Abstract
In this paper, we propose an effective method to solve partial differential equations dependent on time with Neumann boundary condition, by examining its effectivity on direct and inverse reaction–diffusion equation. This method merges the radial point interpolation and the Hermite-type interpolation techniques to provide us suitable tools to impose the boundary condition. This technique is called meshless radial point Hermite interpolation “MRPHI” which utilizes the radial basis function and its derivative to prepare suitable shape functions that are the key for expanding the high-order derivative. This procedure is tested on some types of two-dimensional diffusion equations to show stability through the time in different arbitrary domains.
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References
Ghehsareh HR, Zaghian A, Zabetzadeh SM (2018) The use of local radial point interpolation method for solving two-dimensional linear fractional cable equation. Neural Comput Appl 29(10):745–754
Asari SS, Amirfakhrian M, Chakraverty S (2019) Application of radial basis functions in solving fuzzy integral equations. Neural Comput Appl 31(10):6373–6381
Liu G-R, Gu Y-T (2005) An introduction to meshfree methods and their programming. Springer, Berlin
Kansa E (1990) Multiquadrics-a scattered data approximation scheme with applications to computational fluid-dynamics. I surface approximations and partial derivative estimates. Comput Math Appl 19(8–9):127–145
Jakobsson S, Andersson B, Edelvik F (2009) Rational radial basis function interpolation with applications to antenna design. J Comput Appl Math 233(4):889–904
Abbasbandy S, Ghehsareh HR, Hashim I (2013) A meshfree method for the solution of two-dimensional cubic nonlinear schrödinger equation. Eng Anal Boundary Elem 37(6):885–898
Abbasbandy S, Ghehsareh HR, Hashim I (2012) Numerical analysis of a mathematical model for capillary formation in tumor angiogenesis using a meshfree method based on the radial basis function. Eng Anal Boundary Elem 36(12):1811–1818
Kamranian M, Dehghan M, Tatari M (2016) Study of the two-dimensional sine- Gordon equation arising in Josephson junctions using meshless finite point method. Int J Numer Model Electr Netw Dev Fields 30(6):e2210
Abbaszadeh M, Khodadadian A, Parvizi M, Dehghan M, Heitzinger C (2019) A direct meshless local collocation method for solving stochastic Cahn–Hilliard–Cook and stochastic Swift–Hohenberg equations. Eng Anal Boundary Elem 98:253–264
Esmaeilbeigi M, Mirzaee F, Moazami D (2017) Radial basis functions method for solving three-dimensional linear Fredholm integral equations on the cubic domains. Iran J Numer Anal Optim 7(2):15–37
Esmaeilbeigi M, Mirzaee F, Moazami D (2017) A meshfree method for solving multidimensional linear Fredholm integral equations on the hypercube domains. Appl Math Comput 298:236–246
Mirzaee F, Samadyar N (2018) Using radial basis functions to solve two dimensional linear stochastic integral equations on non-rectangular domains. Eng Anal Boundary Elem 92:180–195
Mirzaee F, Samadyar N (2019) Numerical solution of time fractional stochastic korteweg-de vries equation via implicit meshless approach. Iran J Sci Technol Trans A Sci 43(6):2905–2912
Samadyar N, Mirzaee F (2019) Numerical solution of two-dimensional weakly singular stochastic integral equations on non-rectangular domains via radial basis functions. Eng Anal Boundary Elem 101:27–36
Assari P, Dehghan M (2018) A meshless local discrete collocation ( MLDC) scheme for solving 2-dimensional singular integral equations with logarithmic kernels, International Journal of Numerical Modelling: Electronic Networks. Devices and Fields 31(3):e2311
Shivanian E (2013) Analysis of meshless local radial point interpolation ( MLRPI) on a nonlinear partial integro-differential equation arising in population dynamics. Eng Anal Boundary Elem 37(12):1693–1702
Abbasbandy S, Shirzadi A (2010) A meshless method for two-dimensional diffusion equation with an integral condition. Eng Anal Boundary Elem 34(12):1031–1037
Abbasbandy S, Shirzadi A (2011) MLPG method for two-dimensional diffusion equation with Neumann’s and non-classical boundary conditions. Appl Numer Math 61:170–180
Dehghan M, Ghesmati A (2010) Numerical simulation of two-dimensional sine-gordon solitons via a local weak meshless technique based on the radial point interpolation method ( RPIM). Comput Phys Commun 181:772–786
Liu G, Yan L, Wang J, Gu Y (2002) Point interpolation method based on local residual formulation using radial basis functions. Struct Eng Mech 14:713–732
