Ranjan Kumar Mohanty
Professor and Dean (FMCS)
Department of Mathematics
SOUTH ASIAN UNIVERSITY
Akbar Bhawan, Chanakyapuri
New Delhi – 110021, INDIA
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Room No.: 325

Profile

 Dr. Ranjan Kumar Mohanty received his Ph.D degree from the Department of Mathematics, IIT Delhi on the topic  ‘Numerical Methods for Partial Differential Equations’, in the year 1990. He is already having more than 25 years of teaching and research experience at University of Indore, University of Delhi and South Asian University. Dr. Mohanty is a recipient of prestigious ‘Fulbright-Nehru Senior Research Fellowship 2013’, ‘Commonwealth Fellowship 2000’, and ‘BOYSCAST Fellowship 1996’. He has worked as a post-doctoral fellow at Louisiana Tech University (USA), Nottingham Trent University (UK) and Loughborough University (UK), under the aforesaid fellowship program. He has also visited Technical University Dresden (Germany), University of Strathclyde (Scotland) and  Nottingham Trent University (England)  under INSA-DFG and INSA-Royal Society bilateral exchange program to carry out advanced research work in his field of research. Dr. Mohanty has already published more than 170 research papers in various international academic Journals of repute. To his credit he has successfully guided 22  Ph.D students. He has delivered several lectures as a resource person and invited speakers at different places in India and abroad. He is a regular reviewer of many national and international Journals. Dr Mohanty has been elected as a Fellow of National Academy of Sciencesis (FNASc), Allahabad in the year 2012. 

Qualifications

  • PhD in Mathematics from Indian Institute of Technology Delhi , in 1990
  • M.Sc. in Mathematics from Utkal University, Bhubaneswar , in 1984

Awards & Honours

  • Elected, Fellow of the National Academy of Sciences, India (FNASc) 2012.
  • DFG Fellowship Awardee, sponsored by The Deutsche Forschungsgemeinschaft (The German Research Foundation) under INSA-DFG Bilateral Exchange Programme.Research in ‘High Order Discretizations in Partial Differential Equations and Singular Perturbation’ at the Institute of Numerical Mathematics, Technical University Dresden, Germany. (01-06-2008 to 31-08-2008).
  • Royal Society Fellowship Awardee, sponsored by The Royal Society of London under INSA-Royal Society Bilateral Exchange Program.Research in ‘Recent Parallel Computational Methods for Differential Equations’ at The Nottingham Trent University, U.K. (24-03-2004 to 23-06-2004).
  • Commonwealth Fellowship Awardee, sponsored by The British Council, London. Research in ‘Applied Numerical Analysis’ at The Nottingham Trent University, U.K. (01-10-2000 to 30-09-2001).
  • BOYSCAST Fellowship Awardee, sponsored by the Department of Science and Technology, Government of India.Research in ‘Parallel Computational Methods’ at Loughborough University, U.K. (01-05-1996 to 30-04-1997).
  • Fulbright Nehru Senior Research Fellowship 2013 Awardee, sponsored by The United States-India Educational Foundation (USIEF), New Delhi.Research in ‘Computational Fluid Dynamics’ at the Louisiana Tech University, USA . (01-11-2013 to 30-04-2014)
  • Royal Society Fellowship Awardee, sponsored by The Royal Society of Edinburgh and INSA, New Delhi. Research in ‘Stability of Numerical Schemes for Second Order Initial Value Problems on Variable Mesh’ at University of Strathclyde, Glasgow, Scotland, U.K. (01-10-2014 to 28-10-2014).

Recent Publications

  • R.K. MOHANTY and SEAN  McKEE, “A  New Two-step Stable High Accuracy Implicit Method for General Second Order Nonlinear Initial Value Problems on a Variable Mesh”, Numerical Algorithms, To appear (2016).
  • JYOTI  TALWAR, R.K. MOHANTY  and  SWARN  SINGH, “A New Algorithm Based on Spline in Tension Approximation for 1D Quasilinear Parabolic Equations on a Variable Mesh”, International Journal of Computer Mathematics, To appear (2016).
  • R.K. MOHANTY  and  RAVINDRA  KUMAR, “A New Numerical Method Based on Non-Polynomial Spline in Tension Approximations for 1D Quasilinear Hyperbolic Equations on a Variable Mesh”, Differential Equations and Dynamical Systems,  To appear (2016).
  • Efficient Algorithms for Fourth and Sixth Order Two Point Nonlinear Boundary Value Problems using Non polynomial Spline Approximations on a Geometric Mesh. NAVNIT JHA, R.K. MOHANTY and VINOD CHAUHAN (2016)
  • R.K. MOHANTY and DEEPTI  KAUR, “High Accuracy Implicit Variable Mesh Methods for Numerical Study of Special Types of Fourth Order Non-linear Parabolic Equations”, Applied Mathematics and Computation, Vol. 273,  pp. 678-696  (2016).
  • M.K. JAIN, SACHIN SHARMA and R.K. MOHANTY, High Accuracy Variable Mesh Method for Nonlinear Two-Point Boundary Value Problems in Divergence Form, Applied Mathematics and Computation, Vol. 273,  pp. 678-696  (2016).
  • R.K. MOHANTY,  WEIZHONG  DAI  and FEI  HAN, “A  New High Accuracy Method for Two-dimensional  Biharmonic Equation with Nonlinear Third Derivative Terms: Application to Navier-Stokes Equations of Motion”, International Journal of Computer Mathematics,  Vol. 92, pp. 1574-1590 (2015).
  • R.K. MOHANTY  and JYOTI  TALWAR, “A New Compact Alternating Group Explicit Iteration Method for the Solution of Nonlinear Time-dependent Viscous Burgers' Equation”, Numerical Analysis and Applications,  Vol. 8, pp. 314-328 (2015).
  • ·         R.K. MOHANTY  and  JYOTI  TALWAR, “A New Coupled Reduced Alternating Group Explicit Method for Non-linear Singular Two Point Boundary Value Problems on a Variable Mesh”, Sibirskii Zhurnal Vychislitel’noi Matematiki  (Numerical Analysis and Applications), Vol. 8, pp. 55-67 (2015).
  • R.K. MOHANTY  and  JYOTI  TALWAR, “A New Coupled Reduced Alternating Group Explicit Method for Non-linear Singular Two Point Boundary Value Problems on a Variable Mesh”, Sibirskii Zhurnal Vychislitel’noi Matematiki  (Numerical Analysis and Applications), Vol. 8, pp. 55-67 (2015).
