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The recursion formulas for this equation will be the following:Mag0,m1 / 2 m2/ 2 m3/ 21 2 3m , m , m = ∑ ∑ ∑ ai,j,l x x x x Ge(i + j +1a 0 = 1a,0,0i,j,l− a− ai,j−1,li,j,l−12= −a3i−1,j,li= 0 j=0l = 02i+2 j+2l1( m2− 2 j + 2)( m2− 2 j + 1)b(2i+ 2 j + 2l)(2i+ 2 j + 2l− 1)( m3− 2l+ 2)( m3− 2l+ 1)b(2i+ 2 j + 2l)(2i+ 2 j + 2l− 1)m234−2i( m1− 2i+ 2)( m1− 2i+ 1)b(2i+ 2i+ 2l)(2i+ 2 j + 2l− 1)−m32−2j−m4−2ll,wx21)Mag1,m1 / 2 m2/ 2 m3/ 21 2 3m , m , m = ∑ ∑ ∑ ai,j,l x x x x Ge(i + j + l + 1, wx1a 0 = 1a,0,0i,j,l− a− a2= −ai,j−1,li,j,l−13i−1,j,li = 0 j=0l = 02i+2 j+2l+ 11( m2− 2 j + 2)( m2− 2 j + 1)b(2i+ 2 j + 2l)(2i+ 2 j + 2l+ 1)( m3− 2l+ 2)( m3− 2l+ 1)b(2i+ 2 j + 2l)(2i+ 2 j + 2l+ 1)m234−2im3( m1− 2i+ 2)( m1− 2i+ 1)b(2i+ 2i+ 2l)(2i+ 2 j + 2l+ 1)−−2j2−m4−2l21)Actually, it was possible to write the solutions alsofor the equations of the larger dimensionality and othertypes, but the author is confident, that the idea of theirformation is already clear.equation of Laplace, which contains the arbitraryfunctions of the variables ( n − 1) :In (Полянин 2001) it is possible to find theexcellent general formula for the solution of thexxu( x ,..., x ) ( 1) [ f ( x ,..., x ) g( x ,..., x )]∞2k2k+ 1k n k n k1 n= ∑ − ∆1 n−1 + ∆1 n−1(35),k=0 (2 k)! (2k+ 1)!f ( x ,..., x −) and g( x1 ,..., xn− 1)- thewhere 1 n 1arbitrary infinitely differentiated functions .If he formulas (14), (22), (25), (29) and (33) areconsidered, then at once it can be seen structuralsimilarity to the formula (35).Annual Proceedings of Vidzeme University College “ICTE in Regional Development”, 2006148
CONCLUSIONSThe basic content of this article - it is the method ofthe dual substitution, which provides the complete setof functions for solving the differential equations in thepartial derivatives of mathematical physics.All the other formulas and solutions given in thearticle, no matter how difficult it was to derive them,are the result of applying the method of dualsubstitution.The method of dual substitution is simple and itsoperations correspond to the usual algebra.Especially this method can be effective forexplaining the content of the solutions of partialdifferential equations when teaching in the educationalinstitutions.The known contemporary analytical methods ofsolving the partial differential equations containcomplex integral expressions and differentcombinations of special functions in their results.The method of dual substitution makes it possible toobtain the simpler functions for solving the equation.The author considers that the best usage of thesesimpler functions is following:• For the concrete partial differential equations,where there are ten or more simpler functionsaccording to the method of dual substitution;• Entire area being considered is divided into thecells (region) of such value that the obtained set ofthe simpler functions would give the necessaryaccuracy in each separate cell;• The system of equations or the objective function,which defines extreme and initial conditions, isconstructed for the optimum search forunrestricted variables in all the cells.the approximation of any polynomial curve – moreoften it is more easily and precise to approximate byten polynomials than one.The author hopes that this method will be usedwidely in the solution of different mathematical andpractical equations.REFERENCESЛюк, Ю. 1980. Специальные математическиефункции и их аппроксимации. Москва, Мир,Пер. с анг. Г.П.Бабенко Под редакциейК.И.Бабенко.Виноградов, И.М. 1977. Математическаяэнциклопедия Т1. Москва, СоветскаяЭнциклопедия.Корн, Г. and Т. Корн. 1977. Справочник поматематике Издание четвёртое, Москва, Наука.Полянин, А.Д. 2001. Справочник по линейнымуравнениям математической физики. Москва,Физико-Математическая литература.Мак-Кракен, Д. and У. Дорн. 1977. Численныеметоды и программирование на фортране, Москва,Мир, Пер. с анг. Б.Н.Казака. Под редакциейБ.М.Наймарка.BIOGRAPHYValerijs Stepuchevs working as Leading Engineer inElectronic Department of Latvian Intelligent Systems,Ltd. In 1980, he graduated the Institute of CivilAviation in Riga and holds Engineer diploma inElectronics. His main interests are electronics andmathematics especially methods for solution of partialequation systems.This method of the solution must give better resultsthan only the grid method or analytical method of thesolution. This solution can be named as the complexgrid. The complex grid comparing to the grid methoduses more information about the differential equation.Using results, it is possible to make differentiation,integration, and other different mathematicaloperations, as this can be done by the simpler functionsof the solution.But difference from the typical analytical methodcan be demonstrated based on the simple example ofAnnual Proceedings of Vidzeme University College “ICTE in Regional Development”, 2006149
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ISBN 9984-633-03-9Annual Proceeding
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“Development of Creative Human -
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TABLE OF CONTENTSINTELLIGENT SYSTEM
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INTELLIGENT SYSTEM FOR LEARNERS’
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LEARNER 1GROUP OF HUMAN AGENTSLEARN
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QuantityQuantityFigure 6. Distribut