Liu G, Gu Y (2001) A local radial point interpolation method ( LR-PIM) for free vibration analyses of 2- D solids. J Sound Vib 246(1):29–46
Shivanian E (2014) Analysis of meshless local and spectral meshless radial point interpolation ( MLRPI and SMRPI) on 3- D nonlinear wave equations. Ocean Eng 89:173–188
Shivanian E (2015) Meshless local petrov- Galerkin ( MLPG) method for three-dimensional nonlinear wave equations via moving least squares approximation. Eng Anal Boundary Elem 50:249–257
Shivanian E, Khodabandehlo HR (2014) Meshless local radial point interpolation (MLRPI) on the telegraph equation with purely integral conditions. Eur Phys J Plus 129(11):241
Shivanian E (2016) On the convergence analysis, stability, and implementation of meshless local radial point interpolation on a class of three-dimensional wave equations. Int J Numer Meth Eng 105(2):83–110
Shivanian E, Khodabandehlo HR (2016) Application of meshless local radial point interpolation (MLRPI) on a one-dimensional inverse heat conduction problem. Ain Shams Eng J 7(3):993–1000
Shivanian E, Rahimi A, Hosseini M (2016) Meshless local radial point interpolation to three-dimensional wave equation with Neumann’s boundary conditions. Int J Comput Math 93(12):2124–2140
Hosseini VR, Shivanian E, Chen W (2015) Local integration of 2-d fractional telegraph equation via local radial point interpolant approximation. Eur Phys J Plus 130(2):33
Hosseini VR, Shivanian E, Chen W (2016) Local radial point interpolation ( MLRPI) method for solving time fractional diffusion-wave equation with damping. J Comput Phys 312:307–332
Shivanian E, Khodayari A (2017) Meshless local radial point interpolation ( MLRPI) for generalized telegraph and heat diffusion equation with non-local boundary conditions. J Theor Appl Mech 55:571–582
Mirzaee F, Samadyar N (2019) Combination of finite difference method and meshless method based on radial basis functions to solve fractional stochastic advection-diffusion equations. Eng Comput. https://doi.org/10.1007/s00366-019-00789-y
Mirzaee F, Samadyar N (2019) Numerical solution based on two-dimensional orthonormal bernstein polynomials for solving some classes of two-dimensional nonlinear integral equations of fractional order. Appl Math Comput 344:191–203
Mirzaee F, Alipour S, Samadyar N (2019) A numerical approach for solving weakly singular partial integro-differential equations via two-dimensional-orthonormal bernstein polynomials with the convergence analysis. Numer Methods Part Differ Equ 35(2):615–637
Mirzaee F, Alipour S (2018) Approximate solution of nonlinear quadratic integral equations of fractional order via piecewise linear functions. J Comput Appl Math 331:217–227
Wang J, Liu G (2002) A point interpolation meshless method based on radial basis functions. Int J Numer Meth Eng 54(11):1623–1648
Liu Y, Hon Y, Liew K (2006) A meshfree hermite-type radial point interpolation method for kirchhoff plate problems. Int J Numer Meth Eng 66(7):1153–1178
Cui X, Liu G, Li G (2011) A smoothed hermite radial point interpolation method for thin plate analysis. Arch Appl Mech 81(1):1–18
Liu G, Kee BB, Chun L (2006) A stabilized least-squares radial point collocation method (ls-rpcm) for adaptive analysis. Comput Methods Appl Mech Eng 195(37–40):4843–4861
Kee BB, Liu G, Lu C (2007) A regularized least-squares radial point collocation method (rls-rpcm) for adaptive analysis. Comput Mech 40(5):837–853
Kee BB, Liu G, Zhang G, Lu C (2008) A residual based error estimator using radial basis functions. Finite Elem Anal Des 44(9–10):631–645
El Seblani Y, Shivanian E (2019) Boundary value identification of inverse cauchy problems in arbitrary plane domain through meshless radial point hermite interpolation. Eng Comput. https://doi.org/10.1007/s00366-019-00755-8
Fasshauer GE (2007) Meshfree approximation methods with MATLAB, vol 6. World Scientific, Singapore
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The author is grateful to two anonymous reviewers for carefully reading this paper and for their comments and suggestions which have improved the paper.
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Seblani, Y.E., Shivanian, E. New insight into meshless radial point Hermite interpolation through direct and inverse 2-D reaction–diffusion equation . Engineering with Computers 37, 3605–3613 (2021). https://doi.org/10.1007/s00366-020-01020-z
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DOI: https://doi.org/10.1007/s00366-020-01020-z