  • R.K. MOHANTY  and  NIKITA  SETIA, “A New High Accuracy Two-level Implicit Off-step Discretization for the System of Three Space Dimensional Quasi-linear Parabolic Partial Differential Equations”, Computers and Mathematics with Applications,  Vol. 69, pp. 1096-1113 (2015).
  • JYOTI  TALWAR, R.K. MOHANTY  and  SWARN  SINGH, “A New Spline in Compression Approximation for One Space Dimensional Quasilinear Parabolic Equations on a Variable Mesh”, Applied Mathematics and Computations, Vol. 260, pp. 82-96  (2015).
  • R.K. MOHANTY, NAVNIT  JHA  and  RAVINDRA  KUMAR, “A New Variable Mesh Method Based on Non-Polynomial Spline in Compression Approximations for 1D Quasilinear Hyperbolic Equations”,  Advances in Difference Equations, Vol. 2015, ID: 337 (2015).
  • JYOTI  TALWAR  and  R.K. MOHANTY, “A Single Sweep AGE Algorithm   based on Off-step Discretization for the Solution of Viscous Burgers' Equation on a Variable Mesh”, Mathematics in Computer Science, Vol. 09, pp. 85-103 (2015).
  • R.K. MOHANTY,  WEIZHONG  DAI  and FEI  HAN, “Compact Operator Method of Accuracy Two in Time  and Four in Space for the Numerical Solution of Coupled Viscous Burgers' Equations”, Applied Mathematics and Computations, Vol. 256, pp. 381-393 (2015).
  • JYOTI  TALWAR  and  R.K. MOHANTY, “Coupled Reduced Alternating Group Explicit Algorithm for Third Order Cubic Spline Method on a Non-Uniform Mesh for Nonlinear Singular Two Point Boundary Value Problems”, Proceedings of the National Academy of Sciences, India Section A: Physical Sciences,  Vol. 85, pp. 71-81 (2015).
  • R.K. MOHANTY and SEAN  McKEE, “On the Stability of Two New Two-step Explicit Methods for the Numerical Integration of Second Order Initial Value Problem on a Variable Mesh”, Applied Mathematics Letters, Vol. 45, pp. 31-36 (2015).
  • R.K. MOHANTY,  WEIZHONG  DAI  and DON  LIU, “Operator Compact Method of Accuracy Two in Time and Four in Space for the Solution of Time Dependent Burgers-Huxley Equation”, Numerical Algorithms, Vol. 70, pp. 591-605 (2015).
  • JYOTI  TALWAR  and  R.K. MOHANTY,  “Spline in Tension method for Non-linear Two Point Boundary Value Problems on a Geometric Mesh”, МАТЕМАТИЧЕСКОЕ МОДЕЛИРОВАНИЕ (Mathematical Models and Computer Simulations), Vol. 27, pp. 33-48 (2015).
  • R.K. MOHANTY  and  NIKITA  SETIA, “A  New Compact Off-step Discretization for the System of 2D Quasi-linear Elliptic Equations on Unequal Mesh”,  Computational Mathematics and Modeling, Vol. 25, pp.381-403 (2014).
  • R.K. MOHANTY  and  RAVINDRA  KUMAR, “A New Fast Algorithm Based on Half-step  Discretization for  One Space Dimensional Quasilinear Hyperbolic Equations”, Applied Mathematics and Computations, Vol. 244, pp.624-641 (2014).
  • VENU  GOPAL, R.K. MOHANTY  and  L.M. SAHA, “A New High Accuracy Non-polynomial Tension Spline Method for the Solution of One Dimensional Wave Equation in Polar Co-ordinates”,  Journal of the Egyptian Mathematical Society, Vol. 22, pp. 280-285 (2014).
  • R.K. MOHANTY, SURUCHI  SINGH and SWARN  SINGH, “A New High Order Space Derivative Discretization for 3D Quasi-linear Hyperbolic Partial Differential Equations”, Applied Mathematics and Computations, Vol. 232, pp.529-541 (2014). 
  • JYOTI  TALWAR  and  R.K. MOHANTY, “A New Modified Group Explicit Iterative Method for the Numerical Solution of Time Dependent Viscous Burgers’ Equation, International Journal of Modelling, Simulation, and Scientific Computing, Vol. 05, ID: 1350029 (2014).
  • R.K. MOHANTY  and  RAVINDRA  KUMAR,  “A Novel Numerical Algorithm of Numerov Type for 2D Quasi-linear Elliptic Boundary Value Problems”,  International Journal for Computational Methods in Engineering Science & Mechanics,  Vol. 15, pp. 473-489  (2014).
  • R.K. MOHANTY and JYOTI  TALWAR, “A Single Sweep AGE Algorithm on a Variable Mesh based on Off-step Discretization for the Solution of Nonlinear Burgers’ Equation”, Journal of Computational Methods in Physics, Vol. 2014, ID: 853198 (2014).
  • R.K. MOHANTY  and  VENU  GOPAL, “High Accuracy Non-polynomial Spline in Compression Method for One-space Dimensional Quasi-linear Hyperbolic Equations with Significant First Order Space Derivative Term”, Applied Mathematics and Computations, Vol. 238, pp.250-265 (2014).
  • R.K. MOHANTY, “New High Accuracy Super Stable Alternating Direction Implicit Methods for Two and Three Dimensional Hyperbolic Damped Wave Equations”, Results in Physics, Vol.04, pp. 156-163 (2014).
  • NAVNIT  JHA  and  R.K. MOHANTY, “Quintic Hyperbolic Nonpolynomial Spline and Finite Difference Method for Nonlinear Second Order Differential Equations and its Application”, Journal of the Egyptian Mathematical Society, Vol. 22, pp. 115-122 (2014).
  • NAVNIT  JHA, R.K. MOHANTY and VINOD  CHAUHAN, “The Convergence of Geometric Mesh Cubic Spline Finite Difference Scheme for Non-linear Higher Order Two Point Boundary Value Problems”, International Journal of Computational Mathematics, Vol. 2014, ID: 527924 (2014). 
  • R.K. MOHANTY  and  VENU  GOPAL, “A Fourth Order Finite Difference Method based on  Spline in Tension Approximation for the Solution of One-space Dimensional Second Order Quasi-linear Hyperbolic Equations” Advances in Difference Equations, Vol. 2013,  ID: 70 (2013).
