Fourth-order exponential time differencing Runge–Kutta scheme and local meshless method to investigate unsteady diffusion–convection problems of anisotropic functionally graded materials

References

• M. Abbaszadeh and M. Dehghan, Direct meshless local Petrov–Galerkin (DMLPG) method for time-fractional fourth-order reaction–diffusion problem on complex domains, Comput. Math. Appl. 79(3) (2020), pp. 876–888.
   Web of Science ®Google Scholar
• M. Abbaszadeh, M. Dehghan, and M. Ivan Azis, The meshless local Petrov–Galerkin method based on moving Taylor polynomial approximation to investigate unsteady diffusion–convection problems of anisotropic functionally graded materials related to incompressible flow, Eng. Anal. Bound. Elem. 132 (2021), pp. 469–480.
   Web of Science ®Google Scholar
• J. Amani Rad and K. Parand, Pricing American options under jump-diffusion models using local weak form meshless techniques, Int. J. Comput. Math. 94(8) (2017), pp. 1694–1718.
   Web of Science ®Google Scholar
• E.O. Asante-Asamani, A. Kleefeld, and B.A. Wade, A second-order exponential time differencing scheme for non-linear reaction-diffusion systems with dimensional splitting, J. Comput. Phys. 415 (2020), pp. 109490.
   Web of Science ®Google Scholar
• W.-T. Ang, A boundary element approach for solving plane elastostatic equations of anisotropic functionally graded materials, Numer. Methods. Partial. Differ. Equ. 35(4) (2019), pp. 1396–1411.
   Web of Science ®Google Scholar
• E.O. Asante-Asamani, A.Q.M. Khaliq, and B.A. Wade, A real distinct poles exponential time differencing scheme for reaction–diffusion systems, J. Comput. Appl. Math. 299 (2016), pp. 24–34.
   Web of Science ®Google Scholar
• S.N. Atluri, The Meshless method (MLPG) for domain and BIE discretizations, Tech. Sci. Pres (2004), pp. 680.
   Google Scholar
• S.N. Atluri and S. Shen, The meshless local Petrov-Galerkin (MLPG) method: a simple & less-costly alternative to the finite element and boundary element methods, Comput. Model. Eng. Sci. 3(1) (2002), pp. 11–51.
   Web of Science ®Google Scholar
• S.N. Atluri and T. Zhu, A new meshless local Petrov–Galerkin (MLPG) approach in computational mechanics, Comput. Mech. 22(2) (1998), pp. 117–127.
   Web of Science ®Google Scholar
• J. Bear and A.H.-D. Cheng, Modeling Groundwater Flow and Contaminant Transport, Vol. 23, Springer Science & Business Media, Dordrecht, 2010.
   Google Scholar
• H.P. Bhatt, J. Joshi, and I. Argyros, Fourier spectral high-order time-stepping method for numerical simulation of the multi-dimensional Allen–Cahn equations, Symmetry 13 (2021), pp. 245.
   Google Scholar
• H.P. Bhatt and A.Q.M. Khaliq, Fourth-order compact schemes for the numerical simulation of coupled Burgers' equation, Comput. Phys. Commun. 200 (2016), pp. 117–138.
   Web of Science ®Google Scholar
• Z.-Q. Cheng and R.C. Batra, Three-dimensional thermoelastic deformations of a functionally graded elliptic plate, Compos. B: Eng. 31 (2000), pp. 97–106.
   Web of Science ®Google Scholar
• S.M. Cox and P.C. Matthews, Exponential time differencing for stiff systems, J. Comput. Phys. 176 (2002), pp. 430–455.
   Web of Science ®Google Scholar
• K. Erbertseder, J. Reichold, B. Flemisch, P. Jenny, and R. Helmig, A coupled discrete/continuum model for describing cancer-therapeutic transport in the lung, PLoS ONE. 7(3) (2012), pp. e31966.
