SANTURIO et al 2020a - Revision history

Rimni at 12:36, 9 July 2020

2020-07-09T12:36:01Z

Rimni at 12:20, 9 July 2020

2020-07-09T12:20:30Z

Rimni at 12:14, 9 July 2020

2020-07-09T12:14:30Z

Rimni at 12:12, 9 July 2020

2020-07-09T12:12:02Z

Rimni at 12:03, 9 July 2020

2020-07-09T12:03:58Z

Rimni at 11:24, 9 July 2020

2020-07-09T11:24:04Z

Rimni at 09:55, 9 July 2020

2020-07-09T09:55:22Z

Rimni at 09:50, 9 July 2020

2020-07-09T09:50:49Z

Rimni at 09:37, 9 July 2020

2020-07-09T09:37:34Z

Rimni at 09:34, 9 July 2020

2020-07-09T09:34:47Z

@@ Line 270: / Line 270: @@
 | <math>\psi \left( r\right) =\, \sum _{j=1}^{N}{\alpha }_{j}\left[ -\frac{4}{{\lambda }^{4}}-\frac{4\, ln{r}_{ij}}{{\lambda }^{4}}-\frac{{r}_{ij}^{2}\, ln{r}_{ij}}{{\lambda }^{2}}-\frac{4{K}_{0}\left( \lambda {r}_{ij}\right) }{{\lambda }^{2}}\right] -</math><math>{\alpha }_{n+1}\frac{1}{{\lambda }^{2}}-{\alpha }_{n+2}\frac{x}{{\lambda }^{2}}-</math><math>{\alpha }_{n+3}\frac{y}{{\lambda }^{2}}\, ,\quad  </math> for 2D and <math display="inline">r\not =0</math>
 |}
-|  style="text-align: right;vertical-align: center;"| (14.a)
+|  style="text-align: right;vertical-align: center;"| (14a)
 |-
+|
@@ Line 277: / Line 277: @@
 |<math>\psi (r)=\, \sum _{j=1}^{N}{\alpha }_{j}\left[ -\frac{4}{{\lambda }^{4}}-\frac{4\, \gamma }{{\lambda }^{4}}+\, \frac{4\, }{{\lambda }^{4}}\mathrm{ln}\,(\frac{\lambda }{2})\right] -</math><math>{\alpha }_{n+1}\frac{1}{{\lambda }^{2}}-{\alpha }_{n+2}\frac{x}{{\lambda }^{2}}-</math><math>{\alpha }_{n+3}\frac{y}{{\lambda }^{2}},\quad </math> for 2D and <math display="inline">r=</math><math>0</math>
 |}
-|  style="text-align: right;vertical-align: center;"| (14.b)
+|  style="text-align: right;vertical-align: center;"| (14b)
 |}

@@ Line 338: / Line 338: @@
 |}
-where <math display="inline">{\lambda }^{2}=\sqrt{\frac{1}{k\Delta t}}</math> and <math display="inline">A\left( \mathit{\boldsymbol{X}}\right) =</math><math>-\frac{{U}^{n}\left( \mathit{\boldsymbol{X}}\right) }{k\Delta t}-\frac{F(\mathit{\boldsymbol{X}})}{k}</math> . It is worth noticing that the term <math display="inline">A\left( \mathit{\boldsymbol{X}}\right)</math>  is well defined and depends on the solution obtained from the previous time and the nonhomogenous condition, which is time-independent and is calculated by DRM proposed in section 3.1. The initial condition should be applied in order to start the procedure.
+where <math display="inline">{\lambda }^{2}=\sqrt{\frac{1}{k\Delta t}}</math> and <math display="inline">A\left( \mathit{\boldsymbol{X}}\right) =</math><math>-\frac{{U}^{n}\left( \mathit{\boldsymbol{X}}\right) }{k\Delta t}-\frac{F(\mathit{\boldsymbol{X}})}{k}</math> . It is worth noticing that the term <math display="inline">A\left( \mathit{\boldsymbol{X}}\right)</math>  is well defined and depends on the solution obtained from the previous time and the nonhomogenous condition, which is time-independent and is calculated by DRM proposed in Section 3.1. The initial condition should be applied in order to start the procedure.
 By using MFS for every time step, the homogenous solution is approximated by:
@@ Line 608: / Line 608: @@
 |}
-In this example, for MSF1 and MSF2, the coefficient <math display="inline">b=0.5</math>, and, for the modified Helmholtz operator, virtual sources are 0.5 from the physical boundary. In order to explore the methodologies, two different types of RBF were considered, referenced as RBF1 the TPS described on section 3.1 and RBF2 the following functions [23]:
+In this example, for MSF1 and MSF2, the coefficient <math display="inline">b=0.5</math>, and, for the modified Helmholtz operator, virtual sources are 0.5 from the physical boundary. In order to explore the methodologies, two different types of RBF were considered, referenced as RBF1 the TPS described on Section 3.1 and RBF2 the following functions [23]:
 {| class="formulaSCP" style="width: 100%; text-align: center;"
@@ Line 692: / Line 692: @@
-[[#img-12|Figure 12]] depicts absolute errors at node A and B, for different time steps in all methods. When compared, MSF1 and MSF based on the modified Helmholtz operator present better responses as time step decreases, and MSF2, a method that is known for its ill posed matrix, presents better results for a bigger time step. All methods showed to be sensitive concerning the field node geometrical position.
+[[#img-12|Figure 12]] depicts absolute errors at nodes A and B, for different time steps in all methods. When compared, MSF1 and MSF based on the modified Helmholtz operator present better responses as time step decreases, and MSF2, a method that is known for its ill posed matrix, presents better results for a bigger time step. All methods showed to be sensitive concerning the field node geometrical position.
 <div id='img-12'></div>
 {| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"

