Optimization of computational methods in functional spaces is one of the main problems of computational mathematics. In the present work algebraic and functional statements for the problem of difference formulas. For optimization of difference formulas, i.e., for construction of optimal difference formulas in functional spaces the important role plays the extremal function of the given difference formula. In this work, in the Sobolev space, the extremal function of the difference formula is explicitly found and the norm of the error functional of the difference formula is calculated. Furthermore, existence and uniqueness of the optimal difference formula are proved.<\/p>\n
Shadimetov Kh.M. , Mirzakabilov R.N.<\/p>\n
Sun Jun 26 12:14:38 2022<\/p>\n
Contents<\/strong><\/p>\n 1\u00a0 Introduction…………………………………………………………………………… <\/strong>??<\/strong><\/p>\n <\/p>\n 2\u00a0 The problem on construction of difference formulas………………….. <\/strong>??<\/strong><\/p>\n <\/p>\n 3\u00a0 The functional statement of the problem. The extremal function of the difference formulas……………………………………………………………………………… <\/strong>??<\/strong><\/p>\n <\/p>\n 4\u00a0 Square of the norm of the error functional……………………………….. <\/strong>??<\/strong><\/p>\n Suppose it is required to find a solution of a differential equation<\/p>\n (1.1)<\/p>\n with the initial condition \u00a0in the interval . We divide this interval by \u00a0pieces with length \u00a0and we will search approximate values \u00a0of the solution \u00a0at nodes , \u00a0The classical example of such type of methods is the Euler method which consists of the following: an approximate value \u00a0of the solution at the point \u00a0is obtained from the approximate value \u00a0of the solution at the point \u00a0by the formula<\/p>\n (1.2)<\/p>\n where . Therefore \u00a0is a linear combination of values of the function and its derivatives at the point . We consider only discrete methods, i.e. methods which define a solution for discrete values of independent variable. A characteristic feature of discrete methods for solution of equation (1.1) is the process of solution consists of repetition of algorithm for obtaining of searching solution \u00a0by using known, previously calculated values \u00a0and<\/p>\n The problem of construction of a numerical algorithm can be separated into three groups:<\/p>\n The problem consists of choosing such a algorithm for calculating of \u00a0that the difference \u00a0will be minimal. Here \u00a0is an approximate value of the exact solution \u00a0and \u00a0are known. Asymptotic estimation of the algorithm\u2019s accuracy for<\/p>\n <\/p>\n is defined by two numbers \u00a0and , if<\/p>\n <\/p>\n It is supposed that the solution is sufficiently smooth, and parameter \u00a0characterize the algebraic degree of exactness. An example of a such algorithm is Euler methods (1.2) for which \u00a0and .<\/p>\n The main problem is choosing such an algorithm that does not lead to the accumulation of errors, i.e.<\/p>\n <\/p>\n as<\/p>\n <\/p>\n Some convergent algorithms may be accompanied by numerical instability. Therefore we consider conditions which are guarantee the stability of algorithms.<\/p>\n <\/p>\n <\/p>\n The problem on construction of difference formulas exact for polynomials of degree , we also consider first in algebraic statement. Further as difference formula we mean the following approximate equality<\/p>\n (2.1)<\/p>\n Here<\/p>\n <\/p>\n and \u00a0are coefficients of the difference formula.<\/p>\n The difference formula of order \u00a0is called an implicit<\/em> if \u00a0and an explicit<\/em> if . The following difference we call the error<\/em> of the formula<\/p>\n (2.2)<\/p>\n In the class of differentiable functions defined on the interval , the equality (2.2) defines additive and homogeneous functional \u00a0which we call the error functional<\/em> of the difference formula (2.1). We say that the difference formula is exact for a function \u00a0if the difference (2.2) is equal to zero. By another words, functions for which the difference formula is exact form the kernel of the error functional . The problem on construction of difference formulas in algebraic statement for the interval \u00a0is as follows:<\/p>\n Find coefficients \u00a0and \u00a0of the difference formula such that the formula becomes exact for all polynomials from the space \u00a0for possible large , where \u00a0is the space of polynomials of degree .