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Construction of an Approximate-Analytical Solution for Boundary Value Problems of a Parabolic Equation
Dalabaev Umurdin,
Hasanova Dilfuza
Issue:
Volume 8, Issue 2, March 2023
Pages:
39-45
Received:
11 February 2023
Accepted:
27 February 2023
Published:
9 March 2023
Abstract: The article considers an approximate analytical solution (AAS) of a linear parabolic equation with initial and boundary conditions with one spatial variable. Many problems in engineering applications are reduced to solving an initial boundary value problem of a parabolic type. There are various analytical, approximate-analytical and numerical methods for solving such problems. Here we consider ways to obtain an AAS based on the movable node method. Three approaches to obtaining an AAS are considered. In all approaches, an arbitrary point (a movable node) is considered inside the area where the solution is being sought. In the first approach, the parabolic equation is approximated by a difference scheme with a movable node in both variables. As a result, the differential problem is reduced to one algebraic equation (in the case of a boundary condition of the first kind), solving which we obtain an AAS. If one of the boundary conditions is of the second or third kind, assuming the fulfillment of the difference equation up to the boundary, we determine the unknown value of the solution on the boundary. The second and third approaches are based on the idea of the method of lines for differential equations in combination with a movable node. In the second approach, in a parabolic equation, the time derivative is approximated by a difference relation with the node being moved, and as a result we obtain an ordinary second-order differential equation with boundary conditions. Solving the obtained ordinary differential equation, we have an AAS. In the third approach, the approximation of the parabolic equation is performed only by the spatial variable. As a result, we obtain an ordinary differential equation of the first order with an initial condition and the solution of which gives an AAS to the original problem. The simplicity of the described approach allows the use of its engineering calculations. Comparisons have been made.
Abstract: The article considers an approximate analytical solution (AAS) of a linear parabolic equation with initial and boundary conditions with one spatial variable. Many problems in engineering applications are reduced to solving an initial boundary value problem of a parabolic type. There are various analytical, approximate-analytical and numerical metho...
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Witten Complex of Transitive Digraph and Its Convergence
Chong Wang,
Xin Lai,
Rongge Yu,
Yaxuan Zheng,
Baowei Liu
Issue:
Volume 8, Issue 2, March 2023
Pages:
46-50
Received:
21 February 2023
Accepted:
24 March 2023
Published:
27 March 2023
Abstract: Digraphs are generalization of graphs in which each edge is given one or two directions. For each digraph, there exists a transitive digraph containing it. Moreover, all the formal linear combinations of allowed elementary paths form a basis of the path complex for a transitive digraph. Hence, the study of discrete Morse theory on transitive digraphs is very important for the further study of discrete Morse theory on general digraphs. As we know, the definition of discrete Morse function on a digraph is different from that on a simplical complex or a cell complex: each discrete Morse function on a digraph is a discrete flat Witten-Morse function. In this paper, we deform the usual boundary operator, replacing it with a boundary operator with parameters and consider the induced Laplace operators. In addition, we consider the eigenvectors of the eigenvalues of the Laplace operator that approach to zero when the parameters approach infinity, define the generation space of these eigenvectors, and further give the Witten complex of digraphs. Finally, we prove that for a transitive digraph, Witten complex approaches to its Morse complex. However, for general digraphs, the structure of Morse complex is not as simple as that of transitive digraphs and the critical path is not directly related to the eigenvector with zero eigenvalue of Laplace operator. This is explained in the last part of the paper.
Abstract: Digraphs are generalization of graphs in which each edge is given one or two directions. For each digraph, there exists a transitive digraph containing it. Moreover, all the formal linear combinations of allowed elementary paths form a basis of the path complex for a transitive digraph. Hence, the study of discrete Morse theory on transitive digrap...