@@ Line 302: / Line 302: @@
 where <math display="inline">\Delta t={t}^{n+1}-{t}^{n}</math> is time step size and <math display="inline">\theta =</math><math>1</math> defining an implicit time marching method.
-Substituting and rearranging terms of Eq. (16) in Eq. (1) and in conditions Eq. (3-4) it yields to the modified Helmholtz equation:
+Substituting and rearranging terms of Eq. (16) in Eq. (1) and in conditions Eqs. (3)-(4) it yields to the modified Helmholtz equation:
 {| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;"

@@ Line 43: / Line 43: @@
 The main issue in MFS is the source positioning in order obtain reasonable responses. The source points can be placed outside the problem domain, avoiding singularities over the space coordinates, and the solution can be approximated over time by Laplace transform [8,9,10] or by finite difference scheme [11], an interesting approach since MFS is well-posed problem in spatial domain. Otherwise, the source points can be placed at different time levels [7], and, by using transient fundamental solutions, they will not generate singularities in time and spatial domains, however generating an ill-posed problem.
-For homogeneous diffusion equations, in Golberg et al. [12] for non-zero initial conditions, rectangular shaped domain and Dirichlet’s boundary conditions prescribed, the Laplace transform method and the finite difference method were studied. For both cases it leads to a non-homogeneous Helmholtz equation. Particular solutions, for different types of problems, are generally found by the dual reciprocity method (DRM), introduced by [13,14], that transform the domain integral into a boundary-type linear combination of radial basis functions (RBFs), and Golberg et al. [15] proposed particular solutions for Helmholtz-type operators. For a similar geometry, in paper [7] transient fundamental solutions were applied for homogeneous Dirichlet’s boundary conditions with more complex initial conditions.
+For homogeneous diffusion equations, in Golberg et al. [12] for non-zero initial conditions, rectangular shaped domain and Dirichlet’s boundary conditions prescribed, the Laplace transform method and the finite difference method were studied. For both cases it leads to a non-homogeneous Helmholtz equation. Particular solutions, for different types of problems, are generally found by the dual reciprocity method (DRM), introduced by [13,14], that transform the domain integral into a boundary-type linear combination of radial basis functions (RBFs), and Golberg et al. [15] proposed particular solutions for Helmholtz-type operators. For a similar geometry, in Young et al. [7] transient fundamental solutions were applied for homogeneous Dirichlet’s boundary conditions with more complex initial conditions.
 In Cao et al. [16], DRM-MFS based on the modified Helmholtz operator and DRM-Trefftz were applied for homogeneous diffusion equation with mixed boundary conditions and constant initial conditions, Chantasiriwan [17] compared MFS based on transient fundamental solutions with MFS based on the modified Helmholtz operator for a square with different types of boundary conditions.