<\/p>\n Therefore, the quality of the difference formula will be higher when the dimension of the space of polynomials \u00a0is larger<\/p>\n Substituting a polynomial<\/p>\n <\/p>\n for \u00a0in (2.2) we get<\/p>\n <\/p>\n Thus, our requirement \u00a0for , which is equivalent to the following system of conditions<\/p>\n (2.3)<\/p>\n is fulfilled in that case when the vectors<\/p>\n <\/p>\n are the solution of the system<\/p>\n <\/p>\n <\/p>\n <\/p>\n System (2.3) has a solution if .<\/p>\n Now we give known difference formulas constructed by algebraic way.<\/p>\n We denote \u00a0then the Adams-Bashforth formula has the form<\/p>\n <\/p>\n and the Adams-Moulton formula has the form<\/p>\n <\/p>\n There are other known formulas, for example, the Nystrom and the Mali-Simpson formulas and others.<\/p>\n <\/p>\n <\/p>\n Now we go to the functional statement of our problem. We consider functions \u00a0which are belong to the Sobolev space \u00a0\u00a0is a Hilbert space of real functions \u00a0which are differ for polynomials of degree \u00a0and square integrable with derivative of order \u00a0on the interval . The inner product in this space is defined as<\/p>\n (3.1)<\/p>\n <\/p>\n Since the space \u00a0embedded to the space \u00a0of continuous functions, the error functional of the difference formula is a linear functional<\/p>\n <\/p>\n <\/p>\n (3.2)<\/p>\n <\/p>\n The problem of construction of the difference formula<\/p>\n <\/p>\n in the functional statement is to find such a functional (3.2) which has minimal norm in the space . For finding the explicit form of the error functional \u00a0we use an extremal function of the given functional, i.e. we use the function \u00a0satisfying the equality<\/p>\n <\/p>\n It is known that in the work of I. Babuska et al [1] finding of the extremal function \u00a0was reduced to the differential equation of order . However, there was not given the solution of the differential equation. Our method of finding \u00a0is differ from the method of I.Babuska and allows to find the extremal function explicitly.<\/p>\n Problems for construction of cubature and difference formulas in functional statement were considered, for example, in the works [2]-[10],[13],[15].<\/p>\n The norm of a function in the space \u00a0is defined as<\/p>\n (3.3)<\/p>\n <\/p>\n Since the functional of the form<\/p>\n <\/p>\n is defined on the space \u00a0then we have<\/p>\n (3.4)<\/p>\n <\/p>\n Since \u00a0is the Hilbert space equipped with the inner product (3.1) and corresponding norm (3.3) then applying the Riesz theorem for any linear functional, in particular for the error functional, we find an explicit form of the norm of the functional using a function \u00a0which is the Riesz element. By the Riesz theorem for any \u00a0the following equalities hold<\/p>\n (3.5)<\/p>\n <\/p>\n <\/p>\n By virtue of (3.1) and (3.5) we obtain the following identity which is true for any function \u00a0of the space of infinitely differentiable finite functions<\/p>\n (3.6)<\/p>\n <\/p>\n By Integrating by parts \u00a0times the left hand side of the (3.6) we get<\/p>\n <\/p>\n <\/p>\n <\/p>\n <\/p>\n where \u00a0is the characteristic function of the interval .<\/p>\n Thus, in the space of generalized functions we have the following equation<\/p>\n (3.7)<\/p>\n The general solution of equation (3.7) is written in the form<\/p>\n (3.8)<\/p>\n <\/p>\n And in the formula (3.8) the sum<\/p>\n <\/p>\n is a polynomial of degree \u00a0with overdetermined coefficients \u00a0corresponding to the following term<\/p>\n <\/p>\n The function \u00a0is a fundamental solution of the equation<\/p>\n <\/p>\n and it has the form<\/p>\n <\/p>\n We consider equation (13) out of the interval . By virtue of (2.3) out of the interval \u00a0the expression \u00a0is a polynomial of degree , because this function is from \u00a0after \u00a0times differentiation it becomes zero. For fulfilling the condition \u00a0for \u00a0it should be<\/p>\n <\/p>\n where \u00a0is a polynomial of degree<\/p>\n So, for any difference formula of the form (2.1) in the space \u00a0its extremal function, i.e. its Riesz element is given by the formula<\/p>\n <\/p>\n where \u00a0is a polynomial of degree . Thus we have proved the following theorem.