International Journal of Mathematics Trends and Technology
Volume 68 | Issue 9 | Year 2022 | Article Id. IJMTT-V68I9P509 | DOI : https://doi.org/10.14445/22315373/IJMTT-V68I9P509
| Received | Revised | Accepted | Published |
|---|---|---|---|
| 05 Aug 2022 | 08 Sep 2022 | 19 Sep 2022 | 30 Sep 2022 |
Ring theory plays a vital role in mathematics, physics, chemistry, and computer science. Ring theory has applications in geometry, symmetry and transformation puzzles like Rubik's Cube. Also the vector space and partial differential equations has many applications in mathematics, engineering etc. Partial differential equations are used in problems involving functions of several variables, such as heat or sound, elasticity, electrodynamics, fluid flow, etc. In this article we have established relation between first order partial differential equations and ring theory, vector space. If g(x,y,z,p,q) is the given first order partial differential equation, the set of all partial differential equations f(x,y,z,p,q) which are compatible with g(x,y,z,p,q) form ring structure under usual addition and multiplication of two functions. Furthermore this ring is commutatve. Also if we use usual vector addition of functions and scalar multiplication then this newly formed set is a vector space.
[1] W. Bruns, J. Herzog, "Cohen-Macaulay Rings", Cambridge University Press, Cambridge, U.K, vol. 2, 1993.
[2] S. V. Duzhin, V. V. Lychagin, "Symmetries of Distributions and Quadrature of Ordinary Differential Equations", Applicandae Mathematicae, vol. 24, pp. 29-51, 1991.
[3] Mr. Sagar Waghmare, Dr. Ashok Mhaske, Mr. Amit Nalvade, Smt. Todmal Shilpa, “New Group Structure of Compatible System of First Order Partial Differential Equations,” International Journal of Scientific & Engineering Research, vol. 67, no. 9, pp. 114-117, 2021. doi:10.14445/22315373/IJMTT-V67I9P513.
[4] O. A. Chalykh and A. P. Veselov, Moscow State University, SU 117234 Moscow, USSR.
[5] Craddock, Mark, "Symmetry Groups of Linear Partial Differential Equations and Representation Theory: The Laplace and Axially Symmetric Wave Equations," Journal of Differential Equations, 2000.
[6] Lahno, P. Basarab–Horwath and V. "Group Classification of Nonlinear Partial Differential Equations: a New Approach to Resolving the Problem," Proceedings of Institute of Mathematics of NAS of Ukraine, vol. 43, pp. 86-92, 2002.
[7] Chandradeepa Chitalkar, Vasant R. Nikam, "Research Paper: Solution of Fractional Partial Differential Equations using Iterative Method," Fractional Calculus and Applied Analysis, 2012
[8] K. L. Bondar and Ashok Mhaske, “Fuzzy Transportation Problem with Error by Using Lagrange’s Polynomial,” The Journal of Fuzzy Mathematics, vol. 24, no. 4, pp. 825-832, 2016.
[9] Ashok S Mhaske, K L Bondar, "Fuzzy Database and Fuzzy Logic for Fetal Growth Condition," Asian Journal of Fuzzy and Applied Mathematics, vol. 03, no. 3, pp. 95-104, 2015.
[10] Ashok S Mhaske, K L Bondar, "Fuzzy Transportation by using Monte Carlo Method," Advances in Fuzzy Mathematics, vol. 12, no. 1, pp. 111-127, 2017
[11] Ambadas Deshmukh, Ashok Mhaske, P.U. Chopade and K.L. Bondar, “Fuzzy Transportation Problem by using Fuzzy Random Number,” International Review of Fuzzy Mathematics, vol. 12, no. 1, pp. 81-94, 2017
[12] Ambadas Deshmukh, Ashok Mhaske, P.U. Chopade and Dr. K.L. Bondar, “Fuzzy Transportation Problem by using Trapezoidal Fuzzy Numbers,” IJRAR- International Journal of Research and Analytical Reviews, vol. 5, no. 3, pp. 261-265, 2018.
