Ritz-Galerkin method for solving a parabolic equation with non-local and time-dependent boundary conditions
The paper is devoted to the investigation of a parabolic partial differential equation with non‐local and time‐dependent boundary conditions arising from ductal carcinoma in situ model. Approximation solution of the present problem is implemented by the Ritz–Galerkin method, which is a first attempt...
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| Published in | Mathematical methods in the applied sciences Vol. 39; no. 5; pp. 1241 - 1253 |
|---|---|
| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
Freiburg
Blackwell Publishing Ltd
01.04.2016
Wiley Subscription Services, Inc |
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| Online Access | Get full text |
| ISSN | 0170-4214 1099-1476 |
| DOI | 10.1002/mma.3568 |
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| Abstract | The paper is devoted to the investigation of a parabolic partial differential equation with non‐local and time‐dependent boundary conditions arising from ductal carcinoma in situ model. Approximation solution of the present problem is implemented by the Ritz–Galerkin method, which is a first attempt at tackling parabolic equation with such non‐classical boundary conditions. In the process of dealing with the difficulty caused by integral term in non‐local boundary condition, we use a trick of introducing the transition function G(x,t) to convert non‐local boundary to another non‐classical boundary, which can be handled with the Ritz–Galerkin method. Illustrative examples are included to demonstrate the validity and applicability of the technique in this paper. Copyright © 2015 John Wiley & Sons, Ltd. |
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| AbstractList | The paper is devoted to the investigation of a parabolic partial differential equation with non-local and time-dependent boundary conditions arising from ductal carcinoma in situ model. Approximation solution of the present problem is implemented by the Ritz-Galerkin method, which is a first attempt at tackling parabolic equation with such non-classical boundary conditions. In the process of dealing with the difficulty caused by integral term in non-local boundary condition, we use a trick of introducing the transition function G(x,t) to convert non-local boundary to another non-classical boundary, which can be handled with the Ritz-Galerkin method. Illustrative examples are included to demonstrate the validity and applicability of the technique in this paper. Copyright © 2015 John Wiley & Sons, Ltd. The paper is devoted to the investigation of a parabolic partial differential equation with non‐local and time‐dependent boundary conditions arising from ductal carcinoma in situ model. Approximation solution of the present problem is implemented by the Ritz–Galerkin method, which is a first attempt at tackling parabolic equation with such non‐classical boundary conditions. In the process of dealing with the difficulty caused by integral term in non‐local boundary condition, we use a trick of introducing the transition function G ( x , t ) to convert non‐local boundary to another non‐classical boundary, which can be handled with the Ritz–Galerkin method. Illustrative examples are included to demonstrate the validity and applicability of the technique in this paper. Copyright © 2015 John Wiley & Sons, Ltd. The paper is devoted to the investigation of a parabolic partial differential equation with non-local and time-dependent boundary conditions arising from ductal carcinoma in situ model. Approximation solution of the present problem is implemented by the Ritz-Galerkin method, which is a first attempt at tackling parabolic equation with such non-classical boundary conditions. In the process of dealing with the difficulty caused by integral term in non-local boundary condition, we use a trick of introducing the transition function G(x,t) to convert non-local boundary to another non-classical boundary, which can be handled with the Ritz-Galerkin method. Illustrative examples are included to demonstrate the validity and applicability of the technique in this paper. |
| Author | Xu, Yongzhi Li, Heng Zhou, Jian-Rong |