  • R. K. MOHANTY  and  NIKITA  SETIA, “A New Compact High Order Off-step Discretization for the System of 2D Quasi-linear Elliptic Partial Differential  Equations”, Advances in Difference Equations, Vol. 2013,  ID: 223 (2013).
  • SWARN  SINGH,  SURUCHI  SINGH  and  R.K. MOHANTY, “A New High Accuracy Off-step Discretization for the Solution of 2D Non-linear Triharmonic Equations”, East Asian Journal on Applied Mathematics, Vol. 03, pp. 228-246 (2013).
  • R.K. MOHANTY and NIKITA  SETIA, “A New High Order Compact Off-step Discretization for the System of 3D Quasi-linear Elliptic Partial Differential Equations”, Applied Mathematical Modelling, Vol. 37, pp. 6870-6883 (2013).
  • R.K. MOHANTY and VENU  GOPAL, “A New Off-step  High Order Approximation for the Solution of Three-space Dimensional Nonlinear Wave Equations”, Applied Mathematical Modelling, Vol. 37, pp. 2802-2815 (2013).
  • R.K. MOHANTY, M.K. JAIN and SURUCHI  SINGH, “A New Three-level Implicit Cubic Spline Method for the Solution of 1D Quasi-linear Hyperbolic Equations”, Computational Mathematics and Modeling, Vol. 24, pp. 452-470 (2013).
  • R.K. MOHANTY , NAVNIT  JHA  and VINOD  CHAUHAN, “Arithmetic Average Geometric Mesh Discretizations for Fourth and Sixth Order Nonlinear Two Point Boundary Value Problems”, Neural Parallel & Scientific Computations, Vol. 21, pp. 393-410 (2013).
  • NAVNIT  JHA, R.K. MOHANTY and VINOD  CHAUHAN, “Geometric Mesh Three Point Discretization for Fourth Order Nonlinear Singular Differential Equations in Polar System”,  Advances in Numerical Analysis, Vol. 2013, ID: 614508 (2013). 
  • NAVNIT  JHA, R.K. MOHANTY and VINOD  CHAUHAN, “Geometric Mesh Three Point Discretization for Fourth Order Nonlinear Singular Differential Equations in Polar System”,  Advances in Numerical Analysis, Vol. 2013, ID: 614508 (2013). 
  • R.K. MOHANTY, M.K. JAIN and DEEPIKA  DHALL, “High Accuracy Cubic Spline Approximation for Two Dimensional Quasi-linear Elliptic Boundary Value Problems”, Applied Mathematical Modelling, Vol. 37, pp. 155-171 (2013).
  • VENU  GOPAL, R.K. MOHANTY  and  NAVNIT  JHA, “New Non-polynomial Spline in Compression Method of O(k2+h4) for the solution of 1D Wave Equation in  Polar Co-ordinates”,  Advances in Numerical Analysis,  Vol. 2013, ID: 470480 (2013). 
  • B.N. MISHRA  and  R.K. MOHANTY, “ Single Cell Numerov Type Discretization for 2D Biharmonic and Triharmonic Equations on Unequal Mesh”,  Journal of Mathematical and Computational Science, Vol. 03, pp. 242-253 (2013).
  • JYOTI  TALWAR  and  R.K. MOHANTY, “Spline in Compression Method for Non-linear Two Point Boundary Value Problems on a Geometric Mesh”, Neural Parallel & Scientific Computations, Vol. 21, pp. 553-570 (2013). 
  • R.K. MOHANTY and JYOTI  TALWAR, “SWAGE Algorithm for the Cubic Spline Solution of Nonlinear Viscous
    Burgers’ Equation on a Geometric Mesh”, Results in Physics. Vol. 03, pp. 195-204 (2013).
  • JYOTI  TALWAR and R.K. MOHANTY, “A Class of Numerical Methods for the Solution of Fourth-Order Ordinary Differential equations in Polar Coordinates”, Advances in Numerical Analysis, Vol. 2012, ID: 626419 (2012).
  • R.K. MOHANTY, “A Combind Arithmetic Average Discretization and TAGE Iterative Method for Non-linear Two Point Boundary Value Problems with a Source Function in Integral Form”, Differential Equations and Dynamical Systems, Vol. 20, pp. 423-440 (2012).
  • R.K. MOHANTY  and  JYOTI  TALWAR, “A Combined Approach Using Coupled Reduced Alternating Group Explicit (CRAGE) Algorithm and Sixth Order Off-step Discretization for the Solution of Two Point Nonlinear Boundary Value Problems”, Applied Mathematics and Computations, Vol. 219,  pp. 248-259 (2012).
  • R.K. MOHANTY and NIKITA  SETIA, “A New Fourth Order Compact Off-step Discretization for the System of 2D Non-linear Elliptic Partial Differential Equations”, East Asian Journal of Applied Mathematics, Vol. 02, pp. 59-82 (2012).
  • R.K. MOHANTY and NIKITA  SETIA, “A New High Accuracy Two-level Implicit Off-Step Discretization for the System of Two Space Dimensional Quasi-linear Parabolic Partial Differential Equations”, Applied Mathematics and Computations, Vol. 219, pp. 2680-2697 (2012).
  • NIKITA  SETIA  and  R.K. MOHANTY, “A New High Accuracy Variable Mesh Discretization for the Solution of the System of 2D Non-linear Elliptic BoundaryValue Problems”, Neural Parallel & Scientific Computations, Vol. 20, pp. 415-436 (2012). 
  • R.K. MOHANTY, M.K. JAIN and B.N. MISHRA, “A Novel Numerical Method of  O(h4)  for Three-dimensional Non-linear Triharmonic Equations”, Communications in Computational Physics, vol. 12, pp. 1417-1433 (2012).
  • R.K. MOHANTY and VENU  GOPAL, “An Off-step  Discretization for the Solution of 1-D Mildly Non-linear Wave Equations with Variable Coefficients”, Journal of Advanced Research in Scientific Computing, Vol. 04, No. 02,  pp. 1-13 (2012).
  • R.K. MOHANTY, JYOTI  TALWAR and NOOPUR  KHOSLA, “Application of  TAGE  Iterative Methods for the Solution of Non-linear Two Point Boundary Value Problems with  Linear Mixed Boundary Conditions on a Non-uniform Mesh”, International Journal for Computational Methods in Engineering Science & Mechanics, Vol. 13, pp. 129-134 (2012).
