(m)n(calculated from Eq.(49)forx=0)are of the following form:

Figure 7: Composition of the EFup2(x,ω) functions in a linear combination to obtain monomials φ(x)=eω·x·2n

4 Practical Use of the Exponential ABF EFupn(x,ω)

For practical use we created the efupnM module to calculate the values of the functionsEFupn(x,ω)and their derivatives at arbitrary points.The use of the software modules comes down to simply describing a function in a similar way to that of,for example,the trigonometric function sine:sin(ωx+φ)→sine(omega,xpoint,fi).Fig.8 shows a graphical interpretation of the variables that need to be specified when using the efupnM module.

dummy=efupnM(NFUP,OMEGA,VERTEX,DELTAX,XPOINT,KOD,NMAX),where:

NFUP=n-the order of the functionEFupn(x,ω);

OMEGA-frequency or tension parameter;

VERTEX-x-local coordinate system coordinates(located in the center of the support);

DELTAX-the real length of the characteristic segment;

XPOINT - the real x-coordinate of the arbitrary point at which the value of the functionEFupn(x,ω)is sought;

KOD-the order of derivation of the function;

NMAX-accuracy parameter(depends on computer characteristics).

Figure 8:Using the efupnM software module to calculate the values of the basis functions Fupn(x)and EFupn(x,ω)

4.1 Determination of the Best Frequency in the Function Approximation

The basis functions of the exponential type,such as trigonometric functions,exponential splines,or ABFs of the exponential type,contain the parameterωthat provides them additional approximation properties.However,their application in numerical analysis is limited by the fact that the value of the parameterωis,in most cases,unknown,and there is no universal criterion for choosing its value.

In this paper, the value of the parameterωis determined using the least squares method by adopting the value of the parameter that gives the smallest deviation between a given function and its approximation at each characteristic segment of lengthΔx.This method proved to be simple and efficient,and is shown in the example of the exponential function below.

Let there be given a function at the sectionAB=[0,1]in the form:

Using the two characteristic segments of the lengthΔx= 0.5 and the formation of the basis functions according to Fig.7, the corresponding approximations are determined using the basis functionsFup2(x)andEFup2(x,ω).

As previously shown in Section 3.6, the linear combination of the basis functionsEFupn(x,ω)identically approximates the exponential monomials(as well as their linear combination),i.e.,the given function (52), for any number of basis functions or characteristic segmentsΔxin the regionAB, as shown in Fig.9.On the other hand,the approximation of the function(52)using the basis functionsFup2(x)shows a significant deviation from the given function on a small number of segments,as shown in Fig.9.

Figure 9: Comparison of the given function (52) and approximations obtained using algebraic and exponential(frequency ω=10)ABFs for n=2

Thus,the criterion for choosing the parameter’s value is in terms of least squares:

wheref (x)is a given function,(x)is an approximation of a given function,andnsis the number of characteristic segmentsΔxin the domainAB.

Fig.10 shows the values of the least squares sum (53) for the approximations obtained by the values of the parameterωin the interval[0,60]with a stepΔω=0.1.

Since the frequency of the given function(52)is known and isω*=10,it is to be expected that,for the given value ofω*,the least squares sum(53)for approximation by the exponential basis functionsEFup2(x,ω*)will be equal to zero;however,according to Section 3.6,for the values of the parameterωi=2m·ω*,m=0,1,2 also,which is confirmed in Fig.10.

This confirms that the least squares method is a reliable,simple,and optimal choice of criteria for the determination of the value of the parameterω.

Figure 10: Values of the least squares sum of the approximation obtained by the basis functions EFup2(x,ωi)for the different values of the parameter ω on the interval I =[0,60]

4.2 Example 1:Approximation of a High-Degree Polynomial

An algebraic polynomial of degree 12 is approximated

Unlike the previous example where the value of the parameterωfollowed from the function itself,here we have a“problem”of choosing the value of the parameterω.The procedure for determining the value of the parameterωis reduced to the simultaneous direct solution of the linear system of Eq.(43)using the point collocation method for different values of the parameterω.Of all the numerical solutions thus obtained,the one that gives the minimum of the least squares function(53)for a given numbernsof characteristic segmentsΔxis adopted.