| Author_xml | – sequence: 1 givenname: Jian-Rong surname: Zhou fullname: Zhou, Jian-Rong organization: Department of Mathematics, Foshan University, Guangdong, Foshan 528000, China – sequence: 2 givenname: Heng surname: Li fullname: Li, Heng organization: Department of Mathematics, University of Louisville, Louisville, KY 40292, USA – sequence: 3 givenname: Yongzhi surname: Xu fullname: Xu, Yongzhi email: Correspondence to: Yongzhi Xu, Department of Mathematics, University of Louisville, Louisville, KY 40292, United States., ysxu0001@louisville.edu organization: Department of Mathematics, University of Louisville, Louisville, KY 40292, USA |
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| Cites_doi | 10.1142/9789812701732_0033 10.1080/17415970802583911 10.1108/09615531211188784 10.1016/0025-5564(94)00117-3 10.1155/S1085337503210010 10.1080/17476930600667825 10.1016/0020-7225(90)90086-X 10.1016/j.amc.2005.08.011 10.1016/S0167-8396(98)00009-0 10.1016/j.cam.2006.05.002 10.1016/j.apnum.2004.02.002 10.1142/2476 10.1016/0025-5564(96)00023-5 10.1007/978-0-8176-8119-7 10.1088/0305-4470/35/46/303 10.1007/s002850050149 10.3934/dcdsb.2004.4.337 10.1016/j.mcm.2008.02.014 10.1080/00036810600835243 10.1016/j.chaos.2005.11.010 10.1016/0020-7225(93)90010-R 10.1016/j.camwa.2013.04.005 10.1080/17415977.2012.701627 10.1093/imammb/20.3.277 10.1002/num.20071 10.1016/j.na.2007.07.008 10.1002/sapm1972514317 10.1016/j.matcom.2005.10.001 10.1016/j.cam.2009.01.012 10.1016/S0377-0427(00)00376-9 10.1002/num.20019 10.1016/0020-7225(90)90087-Y 10.1002/num.20297 |
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| References | Ang W. Numerical solution of a non-classical parabolic problem: anintegro-differential approach. Applied Mathematics and Computation 2006; 175(2): 969-979. Friedman A, Reitich F. Analysis of a mathematical model for the growth of tumours. Journal of Mathematical Biology 1999; 38(3): 262-284. Bouziani A, Merazga N, Benamira S. Galerkin method applied to a parabolic evolution problem with nonlocal boundary conditions. Nonlinear Analysis: Theory, Methods & Applications 2008; 69(5): 1515-1524. Adam JA, Bellomo N. A Survey of Models for tumour-Immune System Dynamics Birkhauser: Boston, 1997. Dehghan M. On the solution of an initial-boundary value problem that combines Neumann and integral condition for the wave equation. Numerical Methods for Partial Differential Equations 2005; 21: 24-40. Cannon JR, Matheson AL. A numerical procedure for diffusion subject to the specification of mass. International Journal of Engineering Science 1993; 31(3): 347-355. Datta KB, Mohan BM. Orthogonal Functions in Systems and Control World Scientific: River Edge, NJ, USA, 1995. Xu Y, Gilbert R. Some inverse problems raised from a mathematical model of ductal carcinoma in situ. Mathematical and Computer Modelling 2009; 49(3): 814-828. Behiry SH. A wavelet-Galerkin method for inhomogeneous diffusion equations subject to mass specification. Journal of Physics A: Mathematical and General 2002; 35(46): 9745-9753. Dehghan M. A computational study of the one-dimensional parabolic equation subject to nonclassical boundary specifications. Numerical Methods for Partial Differential Equations 2006; 22(1): 220-257. Dehghan M. The one-dimensional heat eqation subject to a boundary integral specification. Chaos, Solitons & Fractals 2007; 32: 661-675. Cannon JR, Lin Y. A Galerkin procedure for diffusion equation with boundary integral conditions. International Journal of Engineering Science 1990; 28(7): 579-587. Burton AC. 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An implicit finite difference scheme for the diffusion equation subject to the specification of mass. International Journal of Engineering Science 1990; 28: 573-578. Dehghan M, Tatari M. Use of radial basis functions for solving the second-order parabolic equation with nonlocal boundary conditions. Numerical Methods for Partial Differential Equations 2008; 24(3): 924-938. Rashedi K, Adibi H, Dehghan M. Application of the Ritz-Galerkin method for recovering the spacewise-coefficients in the wave equation. Computers & Mathematicas with Applications 2013; 65(12): 1990-2008. Bhatti MI, Bracken P. Solution of differential equations in a Bernstein polynomial basis. Journal of Computational and Applied Mathematics 