  • R.K. MOHANTY and JYOTI  TALWAR, “Compact  Alternating Group Explicit  Method for the Cubic Spline Solution of Two Point Boundary Value Problems with Significant Nonlinear First Derivative Terms”, Mathematical Sciences, Vol. 6,ID: 58 (2012).
  • R.K. MOHANTY, RAJIVE  KUMAR  and  VIJAY  DAHIYA, “Cubic Spline Iterative Method for Poisson’s Equation in Cylindrical Polar Coordinates”, ISRN  Mathematical Physics, Vol. 2012, ID: 234516 (2012).
  • R.K. MOHANTY, RAJIVE  KUMAR  and  VIJAY  DAHIYA, “Cubic Spline Method for 1D Wave Equation in Polar  Coordinates”, ISRN  Computational Mathematics, Vol. 2012, ID: 302923 (2012).
  • R.K. MOHANTY and VENU  GOPAL, “High Accuracy Arithmetic Average Type Discretization for the Solution of Two-space Dimensional Non-linear Wave Equations”, International Journal of Modeling, Simulation, and Scientific Computing, Vol. 03, ID: 1150005 (2012).
  • SURUCHI  SINGH, SWARN  SINGH and R.K. MOHANTY, “High Accuracy Cubic Spline Approximation on a Geometric Mesh for the Solution of 1D Non-linear Wave Equations”, Journal of Mathematical and Computational Science, Vol. 02, pp. 1126-1143 (2012).
  • R.K. MOHANTY and SURUCHI  SINGH, “High Order Variable Mesh Approximation for the Solution of 1D Non-linear Hyperbolic Equation”, International Journal of Nonlinear Science, Vol. 14, pp. 220-227 (2012).
  • JYOTI  TALWAR  and  R.K. MOHANTY, “Smart Alternating Group Explicit (SMAGE) Method for the Cubic Spline Solution of Non-linear Two Point Boundary Value Problems”, Neural Parallel & Scientific Computations, Vol.20, pp. 399-414 (2012).
  • R.K. MOHANTY, VIJAY  DAHIYA  and  NOOPUR KHOSLA, “Spline in Compression Methods for Singularly Perturbed 1D Parabolic Equations with Singular   Coefficients”,  Journal of Discrete Mathematics, Vol. 02, pp. 70-77 (2012).
  • R.K. MOHANTY, RAJIVE  KUMAR  and  VIJAY  DAHIYA, “Spline in Tension Methods for Singularly Perturbed One  Space Dimensional Parabolic Equations with Singular   Coefficients”, Neural Parallel & Scientific Computations, Vol. 20, pp. 81-92  (2012).
  • R.K. MOHANTY, M.K. JAIN and B.N. MISHRA, “A Compact Discretization of O(h4)  for Two-dimensional Non-linear Triharmonic Equations”, Physica Scripta, Vol. 84, ID: 025002 (2011).
  • R.K. MOHANTY, M.K. JAIN and DEEPIKA  DHALL, “A Cubic Spline Approximation and Application of TAGE Iterative Method for the Solution of Two-Point Boundary Value Problems with Forcing Function in Integral Form”, Applied Mathematical Modelling, Vol. 35, pp. 3036-3047 (2011).
  • CHRISTIAN GROSSMANN, R.K. MOHANTY and HANS-GOERG ROOS, “A Direct Higher Order Discretization in Singular Perturbations via Domain Split - A Computational Approach”, Applied Mathematics and Computations, Vol. 217, pp. 9302- 9312 (2011).
  • R.K. MOHANTY, M.K. JAIN  and  B.N. MISHRA, “A New Fourth Order Difference Approximation for the Solution of Three-dimensional Non-linear Biharmonic Equations using Coupled Approach”, American Journal of Computational Mathematics, Vol. 01, pp. 318-327 (2011).
  • R.K. MOHANTY and  SURUCHI  SINGH, “A New High Order Approximation for the Solution of Two-space Dimensional Quasi-linear Hyperbolic Equations”, Advances in Mathematical Physics, Vol. 2011, ID: 420608 (2011).
  • R.K. MOHANTY and VIJAY  DAHIYA, “An O(k2+kh2+h4) Accurate Two-lelel Implicit Cubic Spline Method for One Space Dimensional Quasi-linear Parabolic Equations”, American Journal of Computational Mathematics, Vol. 01, pp. 11-17 (2011).
  • R.K. MOHANTY  and  SURUCHI  SINGH, “High  Accuracy  Numerov  Type Discretization for the Solution of One Space Dimensional  Non-linear Wave Equations with Variable Coefficients”, Journal of Advanced Research in Scientific Computing,  Vol. 03, No.01,  pp. 53-66 (2011).
  • R.K. MOHANTY and DEEPIKA DHALL, “High Accuracy Arithmetic Average Discretization for Non-linear Two Point Boundary Value Problems with a Source Function in Integral Form”, Applied Mathematics, Vol. 02, pp.1243-1251 (2011).
  • R.K. MOHANTY and VENU  GOPAL, “High Accuracy Cubic Spline Finite Difference Approximation  for the Solution of One-space Dimensional Non-linear Wave Equations”, Applied Mathematics and Computations, Vol. 218, pp. 4234-4244 (2011).
  • NAVNIT  JHA  and  R.K. MOHANTY, “TAGE iterative algorithm and non-polynomial spline basis for the solution of non-linear singular second order ordinary differential equations”, Applied Mathematics and Computations, Vol. 218, pp. 3289-3296 (2011).
  • R.K. MOHANTY, “A New High Accuracy Finite Difference Discretization for the Solution of  2D Non-linear Biharmonic  Equations Using Coupled Approach”, Numerical Methods for Partial Differential Equations, Vol. 26, pp. 931-944 (2010).
  • R.K. MOHANTY, “Application of AGE Method to High Accuracy Variable Mesh Arithmetic Average type Discretization for 1D Non-linear Parabolic Initial Boundary Value Problems”, International Journal for Computational Methods in Engineering Science & Mechanics, Vol. 11, pp. 133-141 (2010).
  • R.K. MOHANTY, “On the Use of AGE Algorithm with a New High Accuracy Numerov type Variable Mesh Discretization for 1D Non-linear Parabolic Equations”, Numerical Algorithms, Vol. 54, pp. 379-393 (2010). 