Fig.11 shows the values of the least squares sum of the approximation obtained by theEFup2(x,ω)basis functions on two characteristic segments for the values of the parameterωon the intervalI=[－30,10]with the stepΔω=0.1.

Figure 11: Values of the least squares sum of the approximation obtained by the basis functions EFup2(x,ωi)for the values of the parameter ω on the interval I =[－30,10]

The minimum value of the least squares sum for two characteristic segments was obtained for the approximation using the exponential basis functionsEFup2(x,ω), when the value of the parameterω=－16.5,and isLs=0.113453941003,while the value of the least squares sum of the approximation obtained by the algebraic basis functionsFup2(x),i.e.,when the value of the parameterω= 0.0,for the same number of sections is 2.62204156514.

Fig.12 shows a comparison of the given function(54)with the approximations obtained by the algebraic basis functionsFup2(x)and the exponential basis functionsEFup2(x,ω)for four different segment lengthsΔx=1/ns,wherens=2,4,8,16.

In Figs.12a and 12b,it can be seen that for a small number of characteristic segments,the approximation by the exponential basis functionsEFup2(x,ω)gives a significantly better approximation to the given function(54)than the approximation obtained by the algebraic functionsFup2(x),while as the number of segments(ns)increases,this difference in approximations decreases,as shown in Figs.12c and 12d.

In order to draw a conclusion regarding the character of the convergence of the mentioned numerical approximations to a given function,it is necessary to perform a calculation by increasing the number of segments to a certain desired accuracy of the results.From Fig.12,it can be seen that the best approximation is achieved using the exponential basis functions with different parameterωvalues depending on the number of segments in the area (ns).Fig.13 shows the values of the parameterωobtained by the least squares method in relation to the number of characteristic segments in the area.It can be seen that the value of the parameterωis sensitive to discretization of the domain only for a small number of sections up tons= 16, while when the number of sections is greater than 16,the parameterωhas a constant value of－19.0.Therefore,the convergence diagram of the numerical solution for the exponential basis functionsEFup2(x,ω)is obtained using the valuesω=－19.0.

Figure 12: Comparison of the approximations with a given function (54) using the algebraic and exponential ABFs for:(a)ns=2,(b)ns=4,(c)ns=8,and(d)ns=16

Figure 13: Dependence of the parameter’s ω value on the number of characteristic segments in the approximation of the given function(54)using the EFup2(x,ω)basis functions

The diagrams in Fig.14 show,on a logarithmic scale,the relationship between the error expressed over the L2-norm and the segment lengthΔxfor the approximations obtained by the basis functionsFup2(x)andEFup2(x,ω).It can be observed that the approximation obtained by the exponential ABFs achieves greater accuracy compared to the approximation obtained by the algebraic ABFs.Both diagrams show that the expected convergence rate is achieved, which, for the problem of the approximation of a given function,isp=n+1.

Figure 14: Convergence diagrams of the accuracy of the numerical approximations obtained by the Fup2(x)and EFup2(x,ω)basis functions

4.3 Example 2:Approximation of a Sudden Jump Function

The following function on the interval[0,1]is analyzed:

Fig.15 shows a comparison of the given function(55)with the approximations obtained by the algebraic basis functionsFup2(x)and the exponential basis functionsEFup2(x,ω)for the segments of the lengthsΔx=1/nswherens=2,4,8,16.It can be observed that exponential ABFs better describe the given function near the jump, while in the parts of the domain where the given function has a constant value,the approximation obtained byEFup2(x,ω)shows higher oscillations compared to the approximation obtained byFup2(x)function.Fig.15 also shows that,for this example of the function with a sudden jump,the exponential basis functionsEFup2(x,ω)achieve a better approximation than the algebraicFup2(x)basis functions for a smaller number of segments in the domain,while for a larger number of segments, the accuracy of the approximation equates that obtained using the functionsFup2(x).

Figure 15: Comparison of the approximations with a given function (55) using the algebraic and exponential basis functions for:(a)ns=2,(b)ns=4,(c)ns=8,and(d)ns=16

4.4 Example 3:Solving the Differential Equation of Conduction

Let a given differential equation of conduction with corresponding boundary conditions be

with known exact solution of the form

Fig.16 shows the dependence of the solution of Eq.(56)on the parameterω,and how,for high values ofω,the solution function shifts to the right boundary.