@@ Line 738: / Line 738: @@
 ==References==
-[1] Gingold, R. A., Monaghan, J. J. Smoothed particle hydrodynamics: theory and application to non-spherical stars. Monthly notices of the royal astronomical society, 181(3), 375-389, 1977.
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-[2] Oñate, E., Idelsohn, S., Zienkiewicz, O. C., Taylor, R. L. A finite point method in computational mechanics. Applications to convective transport and fluid flow. International journal for numerical methods in engineering, 39(22), 3839-3866, 1996.
+[2] Oñate E., Idelsohn S., Zienkiewicz O.C., Taylor R.L. A finite point method in computational mechanics. Applications to convective transport and fluid flow. International Journal for Numerical Methods in Engineering, 39(22):3839-3866, 1996.
-[3] Belytschko, T., Lu, Y. Y., Gu, L. Element‐free Galerkin methods. International journal for numerical methods in engineering, 37(2), 229-256, 1994.
+[3] Belytschko T., Lu Y.Y., Gu L. Element‐free Galerkin methods. International Journal for Numerical Methods in Engineering, 37(2):229-256, 1994.
-[4]  Atluri, S. N., Zhu, T. A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics. Computational mechanics, 22(2), 117-127, 1998.
+[4]  Atluri S.N., Zhu T. A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics. Computational Mechanics, 22(2):117-127, 1998.
-[5] Chen, W. Meshfree boundary particle method applied to Helmholtz problems. Engineering Analysis with Boundary Elements, 26(7), 577-581, 2002.
+[5] Chen W. Meshfree boundary particle method applied to Helmholtz problems. Engineering Analysis with Boundary Elements, 26(7):577-581, 2002.
-[6] Wang, H., & Qin, Q. H. Methods of Fundamental Solutions in Solid Mechanics. Elsevier, 2019.
+[6] Wang H., Qin Q.H. Methods of fundamental solutions in solid mechanics. Elsevier, 2019.
-[7] Young, D. L., Tsai, C. C., Murugesan, K., Fan, C. M., Chen, C. W. Time-dependent fundamental solutions for homogeneous diffusion problems. Engineering Analysis with Boundary Elements, 28(12), 1463-1473, 2004.
+[7] Young D.L., Tsai C.C., Murugesan K., Fan C.M., Chen C.W. Time-dependent fundamental solutions for homogeneous diffusion problems. Engineering Analysis with Boundary Elements, 28(12):1463-1473, 2004.
-[8] Zhu, S. P. Solving transient diffusion problems: time-dependent fundamental solution approaches versus LTDRM approaches. Engineering analysis with boundary elements, 21(1), 87-90, 1998.
+[8] Zhu S.P. Solving transient diffusion problems: time-dependent fundamental solution approaches versus LTDRM approaches. Engineering Analysis with Boundary Elements, 21(1):87-90, 1998.
-[9] Zhu, S. P., Liu, H. W., Lu, X. P. A combination of LTDRM and ATPS in solving diffusion problems. Engineering analysis with boundary elements, 21(3), 285-289, 1998.
+[9] Zhu S.P., Liu H.W., Lu X.P. A combination of LTDRM and ATPS in solving diffusion problems. Engineering Analysis with Boundary Elements, 21(3):285-289, 1998.
-[10] Sutradhar, A., Paulino, G. H., Gray, L. J. Transient heat conduction in homogeneous and non-homogeneous materials by the Laplace transform Galerkin boundary element method. Engineering Analysis with Boundary Elements, 26(2), 119-132, 2002.
+[10] Sutradhar A., Paulino G.H., Gray L.J. Transient heat conduction in homogeneous and non-homogeneous materials by the Laplace transform Galerkin boundary element method. Engineering Analysis with Boundary Elements, 26(2):119-132, 2002.
-[11] Bulgakov, V., Šarler, B., Kuhn, G. Iterative solution of systems of equations in the dual reciprocity boundary element method for the diffusion equation. International Journal for Numerical Methods in Engineering, 43(4), 713-732, 1998.
+[11] Bulgakov V., Šarler B., Kuhn G. Iterative solution of systems of equations in the dual reciprocity boundary element method for the diffusion equation. International Journal for Numerical Methods in Engineering, 43(4):713-732, 1998.
-[12] Golberg, M. A., Chen, C. S., Muleshkov, A. S. The method of fundamental solutions for time-dependent problems. WIT Transactions on Modelling and Simulation, 23, 1970.
+[12] Golberg M.A., Chen C.S., Muleshkov A.S. The method of fundamental solutions for time-dependent problems. WIT Transactions on Modelling and Simulation, 23, 1970.