   PubMed Web of Science ®Google Scholar
• S. Guenneau, C. Amra, and D. Veynante, Transformation thermodynamics: cloaking and concentrating heat flux, Opt. Express. 20(7) (2012), pp. 8207–8218.
   PubMed Web of Science ®Google Scholar
• M. Ilati, DMLPG method for specifying a control function in two-dimensional parabolic inverse PDEs, Comput. Math. Appl. 80 (2020), pp. 604–616.
   Web of Science ®Google Scholar
• M. Ilati and M. Dehghan, Remediation of contaminated groundwater by meshless local weak forms, Comput. Math. Appl. 72 (2016), pp. 2408–2416.
   Web of Science ®Google Scholar
• M. Ilati and M. Dehghan, DMLPG method for numerical simulation of soliton collisions in multi-dimensional coupled damped nonlinear Schrödinger system which arises from Bose–Einstein condensates, Appl. Math. Comput. 346 (2019), pp. 244–253.
   Web of Science ®Google Scholar
• M. Ivan Azis, M. Abbaszadeh, and M. Dehghan, An LT-BEM for an unsteady diffusion–convection problem of another class of anisotropic FGMs, Int. J. Comput. Math. 99(3) (2022), pp. 575–590.
   Web of Science ®Google Scholar
• O.S. Iyiola, E.O. Asante-Asamani, K.M. Furati, A.Q.M. Khaliq, and B.A. Wade, Efficient time discretization scheme for nonlinear space fractional reaction–diffusion equations, Int. J. Comput. Math.95 (2018), pp. 1274–1291.
   Web of Science ®Google Scholar
• A.Q.M. Khaliq, J. Martin-Vaquero, B.A. Wade, and M. Yousuf, Smoothing schemes for reaction–diffusion systems with nonsmooth data, J. Comput. Appl. Math. 223 (2009), pp. 374–386.
   Web of Science ®Google Scholar
• U.F. Kocks, C.N. Tomé, and H.-R. Wenk, Texture and Anisotropy: preferred Orientations in Polycrystals and Their Effect on Materials Properties, Cambridge University Press, Cambridge, 1998.
   Google Scholar
• H.-Y. Kuo and T. Chen, Steady and transient Green's functions for anisotropic conduction in an exponentially graded solid, Int. J. Solids Struct. 42(3-4) (2005), pp. 1111–1128.
   Web of Science ®Google Scholar
• Z. Lei, B. Yin, and K. Liew, Bending and vibration behaviors of matrix cracked hybrid laminated plates containing CNTR-FG layers and FRC layers, Compos. Struct. 184 (2018), pp. 314–326.
   Web of Science ®Google Scholar
• X. Li, Symmetric coupling of the meshless Galerkin boundary node and finite element methods for elasticity, Comput. Model. Eng. Sci. 97(6) (2014), pp. 483–507.
   Web of Science ®Google Scholar
• X. Li and J. Zhu, A Galerkin boundary node method for two-dimensional linear elasticity, Comput. Model. Eng. Sci. (CMES) 45(1) (2009), pp. 1.
   Web of Science ®Google Scholar
• X. Liang, A.Q.M. Khaliq, and Y. Xing, Fourth order exponential time differencing method with local discontinuous Galerkin approximation for coupled nonlinear Schrödinger equations, Commun. Comput. Phys. 17 (2015), pp. 510–541.
   Web of Science ®Google Scholar
• K. Liew, Z. Pan, and L. Zhang, An overview of layerwise theories for composite laminates and structures: development, numerical implementation and application, Compos. Struct. 216(3) (2012), pp. 240–259.
   Google Scholar
• D. Mirzaei and R. Schaback, Direct meshless local Petrov–Galerkin (DMLPG) method: a generalized MLS approximation, Appl. Numer. Math. 68 (2013), pp. 73–82.