2007; 205: 272-280. Greenspan HP. Models for the growth of tumour by diffusion. Studies in Applied Mathematics 1972; 52(4): 317-340. Xu Y. A free boundary problem model of ductal carcinoma in situ. Discrete and Continuous Dynamical Systems-Series B. 2004; 4(1): 337-348. Yousefi SA. Finding a control parameter in a one-dimensional parabolic inverse problem by using the Bernstein Galerkin method. Inverse Problems in Science and Engineering 2009; 17(6): 821-828. Farouki RT. Legendre-Bernstein basis transformations. Journal of Computational and Applied Mathematics 2000; 119(1- 2): 145-160. Dehghan M, Yousefi SA, Rashedi K. Ritz-Galerkin method for solving an inverse heat conduction problem with a nonlinear source term via Bernstein multi-scaling functions and cubic B-spline functions. Inverse Problems in Science and Engineering 2013; 21(3): 500-523. Byrne HM, Chaplain MAJ. Growth of onnecrotic tumours in the presence and absence of inhibitors. Mathematical Biosciences 1995; 130(2): 151-181. Yousefi SA, Barikbin Z. Ritz-Galerkin method with Bernstein polynomial basis for finding the product solution form of heat equation with non-classic boundary conditions. International Journal of Numerical Methods for Heat & Fluid Flow 2012; 22(1): 39-48. Xu Y. A free boundary problem of diffusion equation with integral condition. Applicable Analysis 2006; 85(9): 1143-1152. Li YM, Zhang XY. Basis conversion among Bezier, Tchebyshev and Legendre. Computer Aided Geometric Design 1998; 15: 637-642. Xu Y. A free boundary problem of parabolic complex equation. Complex Variables and Elliptic Equations 2006; 51(8- 11): 945-951. Zhou Y, Cui M, Lin Y. Numerical algorithm for parabolic problems with non-classical conditions. Journal of Computational and Applied Mathematics 2009; 230(2): 770-780. 2006; 71 2006; 51 2007; 205 2013; 21 2013; 65 2002; 35 2000; 119 2004; 4 1997 2006; 175 2005; 21 1995 2005 2009; 230 2007; 32 1995; 130 1966; 30 2009; 49 2003; 10 1998; 15 2006; 85 1990; 28 2006; 22 1999; 38 1993; 31 2008; 69 2005; 52 2008; 24 1972; 52 1996; 135 2012; 22 2003; 20 2009; 17 e_1_2_9_30_1 e_1_2_9_31_1 e_1_2_9_11_1 e_1_2_9_34_1 e_1_2_9_10_1 e_1_2_9_35_1 e_1_2_9_13_1 e_1_2_9_32_1 e_1_2_9_12_1 e_1_2_9_33_1 Xu Y (e_1_2_9_14_1) 2005 e_1_2_9_15_1 e_1_2_9_17_1 e_1_2_9_36_1 e_1_2_9_16_1 e_1_2_9_19_1 e_1_2_9_18_1 e_1_2_9_20_1 e_1_2_9_22_1 e_1_2_9_21_1 e_1_2_9_24_1 e_1_2_9_23_1 e_1_2_9_8_1 e_1_2_9_7_1 e_1_2_9_5_1 e_1_2_9_4_1 e_1_2_9_3_1 e_1_2_9_2_1 Burton AC (e_1_2_9_6_1) 1966; 30 e_1_2_9_9_1 e_1_2_9_26_1 e_1_2_9_25_1 e_1_2_9_28_1 e_1_2_9_27_1 e_1_2_9_29_1 |
| References_xml | – reference: Dehghan M. On the solution of an initial-boundary value problem that combines Neumann and integral condition for the wave equation. Numerical Methods for Partial Differential Equations 2005; 21: 24-40. – reference: Dehghan M, Yousefi SA, Rashedi K. Ritz-Galerkin method for solving an inverse heat conduction problem with a nonlinear source term via Bernstein multi-scaling functions and cubic B-spline functions. Inverse Problems in Science and Engineering 2013; 21(3): 500-523. – reference: Behiry SH. A wavelet-Galerkin method for inhomogeneous diffusion equations subject to mass specification. Journal of Physics A: Mathematical and General 2002; 35(46): 9745-9753. – reference: Zhou Y, Cui M, Lin Y. Numerical algorithm for parabolic problems with non-classical conditions. Journal of Computational and Applied Mathematics 2009; 230(2): 770-780. – reference: Byrne HM, Chaplain MAJ. Growth of onnecrotic tumours in the presence and absence of inhibitors. Mathematical Biosciences 1995; 130(2): 151-181. – reference: Rashedi K, Adibi H, Dehghan M. Application of the Ritz-Galerkin method for recovering the spacewise-coefficients in the wave equation. Computers & Mathematicas with Applications 2013; 65(12): 1990-2008. – reference: Ang W. Numerical solution of a non-classical parabolic problem: anintegro-differential approach. Applied Mathematics and Computation 2006; 175(2): 969-979. – reference: Dehghan M. A computational study of the one-dimensional parabolic equation subject to nonclassical boundary specifications. Numerical Methods for Partial Differential Equations 2006; 22(1): 220-257. – reference: Adam JA, Bellomo N. A Survey of Models for tumour-Immune System