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Finite Difference Method for Solving Partial Integro-Differential Equations
Dr Majid Erfanian, Dr Hamed Zeidabadi,
Volume 6, Issue 1 (Vol. 6, No. 1 2020)
Abstract
Introduction
In this paper, we have introduced a new method for solving a class of the partial integro-differential equation with the singular kernel by using the finite difference method. One of the best subjects in the numerical analysis is a finite difference method (FDM). We used (FDM) to solve problems in mathematical physics, integral equations, and  engineering, such as electromagnetic potential, fluid flow,  radiation heats transfer, laminar boundary-layer theory and mass transport, Abel integral equations, and problem of mechanics or physics. Also in some physical problems such as fluid flow and heat transfer problems, the Laplace equations and the Poisson equations are describe by (FDM).  In real life most phenomena are modelled by partial differential equations.
Material and methods
First, we employing an algorithm for solving the problem based on the Crank-Nicholson scheme with given conditions. Furthermore, we discrete the singular integral for solving of the problem. Also, the numerical results obtained here can be compared with the cubic B-spline method.
Results and discussion
In addition, solving some examples demonstrates the validity and applicability of the approached method, so that the results are reported in the tables and their figures are shown. The high speed of the calculations, and the assurance of having an approximate solution are obtain by proving the stability of the method.
Conclusion
The following conclusions were drawn from this research.
  • Coefficients of the approximate function via Crank-Nicholson scheme are found very easily and therefore many calculations are reduced.
  • The numerical results obtained here can be compared with the cubic B-spline method
  • The assurance of having an approximate solution are obtain by proving the stability of the method../files/site1/files/61/8.pdf

 
 
Topology coloring
Hmaid Erfanianoraei Dehrokhi, Majid Erfanian Oraei,
Volume 8, Issue 2 (Vol. 8,No. 2, 2022)
Abstract
Introduction
The aim of this study, is painting of topological surfaces with the least number of colors without the distance, and the colors have a border. For this purpose, we need a color mapping. In this mapping, we have not any fixed point, and we can colorable the map with least colors.
Definition: Let f∶ X → X be a graph without a fixed point. f is colorable with k colors, if there is C={C_1,…,C_K}, where all C_i do not include {(x, f(x)}. Or similarly, for every i=1,…, k, there is the equation C_i ∩ f(C_i )=∅.
Also, we define some concepts such as Compression, Metric, or non-Compression of space. Also, to achieve the desired result of each space, we change the properties of the maps.
Material and methods
In this work, first, we define the properties and conditions of the color mapping and color number. Also, by the study of properties of each space, we choose the best of space. One of the best conditions of this space is the lowest color number and higher efficiency. Finally, we proved  that this number is finite, and we can do coloring space with some maps and conversely.
Results and discussion
In this work, we define the properties and conditions of the color mapping and color number. We presented some theorems and Lemma in the article and proved them for coloring of any space by coloring map, the coloring number is at least 3 and at most is a n+3. Also, we proved the coloring number finite and we can  do coloring space with some maps and conversely.
Conclusion
The following conclusions were drawn from this research.
    • the coloring number is at least 3.
    • the coloring number is at most n+3.
    • coloring number is finite and we can do coloring space with some maps.
    • We can do the coloring of any space by the finite coloring map.

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