  • R.K. MOHANTY , “Single Cell Compact Finite Difference Discretizations  of Order Two and Four for Multi-dimensional Triharmonic Problems”, Numerical Methods for Partial Differential Equations, Vol. 26, pp. 1420-1426 (2010).
  • SWARN  SINGH, DINESH  KHATTAR and R.K. MOHANTY, “A New Coupled Approach High Accuracy Numerical Method for the Solution of 2D Non-linear Biharmonic Equations”, Neural Parallel and Scientific Computations, Vol. 17,  pp. 239-256 (2009).
  • DINESH  KHATTAR, SWARN  SINGH and R.K. MOHANTY, “ A New Coupled Approach High Accuracy Numerical Method for the Solution of 3D Non-linear Biharmonic Equations”, Applied Mathematics and Computations, Vol. 215,  pp. 3036-3044 (2009).
  • R.K. MOHANTY, “A Variable Mesh C-SPLAGE Method of Accuracy   for 1D Nonlinear Parabolic Equations”, Applied Mathematics and Computations, Vol. 213, pp. 79-91 (2009).
  • NAVNIT  JHA, R.K. MOHANTY and B.K. MISHRA, “Alternating Group Explicit Iterative Method for Non-linear Singular Fredholm Integro-differential Boundary Value Problems”, International Journal of Computer Mathematics, Vol. 86, pp. 1645-1656 (2009).
  • R.K. MOHANTY and M.K. JAIN, “High  Accuracy  Cubic Spline Alternating  Group  Explicit  Methods  for  1D  Quasi-linear Parabolic  Equations”, International Journal of Computer Mathematics, Vol. 86, pp. 1556-1571 (2009).
  • R.K. MOHANTY, “New Unconditionally Stable Difference Schemes for the Solution of Multi-dimensional Telegraphic Equations”, International Journal of Computer Mathematics, Vol. 86, pp. 2061-2071 (2009).
  • R.K. MOHANTY  and  DEEPIKA  DHALL, “Third Order Accurate Variable Mesh Discretization and Application of TAGE Iterative Method for the Non-linear Two-point Boundary Value Problems with Homogeneous Functions in Integral Form”, Applied Mathematics and Computations, Vol. 215, pp. 2024-2034 (2009).
  • R.K. MOHANTY  and  SWARN  SINGH, “A New High Order Two Level Implicit Discretization for the Solution of Singularly Perturbed Three Space Dimensional Non-linear Parabolic Equations”, International Journal of Numerical Analysis and Modelling, Vol. 05,  pp. 40-54 (2008).
  • R.K. MOHANTY, “A Two-level Implicit Non-uniform Mesh Cubic Spline Method of    for the Parabolic Equation  ”, Neural Parallel and Scientific Computations, Vol. 16, pp. 449-466 (2008).
  • R.K. MOHANTY,  NOOPUR  KHOSLA and A.K. OJHA, “Arithmetic Average Discretization and Two-step BLAGE  Iterative Method for the Solution of Elliptic Partial Differential Equations”, Computing Letters, Vol. 4, pp. 79-90. (2008).
  • R.K. MOHANTY  and  SWARN  SINGH, “A New Two Level Implicit Discretization of O(k2+kh2+h4) for the Solution of Singularly Perturbed Two Space Dimensional Non-linear Parabolic Equations”, Journal of Computational and Applied Mathematics, Vol. 208, pp. 391-403 (2007).
  • R.K. MOHANTY, SAMIR  KARAA  and  U. ARORA, “An  O(k2+kh2+h4) Arithmetic Average Discretization for the solution of 1-D Non-linear Parabolic Equations”, Numerical Methods for Partial Differential Equations, Vol. 23,  pp. 640-651 (2007).
  • R.K. MOHANTY, “An Implicit High Accuracy Variable Mesh Scheme for 1-D Non-linear Singular Parabolic Partial Differential Equations”, Applied Mathematics and Computations, Vol. 186, pp. 219-229 (2007).
  • P.K. PANDEY and  R.K. MOHANTY, “An Order  h4  Numerical Technique for solving Biharmonic Equation”, Neural Parallel and Scientific Computations, Vol. 15, pp. 59-74 (2007).
  • R.K. MOHANTY, “Stability Interval for Explicit Difference Schemes for Multi-dimensional Second Order Hyperbolic Equations with Significant First Order Space Derivative Terms”, Applied Mathematics and Computations, Vol. 190, pp.1683-1690 (2007).
  • R.K. MOHANTY, “ The Smart-BLAGE Algorithm for Singularly Perturbed 2D Elliptic Partial Differential Equations”, Applied Mathematics and Computations,  Vol. 190,  pp. 321-331 (2007).
  • R.K. MOHANTY, “Three-step BLAGE Iterative Method for Two-dimensional Elliptic Boundary Value Problems with Singularity”, International Journal of Computer Mathematics, Vol. 84, pp. 1613 – 1624 (2007).
  • U. ARORA, SAMIR  KARAA and R.K. MOHANTY,  “A  New Stable Variable Mesh Method for 1-D Non-linear Parabolic Partial Differential Equations”, Applied Mathematics and Computations, Vol. 181, pp. 1423-1430 (2006).
  • R.K. MOHANTY, “A Class of Non-Uniform Mesh Three Point Arithmetic Average Discretization for    and the Estimates of   ”, Applied Mathematics and Computations, Vol. 183, pp. 477-485 (2006).
  • R.K. MOHANTY  and  U. ARORA, “A Family of Non-uniform Mesh Tension Spline Methods for Singularly Perturbed Two Point Singular Boundary Value Problems with Significant First Derivatives”, Applied Mathematics and Computations, Vol. 172,  pp. 531-544 (2006).
  • R.K. MOHANTY  and  SWARN  SINGH, “A New Fourth Order Discretization for Singularly Perturbed Two Dimensional Non-linear Elliptic Boundary Value Problems”, Applied Mathematics and Computations, Vol. 175, pp. 1400-1414 (2006).
  • R.K. MOHANTY  and  SWARN  SINGH, “A New Highly Accurate Discretization for Three Dimensional Singularly Perturbed Non-linear Elliptic Partial Differential Equations”, Numerical Methods for Partial Differential Equations, Vol. 22,  pp. 1379-1395 (2006).
  • R.K. MOHANTY, D.J. EVANS  and  NAVNIT  JHA, “A Sixth Order Accurate AGE Iterative Method for Non-linear Singular Two Point Boundary Value Problems”, Journal of Computational Methods in Science and Engineering, Vol. 06, pp. 57-69 (2006).