<\/p>\n <\/p>\n \u0422\u0435\u043e\u0440\u0435\u043c\u0430<\/strong> 1 <\/strong>\u00a0The extremal function of the differential formula (2.1) in the space <\/em>\u00a0is defined by the formula <\/em><\/p>\n (3.9)<\/p>\n <\/p>\n <\/p>\n <\/p>\n We know that by Riesz theorem and definition of the extremal function it follows<\/p>\n <\/p>\n Hence we get the following theorem.<\/p>\n <\/p>\n \u0422\u0435\u043e\u0440\u0435\u043c\u0430<\/strong> 2 <\/strong>\u00a0Square of the norm of the error functional of the difference formula (2.1) is defined by the formula <\/em><\/p>\n <\/p>\n <\/p>\n <\/p>\n On the strength of Theorem 2, using the extremal function and orthogonality conditions (3.4) of the error functional \u00a0to polynomials of degree , after some calculations we get<\/p>\n <\/p>\n <\/p>\n where<\/p>\n <\/p>\n It is known that the stability of a difference formula in the sense of Dahlquits, as the strong stability, is only defined by coefficients<\/p>\n By this reason our search of the optimal formula is only connected by variation of coefficients<\/p>\n <\/p>\n The difference formula with the error functional \u00a0in the space \u00a0can be characterized by two methods. On the one hand it is defined by coefficients \u00a0with conditions (3.4), on the other hand by the extremal function .<\/p>\n Now we go to minimization of square of the norm for the error functional \u00a0of the difference formula for the given \u00a0by coefficients \u00a0with conditions (3.4).<\/p>\n We apply the indefinite multipliers method of Lagrange. For this we consider the function<\/p>\n <\/p>\n Partial derivatives of the function \u00a0by \u00a0and \u00a0equating to zero<\/p>\n <\/p>\n <\/p>\n <\/p>\n This gives the following system of equations<\/p>\n (4.1)<\/p>\n <\/p>\n (4.2)<\/p>\n where<\/p>\n <\/p>\n <\/p>\n <\/p>\n and<\/p>\n <\/p>\n The solution of the system (4.1), (4.2) which we denote as , \u00a0is a critical point for the function . From Lagrange method it follows that \u00a0are the searching values of the coefficients of the optimal difference formula. They give the conditional minimum to the norm \u00a0under the conditions (9).<\/p>\n Now we consider the formula (3.9). Taking into account arbitrariness of coefficients of the polynomial , we choose them such that the derivative \u00a0coincides with the polynomial \u00a0from (4.1). In such a choice for values of the derivative of the extremal function \u00a0at points \u00a0we get \u00a0which follows from the formula (4.1). Therefore, we have proved Babuska\u2019s theorem by algebraic way.<\/p>\n \u0422\u0435\u043e\u0440\u0435\u043c\u0430<\/strong> 3 <\/strong>\u00a0Difference formulas <\/em><\/p>\n <\/p>\n <\/p>\n <\/p>\n are optimal if and only if when the function , defined by the formula (3.9), can be selected such that<\/p>\n <\/p>\n and<\/p>\n <\/p>\n for implicit and explicit formulas, respectively.<\/p>\n <\/p>\n As a results for construction of implicit optimal difference formulas it is necessary to solve the following system of linear equations<\/p>\n where \u00a0are defined from the stability conditions of the difference formula in the sense of Dahlquits and<\/p>\n <\/p>\n \u041e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435<\/strong> 1 <\/strong>The difference formula (2.1) is stable in the sense of Dahlquits if all roots <\/em>\u00a0of the characteristic polynomial <\/em>\u00a0satisfy <\/em>\u00a0and roots, for which <\/em>, are simple [11], [12], [14]. <\/em><\/p>\n <\/p>\n For example,<\/p>\n <\/p>\n for<\/p>\n Similarly, for the explicit difference formula we have<\/p>\n where \u00a0are defined from stability of the difference formula and from the equality<\/p>\n Here we suppose that the system (4.1), (4.2) is solvable. Its solvability follows from the general theory of Lagrange multipliers. However one can verify solvability directly by algebraic way.