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Some Upper Bounds of Maximum E-Eigenvalues of Uniform Hypergraphs
Hongyu Zhang,
Feng Fu,
Caoji Yin
Issue:
Volume 8, Issue 2, March 2023
Pages:
51-56
Received:
25 February 2023
Accepted:
16 March 2023
Published:
28 March 2023
Abstract: A hypergraphs, as a generalization of a general graph, is often used as an effective tool to describe complex structures in discrete mathematics, computer science and other fields. Hypergraph theory and related parameters of hypergraph are important research topics in hypergraph theory. In particular, the problem of spectral extremum of graphs has been widely concerned. This problem originates from the problem proposed by Brualdi and Solheid in 1986. That is to find the upper and lower bounds of spectral radius of a given graph class and characterize the polar graph that reaches the upper and lower bounds. Let H be a uniform hypergraph. Let A(H) be the adjacency tensor of H. In this work, by using Perron-Frobenius theorem, Hölder’s inequality and inequality of arithmetic and geometric means, we establish some upper bounds for the maximum E-eigenvalue of a uniform hypergraph instead of the degrees of vertices and edge number of hypergraph H. In addition, we characterize the extremal hypergraphs that reach the upper bounds.
Abstract: A hypergraphs, as a generalization of a general graph, is often used as an effective tool to describe complex structures in discrete mathematics, computer science and other fields. Hypergraph theory and related parameters of hypergraph are important research topics in hypergraph theory. In particular, the problem of spectral extremum of graphs has ...
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Analysis and Optimization of Improved Index Calculus Algorithm
Issue:
Volume 8, Issue 2, March 2023
Pages:
57-61
Received:
5 April 2023
Accepted:
30 March 2023
Published:
13 April 2023
Abstract: IC (Index Calculus) algorithm is the most effective probability algorithm for solving discrete logarithm of finite prime fields, and IICA (improved Index Calculus algorithm) is an improved algorithm based on IC in the third stage. The essence of IICA is to convert the number required to solve the discrete logarithm into the product of the power of prime factors, and then multiply every prime factor larger than the smooth bound by a smooth number approximating a large prime p from the right end, that is, to perform congruence transformation for every prime factor larger than the smooth bound. If all prime factors larger than the smooth bound fall within the smooth bound, the number required to solve the discrete logarithm is successfully solved. Unfortunately, given a large prime number p, some prime factors do not have congruent transformations of smooth numbers. For this, this paper analyzes the features of the IICA algorithm, based on the characteristics of IICA algorithm, is given when the IICA algorithm cannot undertake congruence transformation termination conditions, namely when ⌈ p/pi ⌉ not smooth algorithm is terminated, where pi is greater than a smooth boundary element factor. According to the ⌈ p/pi ⌉ not smooth algorithm was terminated when judging conditions, optimized the IICA algorithm, and the correctness of the optimization algorithm is verified by an example.
Abstract: IC (Index Calculus) algorithm is the most effective probability algorithm for solving discrete logarithm of finite prime fields, and IICA (improved Index Calculus algorithm) is an improved algorithm based on IC in the third stage. The essence of IICA is to convert the number required to solve the discrete logarithm into the product of the power of ...
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The Free Boundary Problem of a Predator-Prey Model with Fear Effect and Stage Structure
Issue:
Volume 8, Issue 2, March 2023
Pages:
62-67
Received:
25 March 2023
Accepted:
15 April 2023
Published:
23 April 2023
Abstract: In this paper, fear effect and stage structure are introduced in a free boundary problem of a prey-predator model. This system simulates the spread of an invasive or newly introduced predator species, taking into account the presence of both immature and mature stages of prey that are affected by fear of the predator. The predator's predation behavior on adult prey induces fear in the prey, which in turn causes the prey to seek out safer habitats. While this short-term survival strategy may be effective, it ultimately leads to a decrease in the prey's long-term survival fitness, including reduced reproductive ability. Consequently, the overall population of prey is expected to decline over the long term. The existence and uniqueness of the solution is given, and the comparison principle is used to discuss the long-term behavior of the solution by constructing a sequence of upper and lower solutions. We obtain a spreading–vanishing dichotomy for this model, in other words, when the predator can only spread in a limited area, the predator will eventually become extinct, the population density of the two stages of prays will tend to two positive constants, and when the predator can spread to infinity, the predator ultimately survives, and their population density, defined as (u, v, w) will eventually tend to (u*, v*, w*) which we defined blow.
Abstract: In this paper, fear effect and stage structure are introduced in a free boundary problem of a prey-predator model. This system simulates the spread of an invasive or newly introduced predator species, taking into account the presence of both immature and mature stages of prey that are affected by fear of the predator. The predator's predation behav...
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