[13] Ashok Sahebrao Mhaske, Kirankumar Laxmanrao Bondar, "Fuzzy Transportation Problem by using Triangular, Pentagonal and Heptagonal Fuzzy Numbers with Lagrange’s Polynomial to Approximate Fuzzy Cost for Nonagon and Hendecagon," International Journal of Fuzzy System Applications, vol. 9, pp. 112-129, 2020
[14] Ashok S. Mhaske, “Ranking Triangular Fuzzy Numbers Using Area of Rectangle at Different Level of α-Cut for Fuzzy Transportation Problem,” Journal of Emerging Technologies and Innovative Research, vol. 8, no. 3, pp. 2202-2209, 2021.
[15] Dr. Ashok S. Mhaske, “Difference between Fuzzy and Crisp Transportation Problem using Pentagonal Fuzzy Numbers with Ranking by α-cut Method,” Journal of Emerging Technologies and Innovative Research, vol. 8, no. 3, pp. 2143-2150, 2021.
[16] Ambadas Deshmukh, Dr. Arun Jadhav, Ashok S. Mhaske, K. L. Bondar, “Fuzzy Transportation Problem by using TriangularFuzzy Numbers with Ranking using Area of Trapezium, Rectangle and Centroid at Different Level of α-Cut," Turkish Journal of Computer and Mathematics Education, vol. 12, no. 12, 2021.
[17] Dr. Ashok Mhaske, Mr. Amit Nalvade,Mr. Sagar Waghmare,Smt. Shilpa Todmal , “Optimum Solution To Fuzzy Game Theory Problem using Triangular Fuzzy Numbers and Trapezoidal Fuzzy Number,” Journal of Information and Computational Science, vol. 12, no. 3, pp. 199-212, 2022.
[18] Ambadas Deshmukh, Dr. Arun Jadhav, Ashok S. Mhaske, K. L. Bondar “Optimum Solution to Fuzzy Transportation Problem using Different Ranking Techniques to Order Triangular Fuzzy Numbers,” Stochastic Modelling & Applications, vol. 26, no. 3, pp. 35-40, 2022.
[19] K. L. Bondar, A. S. Mhaske & S. G.Purane, “Fuzzy Unbalanced Transportation Problem by using Monte Carlo Method,” Aayushi International Interdisciplinary Research Journal (AIIRJ), no. 25, pp. 6-20, 2018.
[20] Shaoshi Chen R. F, "On the Structure of Compatible Rational Functions," Semantic Scholar, 2011.
[21] E. Kamke, "Differential Equations, Solution Methods and Solvunden. II, " Germain, Leipzig, 1959.
[22] I.S. Krasilschik, V. V. Lychagin, A.M. Vinogradov, "Geometry of Jet Spaces and Differential Equations," Gordon and Breach, 1986.
[23] B.S. Kruglikov, V.V. Lychagin, "On Equivalence of Differential Equations," Journals and Commentaries of the University of Tartu on Mathematics, vol. 3, pp. 7-29, 1999.
[24] B.S. Kruglikov, V.V. Lychagin, "Mayer Brackets and Solvability of PDEs – I," Differential Geometry and its Applications, vol. 17, pp. 251-272, 2002.
[25] B.S. Kruglikov, V.V. Lychagin, "Mayerbrackets and Solvability of PDEs –II", Trans.A.M.S., to Appear Shaoshi Chen, R. F. On the Structure of Compatible Rational Functions, Semantic Scholar, 2004.
Sagar Waghmare, Ashok Mhaske, Amit Nalvade, Smt. Shilpa Todmal, "New Ring and Vector Space Structure of Compatible Systems of First Order Partial Differential Equations," International Journal of Mathematics Trends and Technology (IJMTT), vol. 68, no. 9, pp. 60-65, 2022. Crossref, https://doi.org/10.14445/22315373/IJMTT-V68I9P509
PDF 
Abstract 
Keywords 
References 
Citation