<\/p>\n From the theory of conditional extremum it is known the sufficient condition in which the solution \u00a0gives local minimum to the function \u00a0on the monifold (3.4). It consists of positive definiteness of the quadratic form<\/p>\n (4.3)<\/p>\n on the set of vectors \u00a0under the condition<\/p>\n (4.4)<\/p>\n Here the matrix \u00a0has the form<\/p>\n <\/p>\n We show that in the considered case the quadratic form \u00a0is strongly positive.<\/p>\n <\/p>\n \u0422\u0435\u043e\u0440\u0435\u043c\u0430 4 <\/strong>\u00a0For any nonzero vector <\/em><\/p>\n <\/p>\n lying in the subspace , the function \u00a0is strictly positive.<\/p>\n <\/p>\n <\/p>\n Proof.<\/em> From the definition of the Lagrange function \u00a0and from equality (4.3) we obtain<\/p>\n <\/p>\n Consider the linear functional of the form<\/p>\n <\/p>\n If we take into account the condition (4.4) then this functional belongs to the space . Thus, it has the extremal function \u00a0which is a solution of the equation<\/p>\n <\/p>\n Clearly, that as a \u00a0one can get the following linear combination of the shifts of the fundamental solution<\/p>\n <\/p>\n Here \u00a0is a solution of the equation<\/p>\n <\/p>\n Square of the norm of the function \u00a0in the space \u00a0coincides with the form :<\/p>\n Hence clear that for non zero \u00a0the function \u00a0is strictly positive. It is known that for \u00a0the system (4.4) always has a solution, i.e. the matrix \u00a0has the right inverse, then the system (4.1), (4.2) has a unique solution.<\/p>\n <\/p>\n <\/p>\n \u0422\u0435\u043e\u0440\u0435\u043c\u0430<\/strong> 5 <\/strong>If the matrix <\/em>\u00a0has the right inverse then the matrix <\/em>\u00a0of the system (4.1), (4.2) is nonsingular. <\/em><\/p>\n <\/p>\n <\/p>\n Proof.<\/em> We denote \u00a0the matrix of the quadratic form (4.3). We write homogeneous system corresponding to the system (4.1), (4.2) in the following form:<\/p>\n (4.5)<\/p>\n We show that a unique solution of (4.5) is identical zero.<\/p>\n Let \u00a0be a solution of (4.5). We consider the following functional for the vector :<\/p>\n <\/p>\n It is clear that this functional belongs to the space . As a extremal function for \u00a0we take the following<\/p>\n <\/p>\n This is possible, because, the function \u00a0belongs to the space \u00a0and is the solution of the equation<\/p>\n <\/p>\n The first \u00a0equations of the system (4.5) mean that the function \u00a0vanishes at all nodes of the difference formula, i.e. for \u00a0\u00a0Then with respect to the norm of the functional \u00a0in \u00a0we have<\/p>\n <\/p>\n that is possible only for \u00a0Taking this into account, from first \u00a0equations of the system (4.5) we obtain<\/p>\n (4.6)<\/p>\n By assertion of the theorem the matrix \u00a0has the right inverse, then \u00a0has the left inverse matrix. Hence and from (4.6) it follows that that the solution \u00a0also is zero. Theorem is proved.<\/p>\n <\/p>\n Thus, the system (4.1), (4.2) has unique solution.<\/p>\n <\/p>\n <\/p>\n References<\/strong><\/p>\n <\/p>\n <\/p>\n [1]\u00a0 Babuska I., Vitasek E., Prager M. \u00a0Numerical processes for solution of differential equations<\/em>: – Mir, Moscow, 1969, 369 p.<\/p>\n <\/p>\n <\/p>\n [2]\u00a0 Babuska I., Sobolev S. \u00a0Optimization of numerical methods<\/em>.: – Apl. Mat., 10, 9-170, 1965<\/p>\n <\/p>\n <\/p>\n [3] \u0421\u043e\u0431\u043e\u043b\u0435\u0432 \u0421. \u00a0\u0412\u0432\u0435\u0434\u0435\u043d\u0438\u0435 \u0430 \u0442\u0435\u043e\u0440\u0438\u044e \u043a\u0443\u0431\u0430\u0442\u0443\u0440\u043d\u044b\u0445 \u0444\u043e\u0440\u043c\u0443\u043b.<\/em>: – \u041c., 1974, 808 \u0441<\/p>\n <\/p>\n <\/p>\n [4]\u00a0 Sobolev S., Vaskevich L. \u00a0Cubature fromulas.<\/em>: – Novosibirsk, 1996, 484 p.<\/p>\n <\/p>\n <\/p>\n [5]\u00a0 Shadimetov Kh. Weighted optimal cubature formulas in the Sobolev periodic space: – \u00a0Siberian journal of computational mathematics<\/em>. Novosibirsk, 1999, v.2, pp. 185-195.