   Web of Science ®Google Scholar
• D. Mirzaei and R. Schaback, Solving heat conduction problems by the direct meshless local Petrov–Galerkin (DMLPG) method, Numer. Algorithms. 65(2) (2014), pp. 275–291.
   Web of Science ®Google Scholar
• A. Mazzia, G. Pini, and F. Sartoretto, Numerical investigation on direct MLPG for 2D and 3D potential problems, Comput. Model. Eng. Sci. (CMES) 88(3) (2012), pp. 183–209.
   Web of Science ®Google Scholar
• D. Mirzaei, R. Schaback, and M. Dehghan, On generalized moving least squares and diffuse derivatives, IMA J. Numer. Anal. 32(3) (2012), pp. 983–1000.
   Web of Science ®Google Scholar
• E.H. Ooi, E.T. Ooi, and W.T. Ang, Numerical investigation of the meshless radial basis integral equation method for solving 2D anisotropic potential problems, Eng. Anal. Bound. Elem. 53 (2015), pp. 27–39.
   Web of Science ®Google Scholar
• V. Petrova and T. Sadowski, Theoretical modeling and analysis of thermal fracture of semi-infinite functionally graded materials with edge cracks, Meccanica 49(11) (2014), pp. 2603–2615.
   Web of Science ®Google Scholar
• F. Ramirez, P.R. Heyliger, and E. Pan, Static analysis of functionally graded elastic anisotropic plates using a discrete layer approach, Compos. B: Eng. 37 (2006), pp. 10–20.
   Web of Science ®Google Scholar
• H. Ren and Y. Cheng, A new element-free Galerkin method based on improved complex variable moving least-squares approximation for elasticity, Int. J. Comput. Mater. Sci. Eng. 1(1) (2012), pp. 1250011.
   Google Scholar
• H. Ren, Y. Cheng, and W. Zhang, An interpolating boundary element-free method (IBEFM) for elasticity problems, Sci. China Phys. Mech. Astron. 53(4) (2010), pp. 758–766.
   Web of Science ®Google Scholar
• F. Sartoretto, A. Mazzia, and G. Pini, The DMLPG meshless technique for Poisson problems, Appl. Math. Sci. 8(164) (2014), pp. 8233–8250.
   Google Scholar
• A. Shirzadi, Numerical solutions of 3D Cauchy problems of elliptic operators in cylindrical domain using local weak equations and radial basis functions, Int. J. Comput. Math. 94(2) (2017), pp. 252–262.
   Web of Science ®Google Scholar
• J. Sladek, V. Sladek, and Y. Hon, Inverse heat conduction problems by meshless local Petrov–Galerkin method, Eng. Anal. Bound. Elem. 30(8) (2006), pp. 650–661.
   Web of Science ®Google Scholar
• J. Sladek, V. Sladek, and C. Zhang, Stress analysis in anisotropic functionally graded materials by the MLPG method, Eng. Anal. Bound. Elem. 29(6) (2005), pp. 597–609.
   Web of Science ®Google Scholar
• J. Sladek, V. Sladek, Ch. Hellmich, and J. Eberhardsteiner, Heat conduction analysis of 3-D axisymmetric and anisotropic FGM bodies by meshless local Petrov–Galerkin method, Comput. Mech.29(3) (2007), pp. 323–333.
   Google Scholar
• H. Wang and Q.-H. Qin, Boundary integral based graded element for elastic analysis of 2D functionally graded plates, Eur. J. Mech. A. Solids 33 (2012), pp. 12–23.
   Web of Science ®Google Scholar
• M. Yousuf, Efficient L-stable method for parabolic problems with application to pricing American options under stochastic volatility, Appl. Math. Comput. 213 (2009), pp. 121–136.
   Web of Science ®Google Scholar
• Z.F. Yuan and H.M. Yin, Elastic Green's functions for a specific graded material with a quadratic variation of elasticity, J. Appl. Mech. 78(2) (2011), pp. 021021.
   Web of Science ®Google Scholar