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LEARNERStructure of theconcept mapL
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WEB-BASED INTELLIGENT TUTORING SYST
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materials to be presented and which
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INFORMATION TECHNOLOGIES AND E-LEAR
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correspondence with the course aim
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projects and through IT. Hence, it
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APPLICATION OF MODELING METHODS IN
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can support configuration managemen
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The EKD is one of the Enterprise mo
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CHANGES TO TRAINING AND PERSPECTIVE
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or an end, yet none of these attitu
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make decisions. It cannot be volunt
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logs), data and video conferencing
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Ability to follow user’s multi-ta
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CONCLUSIONSEDUSA method gives us a
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in successful SD. Given this situat
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SPATIAL INFORMATIONFor the visualis
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MOBILE TECHNOLOGIES USE IN SERVICES
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learning environment (Learning Mana
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ago only some curricula on Logistic
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The Web-based version can be access
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Web-portal, which incorporates diff
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DO INTELLIGENT OBJECTS AUTOMATICALL
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Table 1. Examples for introducing R
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workable influencing of the process
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are handed over to the objects and
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• Basic processes, such as wareho
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THE ECR E-COACH: A VIRTUAL COACHING
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participating in the workshops and
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• Assessment modules enable indiv
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with pictures and illustrated graph
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ECR Question Banknumber category su
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educational programme that follows
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DEVELOPMENT OF WEB BASED GRAVITY MO
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These results of a model require a
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CONCLUSIONSThe main goal of work ha
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dimension and included within any o
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• Resources sharing by providing
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Pursuant to the guidelines of elect
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tariffs of regulated services have
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INFORMATION TECHNOLOGY FOR MOTIVATI
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difficult to predict when and for w
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Listeners' workon the WebListenersS
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PERSPECTIVES OF WEB PAGE AND E-MAIL
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INCREASE IN THE NUMBER OF INTERNETU
Page 103 and 104: tourism accommodations (home pages
Page 105 and 106: interactive relationships with clie
Page 107 and 108: • The data obtained by the resear
Page 109 and 110: Central Statistical Bureau of Latvi
Page 111 and 112: departures for 1995 are taken from
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Page 115 and 116: 140120maximumworldminimum1008060402
Page 117 and 118: would be a promising extension. Cur
Page 119 and 120: AN OVERVIEW OF THE AGENT − BASED
Page 121 and 122: Suitability for social system simul
Page 123 and 124: 6. MASONDescription:MASON is a fast
Page 125 and 126: Suitability for social system simul
Page 127 and 128: could be bad particularly when over
Page 129 and 130: (for 10 repeat &| CCar[]->runfor);P
Page 131 and 132: • Streaming audio• Collaboratio
Page 133 and 134: NECESSITY OF NEW LAYERED APPROACH T
Page 135 and 136: Up to now, there has only been limi
Page 137 and 138: aaaaa6= −aa2,1 = − a0,3226= −
Page 139 and 140: ∂ u∂x∂ u∂y2 2+ b = 02 2wher
Page 141 and 142: a6,3= −2030a4,5−130a4,3- - - -
Page 143 and 144: 0,10,20,30,4( )Mag x y y Ge wx2, =
Page 145 and 146: Example 1. To understand better the
Page 147 and 148: Therefore, further the following co
Page 149 and 150: SOLUTION OF THE THREE-DIMENSIONALEQ
Page 151 and 152: Mag1, m , m , m1 2 3= mm1 m2m32 2 2
Page 153: MagMag0, m , m , m1 2 31, m , m , m
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