online differentiation
This application enables finding derivatives for complex functions that depend on more than one variable. Let us consider a simple case. For instance, our function has the following form: f(x, y, z) = x^2 + y^2 + z^2. As the first step, an user should choose which variable among all three (x, y and z) is its differentiation variable i. e. over which variable the f function will be differentiated. Let us assumed that it is x variable. The rest of variables (y and z) are treated as parameters and kept constant during differentiation. After successful calculations, the program should display an obtained result in two forms. The first of them is longer and allows to conclude a way in which the final result is obtained. On the other hand, the second formula is shorter and simplified (if a simplification is available). When the final result is known one can plot the function under consideration (i. e. f function), its analytic and numerical derivative. The latter one, is obtained numerically from the base function f. To make these plots one need to set numerical values of our parameters: y and z and, of course, set the numerical range in which the variable x varies. Finally, one can compare both analytic and numerical derivative to find out if the obtained solution is correct or not.
Advanced Level
Scientific program (2d FEM) - online 2d mesh generator
To create two-dimensional mesh one should define a figure's outer contour. In the case of regular figures i. e. having the circular or the rectangular shape, to define them one need to set a radius and a number of corners. In the case of non-regular ones, one should mark each corner within the domain represented by the white area. When the figure's edge is defined the program add one more node in the centre of the figure. In that way, the initial mesh is done. To generate the final mesh, one should define the parameters of the algorithm that divides existing elements into smaller ones to reach the assumed mesh size. To understand the meaning of used parameters one should read carefully the article.
Scientific program (3d FEM) - online 3d mesh generator
To create three-dimensional mesh one should define a figure's outer contour. In the case of regular figures i. e. having the spherical or the cubic shape, to define them one need to set a radius and a number of corners. When a figure has a non-regular shape it is divided into layers. The number of layers is set by an user. For each layer, the user must mark each edge corner within the domain represented by the white area. When the figure's edge is defined the program add more nodes on the symmetry line of the figure (i. e. in the middle of each layer and between layers). The initial mesh is done. To generate the final mesh, one should define the parameters of the algorithm that divides existing elements into smaller ones to reach the assumed mesh size. To understand the meaning of used parameters one should read carefully the article.
Computer Simulations
Scientific program (Optfinder - Genetic Algorithms) - online genetic algorithms simulator
To find minimum or maximum of a function using this online application, one should provide the mathematical formula of this function. Then one should define a kind of optimisation i. e. whether function minimum or maximum is sought.
The used method is a genetic algorithms simulation so is based on an evolution of a population of genotypes. That is why, one must define the size of the population. The initial genotypes are represented by coding sequences (i. e. sequences of zeros and ones) and are created randomly, however, one must set the range from which values represented by genotypes are taken. The numerical value of each genotype gives its characteristic called phenotype.
This population changes in next generations. The number of all generations is set by an user. Creation of the next generation is done by a random choice of pairs of parents. A selection model used in reproduction is set by the user. The roulette method is the most popular one, however, the application gives another choices. In the roulette method, genotypes coding numerical values near the aim of the simulation i. e. near the function minimum or maximum, respectively, are more likely to be chosen. They fit better to the assumed condition. Since the parents are chosen the reproduction is done by the 'crossover' mechanism with assumed probability defined by the user. To keep some level of randomness in the simulation the mechanism of mutations is applied. The mutation frequency must also be set by the user.