<\/p>\n <\/p>\n <\/p>\n [6]\u00a0 Shadimetov Kh. On optimal lattice quadrature and cubature formulas.:- \u00a0Dokl. Russian Academy of Sciences.<\/em> Moscow, 2001, v.376, no. 5, pp. 597-599.<\/p>\n <\/p>\n <\/p>\n [7]\u00a0 Shadimetov Kh. Functioanal statement of the problem of optimal difference formulas: – Uzbek mathematical Journal<\/em>, 2015, no. 4, pp.179-183.<\/p>\n <\/p>\n <\/p>\n [8]\u00a0 Shadimetov Kh., Mirzakabilov R.N. The problem on construction of difference formulas: – Problems of Computational and Applied Mathematics.<\/em>– 2018, no. 5 (17). pp. 95-101.<\/p>\n <\/p>\n <\/p>\n [9]\u00a0 Akhmedov D.M., Hayotov A.R., Shadimetov Kh.M. Optimal quadrature formulas with derivatives for Cauchy type singular integrals:- \u00a0Applied Mathematics and Computation.<\/em> 2018, N317. P.150-159.<\/p>\n <\/p>\n <\/p>\n [10]\u00a0 Boltaev N.D., Hayotov A.R., Shadimetov Kh.M. Construction of Optimal Quadrature Formula for Numerical Calculation of Fourier Coefficients in Sobolev space : – \u00a0American Journal of Numerical Analysis<\/em>, 2016, v. 4, no. 1, pp. 1-7.<\/p>\n <\/p>\n <\/p>\n [11]\u00a0 Dahlquits G. Convergence and stability in the numerical integration of ordinary differential equations:- Math. Scand.<\/em>, 1956, v. 4, pp. 33-52.<\/p>\n <\/p>\n <\/p>\n [12]\u00a0 Dahlquits G. Stability and error bounds in the numerical integration of ordinary differential equations:- \u00a0Trans. Roy. Inst. Technol. Stockholm<\/em>, 1959.<\/p>\n <\/p>\n <\/p>\n [13]\u00a0 Shadimetov Kh.M., Hayotov A.R. Optimal quadrature formulas in the sense of Sard in \u00a0space: – \u00a0Calcolo<\/em> (2014) 51:211-243.<\/p>\n <\/p>\n <\/p>\n [14]\u00a0 Henrici P., \u00a0Discrete variable methods in ordinary differential equations<\/em>:- J.Wihey Sons, Ins., New York, London.<\/p>\n <\/p>\n <\/p>\n [15]\u00a0 Shadimetov Kh.M., Hayotov A.R., Akhmedov D.M. Optimal quadrature formulas for Cauchy type singular integrals in Sobolev space:- \u00a0Applied Mathematics and Computation<\/em>. 2015, v. 263, pp. 302-314.<\/p>\n <\/p>","protected":false},"excerpt":{"rendered":" Optimization of computational methods in functional spaces is one of the main problems of computational mathematics. In the present work algebraic and functional statements for the problem of difference formulas. For optimization of difference formulas, i.e., for construction of optimal difference formulas in functional spaces the important role plays the extremal function of the given […]<\/p>\n","protected":false},"author":11,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[1],"tags":[],"sdg_goal":[],"class_list":["post-39456","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"acf":[],"views":772,"_links":{"self":[{"href":"https:\/\/jdpu.uz\/en\/wp-json\/wp\/v2\/posts\/39456","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/jdpu.uz\/en\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/jdpu.uz\/en\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/jdpu.uz\/en\/wp-json\/wp\/v2\/users\/11"}],"replies":[{"embeddable":true,"href":"https:\/\/jdpu.uz\/en\/wp-json\/wp\/v2\/comments?post=39456"}],"version-history":[{"count":1,"href":"https:\/\/jdpu.uz\/en\/wp-json\/wp\/v2\/posts\/39456\/revisions"}],"predecessor-version":[{"id":39457,"href":"https:\/\/jdpu.uz\/en\/wp-json\/wp\/v2\/posts\/39456\/revisions\/39457"}],"wp:attachment":[{"href":"https:\/\/jdpu.uz\/en\/wp-json\/wp\/v2\/media?parent=39456"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/jdpu.uz\/en\/wp-json\/wp\/v2\/categories?post=39456"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/jdpu.uz\/en\/wp-json\/wp\/v2\/tags?post=39456"},{"taxonomy":"sdg_goal","embeddable":true,"href":"https:\/\/jdpu.uz\/en\/wp-json\/wp\/v2\/sdg_goal?post=39456"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}1\u00a0 Introduction<\/h2>\n
\n
\n
\n
2\u00a0 The problem on construction of difference formulas<\/h2>\n
3\u00a0 The functional statement of the problem. The extremal function of the difference formulas<\/h2>\n
4\u00a0 Square of the norm of the error functional<\/h2>\n