  • R.K. MOHANTY  and  U. ARORA, “A TAGE Iterative Method for the Solution of Non-linear Singular Two Point Boundary Value Problems using a Sixth Order Discretization”,  Applied Mathematics and Computations, Vol. 180, pp. 538-548 (2006).
  • R.K. MOHANTY  and  NOOPUR  KHOSLA, “Application of TAGE Iterative Algorithms to an Efficient Third Order Arithmetic Average Variable Mesh Discretization for Two Point Non-linear Boundary Value Problems”, Applied Mathematics and Computations,  Vol. 172, pp. 148-162 (2006).
  • R.K. MOHANTY, “Comparison of TAGE and SOR Methods for Variable Mesh Arithmetic Average Discretization for Non-linear Two Point Boundary Value Problems with Mixed Boundary Conditions”, International Journal of Applied Mathematics  &  Statistics, Vol. 06, pp. 87-100 (2006).
  • R.K. MOHANTY, SAMIR  KARAA  and  U. ARORA, “Fourth Order Nine Point Unequal Mesh Discretization for the Solution of 2D Nonlinear Elliptic Partial Differential Equations”, Neural Parallel and Scientific Computations, Vol. 14, pp. 453-470 (2006).
  • R.K. MOHANTY  and  NAVNIT  JHA, “A Class of Variable Mesh Spline in Compression Methods for Singularly Perturbed Two Point Singular Boundary Value Problems”, Applied Mathematics and Computations, Vol. 168,  pp. 704-716 (2005).
  • R.K. MOHANTY, “A Family of Variable Mesh Methods for the Estimates of (du/dr) and the Solution of Non-linear Two Point Boundary Value Problems with Singularity”, Journal of Computational and Applied Mathematics, Vol. 182,  pp. 173-187 (2005).
  • R.K. MOHANTY  and  NOOPUR  KHOSLA, “A Third Order Accurate Variable Mesh TAGE Iterative Method for the Numerical Solution of Two Point Non-linear Singular Boundary Value Problems”, International Journal of  Computer Mathematics, Vol. 82, pp. 1261-1273 (2005).
  • R.K. MOHANTY  and  D.J. EVANS, “Alternating Group Explicit Parallel Algorithms for the Solution of One Space Dimensional Non-linear Singular Parabolic Equations using an  O(k2+h4) Difference Method”, International Journal of  Computer Mathematics, Vol. 82, pp. 203-218 (2005).
  • R.K. MOHANTY, D.J. EVANS  and  NOOPUR  KHOSLA, “An    Non-uniform Mesh Cubic Spline TAGE Method for Non-linear Singular Two-point Boundary Value Problems”, International Journal of  Computer Mathematics, Vol. 82, pp.1125-1139 (2005).
  • R.K. MOHANTY, “An Operator Splitting Technique for an Unconditionally Stable Difference Method for a Linear Three Space Dimensional Hyperbolic Equation with Variable Coefficients”, Applied Mathematics and Computations, Vol. 162, pp. 549-557 (2005).
  • R.K. MOHANTY, “An Unconditionally Stable Finite Difference Formula for a Linear Second Order One Space Dimensional Hyperbolic Equation with Variable Coefficients”, Applied Mathematics and Computations, Vol. 165, pp. 229-236 (2005)
  • R.K. MOHANTY,  D.J. EVANS  and  U. ARORA, “Convergent Spline in Tension Methods for Singularly Perturbed  Two Point  Singular  Boundary Value Problems”, International Journal of  Computer Mathematics, Vol. 82,  pp. 55-66 (2005).
  • R.K. MOHANTY  and  D.J. EVANS, “Highly Accurate Two Parameter  CAGE  Parallel Algorithms for Non-linear Singular Two Point Boundary Value Problems”, International Journal of  Computer Mathematics, Vol. 82,  pp. 433-444 (2005).
  • R.K. MOHANTY  and  SWARN  SINGH, “Non-uniform Mesh Arithmetic Average Discretization for Parabolic Initial Boundary Value Problems”, Neural Parallel and Scientific Computations, Vol. 13, pp. 411-426 (2005).
  • D.J. EVANS  and  R.K. MOHANTY, “On the Application of the SMAGE  Parallel Algorithms on a Non-uniform Mesh for the Solution of Non-linear Two Point Boundary Value Problems with Singularity”, International Journal of  Computer Mathematics, Vol. 82,  pp. 341-353 (2005).
  • R.K. MOHANTY, P.L. SACHDEV  and  NAVNIT JHA, “An O(h4) Accurate Cubic Spline  TAGE Method for Non-linear Singular Two Point Boundary Value Problems”, Applied Mathematics and Computations,  Vol. 158,  pp. 853-868 (2004).
  • R.K. MOHANTY, “An Operator Splitting Method for an Unconditionally Stable Difference Scheme for a Linear Hyperbolic Equation with Variable Coefficients in Two Space Dimensions”, Applied Mathematics and Computations, Vol. 152, pp. 799 – 806 (2004).
  • R.K. MOHANTY, “An Unconditionally Stable Difference Scheme for the One Space Dimensional Linear Hyperbolic Equation”, Applied Mathematics Letters, Vol. 17,  pp. 101-105 (2004).
  • R.K. MOHANTY  and  D.J. EVANS, “Fourth Order Accurate BLAGE Iterative Method for the Solution of Two Dimensional Elliptic Equation in Polar Coordinates”, International Journal of  Computer Mathematics, Vol. 81,  pp. 1537-1548 (2004).
  • R.K. MOHANTY, NAVNIT JHA  and  D.J. EVANS, “Spline in Compression Method for the Numerical Solution of Singularly Perturbed Two Point Singular Boundary Value Problems”, International Journal of  Computer Mathematics, Vol. 81, pp. 615-627 (2004).
  • R.K. MOHANTY  and  D.J. EVANS, “A  Fourth Order Accurate Cubic Spline Alternating Group Explicit Method for Non-linear Singular Two Point Boundary Value Problems”, International Journal of  Computer Mathematics, Vol. 80, pp. 479-492 (2003)
  • R.K. MOHANTY, “A Two-level Implicit Difference Formula of O(k2+h4)  for the Numerical Solution of One Space Dimensional Unsteady Quasi-linear Biharmonic Problem of First Kind”, Journal of Computational Methods in Science and Engineering, Vol. 3,  pp. 193-208 (2003).