-[13] Nardini, D., Brebbia, C. A. A new approach to free vibration analysis using boundary elements. Applied mathematical modelling, 7(3), 157-162, 1983.
+[13] Nardini D., Brebbia C.A. A new approach to free vibration analysis using boundary elements. Applied Mathematical Modelling, 7(3):157-162, 1983.
-[14] Partridge, P. W., Brebbia, C. A. Dual reciprocity boundary element method. Springer Science & Business Media, 2012.
+[14] Partridge P.W., Brebbia C.A. Dual reciprocity boundary element method. Springer Science & Business Media, 2012.
-[15] Golberg, M. A., Chen, C. S., Rashed, Y. F. The annihilator method for computing particular solutions to partial differential equations. Engineering analysis with boundary elements, 23(3), 275-279, 1999.
+[15] Golberg M.A., Chen C.S., Rashed Y.F. The annihilator method for computing particular solutions to partial differential equations. Engineering Analysis with Boundary Elements, 23(3):275-279, 1999.
-[16] Cao, L., Qin, Q. H., Zhao, N. Application of DRM-Trefftz and DRM-MFS to transient heat conduction analysis. Recent Patents on Space Technology, 2, 41-50, 2010.
+[16] Cao L., Qin Q.H., Zhao N. Application of DRM-Trefftz and DRM-MFS to transient heat conduction analysis. Recent Patents on Space Technology, 2:41-50, 2010.
-[17] Chantasiriwan, S. Methods of fundamental solutions for time‐dependent heat conduction problems. International journal for numerical methods in engineering, 66(1), 147-165, 2006.
+[17] Chantasiriwan S. Methods of fundamental solutions for time‐dependent heat conduction problems. International Journal for Numerical Methods in Engineering, 66(1):147-165, 2006.
-[18] Young, D. L., Tsai, C. C., Fan, C. M. Direct approach to solve nonhomogeneous diffusion problems using fundamental solutions and dual reciprocity methods. Journal of the Chinese Institute of Engineers, 27(4), 597-609, 2004.
+[18] Young D.L., Tsai C.C., Fan C.M. Direct approach to solve nonhomogeneous diffusion problems using fundamental solutions and dual reciprocity methods. Journal of the Chinese Institute of Engineers, 27(4):597-609, 2004.
-[19] Young, D. L., Chen, C. W., Fan, C. M., Tsai, C. C. The method of fundamental solutions with eigenfunction expansion method for nonhomogeneous diffusion equation. Numerical Methods for Partial Differential Equations: An International Journal, 22(5), 1173-1196, 2006.
+[19] Young D.L., Chen C.W., Fan C.M., Tsai C.C. The method of fundamental solutions with eigenfunction expansion method for nonhomogeneous diffusion equation. Numerical Methods for Partial Differential Equations: An International Journal, 22(5):1173-1196, 2006.
-[20] Golberg, M. A., Chen, C. S. The theory of radial basis functions applied to the BEM for inhomogeneous partial differential equations. Boundary Elements Communications, 5(2), 57-61, 1994.
+[20] Golberg M.A., Chen C.S. The theory of radial basis functions applied to the BEM for inhomogeneous partial differential equations. Boundary Elements Communications, 5(2):57-61, 1994.
-[21] Chen, C. S., Rashed, Y. F. Evaluation of thin plate spline based particular solutions for Helmholtz-type operators for the DRM. Mechanics Research Communications, 25(2), 195-201, 1998.
+[21] Chen C.S., Rashed Y.F. Evaluation of thin plate spline based particular solutions for Helmholtz-type operators for the DRM. Mechanics Research Communications, 25(2):195-201, 1998.
-[22] Golberg, M. A., Chen, C. S. The method of fundamental solutions for potential, Helmholtz and diffusion problems. Boundary integral methods: numerical and mathematical aspects, 1, 103-176, 1998.
+[22] Golberg M.A., Chen C.S. The method of fundamental solutions for potential, Helmholtz and diffusion problems. Boundary Integral Methods: Numerical and Mathematical Aspects, 1:103-176, 1998.
-[23] Balakrishnan, K., Sureshkumar, R., Ramachandran, P. A. An operator splitting-radial basis function method for the solution of transient nonlinear Poisson problems. Computers & Mathematics with Applications, 43(3-5), 289-304, 2002.
+[23] Balakrishnan K., Sureshkumar R., Ramachandran P. A. An operator splitting-radial basis function method for the solution of transient nonlinear Poisson problems. Computers & Mathematics with Applications, 43(3-5):289-304, 2002.