Figure 16:Dependence of the exact solution of the conduction problem(56)on the frequency ω

Fig.17 compares the exact solution (57) of the conduction problem (56) with the solutions obtained using the basis functionsFup2(x)andEFup2(x,ω)with the point collocation method for the characteristic segmentΔx=0.25,i.e.,with a total of seven basis functions on the domain.

Approximation using the basis functions of the algebraic type is limited by the Peclet numberPe=Δx·ω ＜2 because, at high values of thePe, there is a numerical error and oscillation in the approximate solution.For the atomic basis functions of the exponential type there is no such a restriction.

In Fig.17a forω=2 andPe=0.5 ?2,the solutions coincide with the exact solution.In Fig.17b forω=5 andPe=1.25＜2,the solution obtained withEFup2(x,ω)fully corresponds to the correct solution,while the solution obtained withFup2(x)shows a deviation from the exact one,but still does not oscillate.In Fig.17c forω= 10 andPe= 2.5＞2,the solution obtained withEFup2(x,ω)still fully corresponds to the exact solution, while the solution obtained withFup2(x)begins to oscillate significantly around the exact solutions.In Fig.17d, forω= 10000 andPe= 2500 ? 2, the solution obtained withEFup2(x,ω)corresponds to the exact solution, while the solution obtained withFup2(x)satisfies the boundary conditions and the differential equation at the collocation points but is completely unusable.

Figure 17:Comparison of the numerical solutions of Eq.(56)obtained by the Fup2(x)and EFup2(x,ω)basis functions with the exact solution

5 Conclusion

The current knowledge regarding algebraic atomic basis functionsFupn(x)is synthesized in the paper.Their basic properties are described, and the expressions for the necessary mathematical operations are presented in a simpler, more understandable, and user-friendly way.Very little was known about the ABFs of the exponential type, and they were developed only at the basic level in[12,27,28].In this paper,the basic properties of the functionsEFupn(x,ω)are shown using the same approach as that for the ABFs of the algebraic type.The expressions for calculating the values of the functions and the desired number of derivatives at arbitrary points of the basis function support and,most importantly,the rules(elements)for their practical use are derived.The EFupnM software module for the practical application of these functions is also shown.

In the examples of the approximations of given functions, namely, a high-degree algebraic polynomial representing an asymmetric function and functions with a sudden jump,the exponential basis functionsEFupn(x,ω)show better properties compared to the basis functions of the algebraic typeFupn(x).This is especially evident in approximations that use a smaller number of basis functions.As the number of basis functions in the region increases, the approximation propertiesEFupn(x,ω)of the functions are equated with the properties of the functionsFupn(x).The advantage of theEFupn(x,ω)function comes to expression especially when solving a differential equation of conduction that has an exact solution in the form of an exponential-type function.The exponential basis functions give a better approximate solution of high accuracy with the absence of the oscillations of the numerical solution.

Algebraic atomic basis functions have been used for many years to solve various numerical problems, and their advantage over other basis functions has become unquestionable.The ABFs of the exponential type show even better approximation properties,as demonstrated in this paper.The only question that still remains open is the choice of the value of the tension parameterω.As with exponential splines, this complex issue requires further research both in one-dimensional problems and in the higher dimensions of space.In this paper,for the parameter selection criterion,we used the least squares sum,which proved to be simple and reliable.However,a disadvantage was the additional CPU time required to simultaneously solve the system of equations for the purpose of obtaining the approximations for different values of the parameterω.This could be reduced by reducing the search interval of the parameter values according to the properties of a given numerical problem,i.e.,whether it is an approximation of a given function or solving a differential equation.

Our further research should include an improvement of the procedure for finding the optimal value of the tension parameter.The natural sequence of development and application of the ABF of the exponential type leads to 2D and 3D numerical analysis.The advantage of ABF of the exponential type can be suitable for the application of adaptive procedures in the problems of computational mechanics.

Funding Statement:This research is partially supported through Project KK.01.1.1.02.0027,a project co-financed by the Croatian Government and the European Union through the European Regional Development Fund-the Competitiveness and Cohesion Operational Programme.

Conflicts of Interest:The authors declare that they have no conflicts of interest to report regarding the present study.