  • R.K. MOHANTY, “An Accurate Three Spatial Grid Point Discretization of  O(k2+h4) for the Numerical Solution of One Space Dimensional Unsteady Quasi-linear Biharmonic Problem of Second Kind”, Applied Mathematics and Computations, Vol. 140,  pp. 1-14 (2003).
  • R.K. MOHANTY, D.J. EVANS  and  DINESH  KUMAR, “High Accuracy Difference Formulae for a Fourth Order Quasi-linear Parabolic Initial Boundary Value Problem of First Kind”, International Journal of  Computer Mathematics, Vol. 80, pp. 381-398 (2003).
  • R.K. MOHANTY,  DINESH  KUMAR  and  M.K. JAIN,   “Single Cell Discretization of  O(kh2+h4) for (u/n)  for Three Space Dimensional Mildly Quasi-linear Parabolic Equation”, Numerical Methods for Partial Differential Equations, Vol. 19,  pp. 327-342 (2003).
  • R.K. MOHANTY, P.L. SACHDEV  and  NAVNIT JHA, “TAGE Method for Non-linear Singular Two Point Boundary Value Problem using a Fourth Order Difference Scheme”,  Neural  Parallel and Scientific Computations,  Vol. 11,  pp. 281-297 (2003).
  • R.K. MOHANTY  and  D.J. EVANS, “The Numerical Solution of Fourth Order Mildly Quasi-linear Parabolic Initial Boundary Value Problem of Second Kind”, International Journal of  Computer Mathematics, Vol. 80, pp. 1147-1159 (2003)
  • R.K. MOHANTY  and  URVASHI  ARORA, “ A New Discretization Method of Order Four for the Numerical Solution of One Space Dimensional Second Order Quasi-linear Hyperbolic Equation”, International Journal of Mathematical Education in  Science and  Technology, Vol. 33,  pp. 829-838 (2002).
  • D.J. EVANS  and  R.K. MOHANTY, “Alternating Group Explicit Method for the Numerical Solution of Non-linear Singular Two Point Boundary Value Problems using a Fourth Order Finite Difference Method”, International Journal of  Computer Mathematics, Vol. 79, pp. 1121-1133 (2002).
  • R.K. MOHANTY, M.K. JAIN  and  URVASHI  ARORA, “An Unconditionally Stable  ADI Method for the Linear Hyperbolic Equation in Three Space Dimensions”, International Journal of Computer Mathematics, Vol. 79,  pp. 133-142 (2002).
  • R.K. MOHANTY  and  SHIVANI DEY, “A New Finite Difference Discretization of Order Four for (u/n) for Two Dimensional Quasi-linear Elliptic Boundary Value Problem”, International Journal of  Computer Mathematics, Vol. 76, pp. 505-516 (2001).
  • R.K. MOHANTY and M.K. JAIN, “An Unconditionally Stable  Alternating Direction Implicit Scheme for the Two Space Dimensional Linear Hyperbolic Equation”, Numerical Methods for Partial Differential Equations, Vol. 17,  pp. 684-688 (2001).
  • R.K. MOHANTY, D.J. EVANS  and  P.K. PANDEY, “Block Iterative Methods for the Numerical Solution of Three Dimensional Mildly Non-linear Biharmonic Problems of First Kind”, International Journal of  Computer Mathematics, Vol. 77, pp. 319-332 (2001).
  • R.K. MOHANTY, URVASHI  ARORA  and  M.K. JAIN,  “Fourth Order Approximation for the Three Space Dimensional Certain Mildly Quasi-linear Hyperbolic Equation”, Numerical Methods for Partial Differential Equations, Vol. 17, pp. 277-289 (2001).
  • R.K. MOHANTY, URVASHI  ARORA  and  M.K. JAIN, “Linear Stability Analysis and Fourth Order Approximations at First Time Level for the Two Space Dimensional Mildly Quasi-linear Hyperbolic Equations”, Numerical Methods for Partial Differential Equations, Vol. 17, pp. 607-618 (2001).
  • R.K. MOHANTY, M.K. JAIN  and  DINESH KUMAR, “Single  Cell Discretization of O(kh2+h4) for the Estimates of (u/n) for Two Space Dimensional Quasi-linear Parabolic Equation”, Numerical Methods for Partial Differential Equations, Vol. 17,  pp. 250-261 (2001).
  • R.K. MOHANTY, D.J. EVANS  and  SHIVANI DEY, “Three Point Discretization of Order Four and Six for (du/dx) of the Solution of Non-linear Singular Two Point Boundary Value Problem”, International Journal of  Computer Mathematics, Vol. 78,  pp. 123-139 (2001).
  • R.K. MOHANTY,  “A Fourth Order Finite Difference Method for the General One-Dimensional Non-linear Biharmonic Problems of First Kind”, Journal of Computational and Applied Mathematics, Vol. 114, pp. 275-290 (2000).
  • R.K. MOHANTY,  M.K. JAIN and  DINESH KUMAR, “Single Cell Finite Difference Approximations of O(kh2+h4) for (u/x) for One Space Dimensional Non-linear Parabolic Equations”, Numerical Methods for Partial Differential Equations, Vol. 16,  pp. 408-415 (2000).
  • R.K. MOHANTY  and  SHIVANI DEY, “Single Cell Fourth Order Difference Approximations  for  (u/x),  (u/y)  and  (u/z)  of  the  Three  Dimensional Quasi-linear Elliptic Equation”,  Numerical Methods for Partial Differential Equations, Vol.16, pp. 417-425 (2000).
  • R.K. MOHANTY and D.J. EVANS, “Block Iterative Methods for One Dimensional Non-linear Biharmonic Problems on a Parallel Computer”, Parallel Algorithms and Applications, Vol. 13,  pp. 239-263 (1999).
  • R.K. MOHANTY  and  D.J. EVANS, “ New  Algorithms  for  the  Numerical Solution of One Dimensional Singular Biharmonic Problems of  Second Kind ”, International Journal of  Computer Mathematics, Vol. 73,  pp. 105-124 (1999).
  • D.J. EVANS and R.K. MOHANTY, “Block Iterative Methods for the Numerical Solution of Two Dimensional Non-linear Biharmonic Equations”, International Journal of  Computer Mathematics, Vol. 69, pp. 371-390 (1998).