@@ Line 419: / Line 419: @@
 {| style="text-align: center; margin:auto;width: 100%;"
 |-
-| <math>{U}^{\ast }\left( \mathit{\boldsymbol{X}},\mathit{\boldsymbol{\xi }};{t}_{F},t\right) =</math><math>\, \frac{1}{{\left( 4\pi k\tau \right) }^{\frac{d}{2}}}exp\left[ \frac{-{r}^{2}\left( \mathit{\boldsymbol{X}},\mathit{\boldsymbol{\xi }}\right) }{4k\tau }\right]</math>
+| <math>{U}^{\ast }\left( \mathit{\boldsymbol{X}},\mathit{\boldsymbol{\xi }};{t}_{F},t\right) =</math><math>\, \frac{1}{{\left( 4\pi k\tau \right) }^{\frac{d}{2}}}\exp\left[ \frac{-{r}^{2}\left( \mathit{\boldsymbol{X}},\mathit{\boldsymbol{\xi }}\right) }{4k\tau }\right]</math>
 |-
-| <math>{Q}^{\ast }\left( \mathit{\boldsymbol{X}},\mathit{\boldsymbol{\xi }};{t}_{F},t\right) =</math><math>\, \frac{-r\left( \mathit{\boldsymbol{X}},\mathit{\boldsymbol{\xi }}\right) \frac{\partial r\left( \mathit{\boldsymbol{X}},\mathit{\boldsymbol{\xi }}\right) }{\partial n}}{8\pi k{\tau }^{2}}exp\left[ \frac{-{r}^{2}\left( \mathit{\boldsymbol{X}},\mathit{\boldsymbol{\xi }}\right) }{4k\tau }\right] \,</math>
+| <math>{Q}^{\ast }\left( \mathit{\boldsymbol{X}},\mathit{\boldsymbol{\xi }};{t}_{F},t\right) =</math><math>\, \frac{-r\left( \mathit{\boldsymbol{X}},\mathit{\boldsymbol{\xi }}\right) \frac{\partial r\left( \mathit{\boldsymbol{X}},\mathit{\boldsymbol{\xi }}\right) }{\partial n}}{8\pi k{\tau }^{2}}\exp\left[ \frac{-{r}^{2}\left( \mathit{\boldsymbol{X}},\mathit{\boldsymbol{\xi }}\right) }{4k\tau }\right] \,</math>
 |}
 |  style="text-align: right;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(23)