Dynamics Birkhauser: Boston, 1997. – reference: Xu Y. A free boundary problem of diffusion equation with integral condition. Applicable Analysis 2006; 85(9): 1143-1152. – reference: Bhatti MI, Bracken P. Solution of differential equations in a Bernstein polynomial basis. Journal of Computational and Applied Mathematics 2007; 205: 272-280. – reference: Datta KB, Mohan BM. Orthogonal Functions in Systems and Control World Scientific: River Edge, NJ, USA, 1995. – reference: Dehghan M, Tatari M. Use of radial basis functions for solving the second-order parabolic equation with nonlocal boundary conditions. Numerical Methods for Partial Differential Equations 2008; 24(3): 924-938. – reference: Byrne HM, Chaplain MAJ. Growth of necrotic tumours in the presence and absence of inhibitors. Mathematical Biosciences 1996; 135(2): 187-216. – reference: Cannon JR, Lin Y. A Galerkin procedure for diffusion equation with boundary integral conditions. International Journal of Engineering Science 1990; 28(7): 579-587. – reference: Farouki RT. Legendre-Bernstein basis transformations. Journal of Computational and Applied Mathematics 2000; 119(1- 2): 145-160. – reference: Xu Y. A free boundary problem of parabolic complex equation. Complex Variables and Elliptic Equations 2006; 51(8- 11): 945-951. – reference: Bouziani A, Merazga N, Benamira S. Galerkin method applied to a parabolic evolution problem with nonlocal boundary conditions. Nonlinear Analysis: Theory, Methods & Applications 2008; 69(5): 1515-1524. – reference: Franks SJ, Byrne HM, Mudhar HS, Underwood JC. Lewis CE mathematical modelling of comedo ductal carcinoma in situ of the breast. Mathematical Medicine and Biology 2003; 20(3): 277-308. – reference: Friedman A, Reitich F. Analysis of a mathematical model for the growth of tumours. Journal of Mathematical Biology 1999; 38(3): 262-284. – reference: Yousefi SA, Barikbin Z. Ritz-Galerkin method with Bernstein polynomial basis for finding the product solution form of heat equation with non-classic boundary conditions. International Journal of Numerical Methods for Heat & Fluid Flow 2012; 22(1): 39-48. – reference: Dehghan M. The one-dimensional heat eqation subject to a boundary integral specification. Chaos, Solitons & Fractals 2007; 32: 661-675. – reference: Xu Y, Gilbert R. Some inverse problems raised from a mathematical model of ductal carcinoma in situ. Mathematical and Computer Modelling 2009; 49(3): 814-828. – reference: Burton AC. Rate of growth of solid tumours as a problem of diffusion growth. Journal of Mathematical Biology 1966; 30(2): 157-176. – reference: Li YM, Zhang XY. Basis conversion among Bezier, Tchebyshev and Legendre. Computer Aided Geometric Design 1998; 15: 637-642. – reference: Yousefi SA. Finding a control parameter in a one-dimensional parabolic inverse problem by using the Bernstein Galerkin method. Inverse Problems in Science and Engineering 2009; 17(6): 821-828. – reference: Xu Y. A free boundary problem model of ductal carcinoma in situ. Discrete and Continuous Dynamical Systems-Series B. 2004; 4(1): 337-348. – reference: Cannon JR, Matheson AL. A numerical procedure for diffusion subject to the specification of mass. International Journal of Engineering Science 1993; 31(3): 347-355. – reference: Greenspan HP. Models for the growth of tumour by diffusion. Studies in Applied Mathematics 1972; 52(4): 317-340. – reference: Dehghan M. Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices Mathematics and Computers in Simulation. 2006; 71: 16-30. – reference: Dehghan M. Efficient techniques for the second-order parabolic equation subject to nonlocal specifications. Applied Numerical Mathematics 2005; 52(1): 39-62. – reference: Bouziani A. On the weak solution of a three-point boundary value problem for a class of parabolic equations with energy specification. Abstract and Applied Analysis 2003; 10: 573-589. – reference: Cannon JR, Lin Y, Wang S. An implicit finite difference scheme for the diffusion equation subject to the specification of mass. International Journal of Engineering Science 1990; 28: 573-578. – volume: 52 start-page: 39 issue: 1 year: 2005 end-page: 62 article-title: Efficient techniques for the second‐order parabolic equation subject to nonlocal specifications publication-title: Applied Numerical Mathematics – volume: 28 start-page: 579 issue: 7 year: 1990 end-page: 587 article-title: A Galerkin procedure for diffusion equation with boundary integral conditions publication-title: International Journal of Engineering Science – year: 2005 – volume: 85 start-page: 1143 issue: 9 year: 2006 end-page: 1152 article-title: A free boundary problem of diffusion equation with integral condition publication-title: Applicable Analysis – volume: 51 start-page: 945 issue: 8– 11 year: 2006 end-page: 951 article-title: A free boundary problem of parabolic complex equation publication-title: Complex Variables and Elliptic Equations – volume: 135 start-page: 187 issue: 2 year: 1996 end-page: 216 article-title: Growth of necrotic tumours in the presence and absence of inhibitors publication-title: Mathematical Biosciences – start-page: 1429 year: 2005 end-page: 1438 – volume: 175 start-page: 969 issue: 2 year: 2006 end-page: 979 article-title: Numerical solution of a non‐classical parabolic problem: anintegro‐differential approach publication-title: Applied Mathematics and Computation – volume: 4 start-page: 337 issue: 1 year: 2004 end-page: 348 article-title: A free boundary problem model of ductal carcinoma in situ publication-title: Discrete and Continuous Dynamical Systems‐Series B – volume: 35 start-page: 9745 issue: 46 year: 2002 end-page: 9753 article-title: A wavelet‐Galerkin method for inhomogeneous diffusion equations subject to mass specification publication-title: Journal of Physics A: Mathematical and General – volume: 31 start-page: 347 issue: 3 year: 1993 end-page: 355 article-title: A numerical procedure for diffusion subject to the specification of mass publication-title: International Journal of Engineering Science – volume: 130 start-page: 151 issue: 2 year: 1995 end-page: 181 article-title: Growth of onnecrotic tumours in the presence and absence of inhibitors publication-title: Mathematical Biosciences – volume: 230 start-page: 770 issue: 2 year: 2009 end-page: 780 article-title: Numerical algorithm for parabolic problems with non‐classical conditions publication-title: Journal of Computational and Applied Mathematics – volume: 52 start-page: 317 issue: 4 year: 1972 end-page: 340 article-title: Models for the growth of tumour by diffusion publication-title: Studies in Applied Mathematics – volume: 65 start-page: 1990 issue: 12 year: 2013 end-page: 2008 article-title: Application of the Ritz–Galerkin method for recovering the spacewise‐coefficients in the wave equation publication-title: Computers & Mathematicas with Applications – volume: 69 start-page: 1515 issue: 5 year: 2008 end-page: 1524 article-title: Galerkin method applied to a parabolic evolution problem with nonlocal boundary conditions publication-title: Nonlinear Analysis: Theory, Methods & Applications – volume: 49 start-page: 814 issue: 3 year: 2009 end-page: 828 article-title: Some inverse problems raised from a mathematical model of ductal carcinoma in situ publication-title: Mathematical and Computer Modelling – volume: 21 start-page: 24 year: 2005 end-page: 40 article-title: On the solution of an initial‐boundary value problem that combines Neumann and integral condition for the wave equation publication-title: Numerical Methods for Partial Differential Equations – volume: 71 start-page: 16 year: 2006 end-page: 30 article-title: Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices publication-title: Mathematics and Computers in Simulation – volume: 10 start-page: 573 year: 2003 end-page: 589 article-title: On the weak solution of a three‐point boundary value problem for a class of parabolic equations with energy 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| SubjectTerms | Approximation approximation solution Bernstein polynomial basis Boundaries Boundary conditions Dealing ductal carcinoma in situ (DCIS) model initial boundary value problem Integrals Mathematical analysis Mathematical models non-local boundary condition parabolic equation Partial differential equations Ritz-Galerkin method time-dependent boundary condition |
| Title | Ritz-Galerkin method for solving a parabolic equation with non-local and time-dependent boundary conditions |
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