  • R.K. MOHANTY and P.K. PANDEY, “Families of Accurate Discretizations of Order Two and Four for 3-D Mildly Non-linear Biharmonic Problems of Second Kind”, International Journal of Computer Mathematics,  Vol. 68,  pp. 363-380 (1998).
  • R.K. MOHANTY, M.K. JAIN and K. GEORGE, “Fourth Order Approximations  at First Time Level, Linear Stability Analysis and the Numerical Solution of Multi-Dimensional Second Order Non-linear Hyperbolic Equations in Polar Coordinates”, Journal of Computational and Applied Mathematics, Vol. 93, pp. 1-12 (1998).
  • R.K. MOHANTY,  “High Accuracy Difference Schemes for a Class of Three Space Dimensional Singular Parabolic  Equations  with Variable Coefficients”,  Journal of Computational and Applied Mathematics, Vol. 89, pp. 39-51 (1997).
  • R.K. MOHANTY, “Order h4 Difference Methods for a Class of Singular Two Space Elliptic Boundary Value Problems”, Journal of Computational and Applied  Mathematics, Vol. 81,  pp. 229-247 (1997).
  • R.K. MOHANTY, “An O(k2 + h4)  Finite Difference Method for One Space Burgers Equation in Polar Coordinates”, Numerical Methods for Partial Differential Equations,  Vol. 12, pp. 579-583 (1996).
  • R.K. MOHANTY  and  P.K. PANDEY, “Difference Methods of Order Two and Four for Systems of Mildly Non-linear Biharmonic Problems of the Second Kind in Two Space Dimensions”, Numerical Methods for Partial Differential Equations, Vol. 12, pp. 707-717 (1996).
  • R.K. MOHANTY, M.K. JAIN  and  P.K. PANDEY, “Finite Difference Methods of Order Two and Four for 2-D Non-linear Biharmonic Problems of First Kind”, International Journal of  Computer Mathematics, Vol. 61, pp. 155-163 (1996).
  • R.K. MOHANTY and M.K. JAIN, “High Accuracy Difference Schemes for the System of Two space Non-linear Parabolic Differential Equations with Mixed Derivatives and Variable Coefficients”, Journal of Computational and Applied Mathematics, Vol. 70,  pp. 15-32 (1996).
  • R.K. MOHANTY, M.K. JAIN  and  K. GEORGE, “High Order Difference Schemes for the System of Two Space Second Order Non-linear Hyperbolic Equations with Variable Coefficients”, Journal of Computational and Applied Mathematics, Vol. 70,  pp. 231-243 (1996).
  • R.K. MOHANTY, M.K. JAIN  and  K. GEORGE, “On the Use of High Order Difference Methods for the System of One Space Second Order Non-linear Hyperbolic Equations with Variable Coefficients”, Journal of Computational and Applied Mathematics,  Vol. 72, pp. 421-431 (1996).
  • R.K. MOHANTY and M.K. JAIN, “The Numerical Solution of the System of   3-D Non-linear Elliptic Equation with Mixed Derivatives and Variable Coefficients using Fourth Order Difference Methods”, Numerical Methods for Partial Differential Equations, Vol. 11, pp. 187-197 (1995).
  • R.K. MOHANTY and M.K. JAIN, “Fourth Order Operator Splitting Method for the Three Space Parabolic Equation with Variable Coefficients”, International Journal of  Computer Mathematics, Vol. 50, pp. 55-64 (1994).
  • M.K. JAIN, R.K. JAIN and R.K. MOHANTY,  “Fourth Order Finite Difference Method for 2-D Parabolic Partial Differential Equations with Non-linear First Derivative Terms”, Numerical Methods for Partial Differential Equations,  Vol. 8, pp. 21-31 (1992).
  • M.K. JAIN, R.K. JAIN and R.K. MOHANTY,  “Fourth Order Finite Difference Method for Three Dimensional Elliptic Equations with Non-linear First Derivative Terms”, Numerical Methods for Partial Differential Equations, Vol. 8, pp. 575-591 (1992).
  • R.K. MOHANTY, “Fourth Order Finite Difference Methods for the System of  2-D Non-linear Elliptic Equations with Variable Coefficients”, International Journal of  Computer Mathematics, Vol. 46, pp. 195-206 (1992).
  • M.K. JAIN, R.K. JAIN and R.K. MOHANTY, “High Order Difference Methods for the System of One Dimensional Second Order Hyperbolic Equations with Non-linear First Derivative Terms”, Journal of Mathematical and Physical Sciences, Vol. 26, pp. 401-411 (1992).
  • M.K. JAIN, R.K. JAIN and R.K. MOHANTY, “A Higher Order Difference Method for 3-D Parabolic Partial Differential Equations with Non-linear First Derivative Terms”, International Journal of  Computer Mathematics,  Vol. 38, pp. 101-112 (1991).
  • M.K. JAIN, R.K. JAIN and R.K. MOHANTY,  “Fourth Order Difference Methods for the System of 2-D Non-linear Elliptic Partial Differential Equations”, Numerical Methods for Partial Differential Equations, Vol. 7, pp. 227-244 (1991).
  • M.K. JAIN, R.K. JAIN and R.K. MOHANTY,  “The Numerical Solution of the Two Dimensional Unsteady Navier-Stokes’ Equations using Fourth Order Difference Method”, International Journal of  Computer Mathematics,  Vol. 39, pp. 125-134 (1991).
  • M.K. JAIN, R.K. JAIN and R.K. MOHANTY, “A Fourth Order Difference Method for the One Dimensional General Quasi-linear Parabolic Partial Differential Equation”, Numerical Methods for Partial Differential Equations, Vol. 6, pp. 311-319 (1990).
  • M.K. JAIN, R.K. JAIN and R.K. MOHANTY, “High Order Difference Methods for System of 1-D Non-linear Parabolic Partial Differential Equations”, International Journal of  Computer Mathematics , Vol. 37, pp. 105-112 (1990).
  • M.K. JAIN, R.K. JAIN and R.K. MOHANTY, “A Fourth Order Difference Method for Elliptic Equations with Non-linear First Derivative Terms”, Numerical Methods for Partial Differential Equations, Vol. 5, pp. 87-95 (1989).

Research Interests

  • Computational Fluid Dynamics; Numerical Methods based on Splines, Galerkin, Collocation, Finite Difference, Finite Element, Spectral, Finite Volume, etc. for the solution of quasi-linear PDEs.

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