@@ Line 605: / Line 605: @@
 {| class="formulaSCP" style="width: 100%; text-align: center;"
 |-
-|  style="vertical-align: top;"|<math>U\left( x,y,t\right) =2\left[ cos\left( \pi x\right) +sin\left( \pi y\right) \right] {e}^{-k{\pi }^{2}t}+</math><math>\frac{{x}^{3}+{y}^{3}+{x}^{2}+y}{12}</math>
+|  style="vertical-align: top;"|<math>U\left( x,y,t\right) =2\left[ \cos\left( \pi x\right) + \sin\left( \pi y\right) \right] {e}^{-k{\pi }^{2}t}+</math><math>\frac{{x}^{3}+{y}^{3}+{x}^{2}+y}{12}</math>
 |}

@@ Line 608: / Line 608: @@
 |}
-In this example, for MSF1 and MSF2, the coefficient <math display="inline">b=</math><math>0.5</math>, and, for the modified Helmholtz operator, virtual sources are 0.5 from the physical boundary. In order to explore the methodologies, two different types of RBF were considered, referenced as RBF1 the TPS described on section 3.1 and RBF2 the following functions [23]:
+In this example, for MSF1 and MSF2, the coefficient <math display="inline">b=0.5</math>, and, for the modified Helmholtz operator, virtual sources are 0.5 from the physical boundary. In order to explore the methodologies, two different types of RBF were considered, referenced as RBF1 the TPS described on section 3.1 and RBF2 the following functions [23]:
 {| class="formulaSCP" style="width: 100%; text-align: center;"
 |-
-|  style="vertical-align: top;"|<math display="inline">\varphi \left( r\right) =\, 1+</math><math>r\quad</math> and <math display="inline">\quad\psi \left( r\right) =\frac{{r}^{2}}{4}+</math><math>\frac{{r}^{3}}{9}\qquad </math> for Laplacian operator
+|  style="vertical-align: top;"|<math display="inline">\varphi \left( r\right) =\, 1+r\quad</math> and <math display="inline">\quad\psi \left( r\right) =\frac{{r}^{2}}{4}+</math><math>\frac{{r}^{3}}{9}\qquad </math> for Laplacian operator
 |-
-|  style="vertical-align: top;"|<math display="inline">\varphi \left( r\right) =\, 1+r-</math><math>{\lambda }^{2}(\frac{{r}^{2}}{4}+\frac{{r}^{3}}{9})\quad</math> and <math display="inline">\quad\psi \left( r\right) =</math><math>\frac{{r}^{2}}{4}+\frac{{r}^{3}}{9}\qquad </math> for modified Helmholtz operator
+|  style="vertical-align: top;"|<math display="inline">\varphi \left( r\right) =\, 1+r-{\lambda }^{2}(\frac{{r}^{2}}{4}+\frac{{r}^{3}}{9})\quad</math> and <math display="inline">\quad\psi \left( r\right) =</math><math>\frac{{r}^{2}}{4}+\frac{{r}^{3}}{9}\qquad </math> for modified Helmholtz operator
 |}
@@ Line 638: / Line 638: @@
-[[#img-9|Figure 9]] presents the results obtained by the three tested methods and the analytical solution for a line of nodes at <math display="inline">y=</math><math>0.5</math> on initial stage and final stage. For initial stages MFS based on modified Helmholtz operator present less accuracy than MFS1 and MFS2, while at final times all methods present good precision. [[#img-10|Figure 10]] shows potential at all nodes at <math display="inline">t=</math><math>3s</math>.
+[[#img-9|Figure 9]] presents the results obtained by the three tested methods and the analytical solution for a line of nodes at <math display="inline">y=0.5</math> on initial stage and final stage. For initial stages MFS based on modified Helmholtz operator present less accuracy than MFS1 and MFS2, while at final times all methods present good precision. [[#img-10|Figure 10]] shows potential at all nodes at <math display="inline">t=3s</math>.
 <div id='img-9'></div>

@@ Line 594: / Line 594: @@
 {| class="formulaSCP" style="width: 100%; text-align: center;"
 |-
-|  style="text-align: center;vertical-align: top;"|<math>I.C.: \,U\left( x,y,t=0\right) =2\left[ cos\left( \pi x\right) +sin\left( \pi y\right) \right] +</math><math>\frac{{x}^{3}+{y}^{3}+{x}^{2}+y}{12}</math>
+|  style="text-align: center;vertical-align: top;"|<math>I.C.: \,U\left( x,y,t=0\right) =2\left[ \cos\left( \pi x\right) + \sin\left( \pi y\right) \right] +</math><math>\frac{{x}^{3}+{y}^{3}+{x}^{2}+y}{12}</math>
 |-
-|<math>B.C.:\, U\left( x,y,t\right) =2\left[ cos\left( \pi x\right) +sin\left( \pi y\right) \right] {e}^{-k{\pi }^{2}t}+</math><math>\frac{{x}^{3}+{y}^{3}+{x}^{2}+y}{12}</math>
+|<math>B.C.:\, U\left( x,y,t\right) =2\left[ \cos\left( \pi x\right) + \sin\left( \pi y\right) \right] {e}^{-k{\pi }^{2}t}+</math><math>\frac{{x}^{3}+{y}^{3}+{x}^{2}+y}{12}</math>
 |-
 |<math>F\left( x,y\right) =\, \frac{1}{m}=-\frac{6x+6y+2}{12}</math>

@@ Line 233: / Line 233: @@
 and with solving the following linear system:
-{| class="formulaSCP" style="width: 100%;border-collapse: collapse;text-align: center;"
+{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;"
 |-
+|
-{| style="width: 80%;text-align:center;"
+{| style="text-align: center; margin:auto;width: 100%;"
 |-
 |  <math>\begin{matrix}{r}_{11}^{2}\mathrm{ln}\,{r}_{11} & {r}_{12}^{2}\mathrm{ln}\,{r}_{12} & {r}_{13}^{2}\mathrm{ln}\,{r}_{13} & \ldots & {r}_{1n}^{2}\mathrm{ln}\,{r}_{1n} & 1 & {x}_{1} & {y}_{1} & {\alpha }_{1} & & \frac{-F\left( X\right) }{k}\\