Tribute to Prof. Dnyanoba Bhaurao Dhaigude

Transcription

1 Tribute to Prof. Dnyanoba Bhaurao Dhaigude M.D. Jahagirdar and B.R.Sontakke Professor Dnyanoba Bhaurao Dhaigude was born on June 10, 1953 in a farmer s family, in small village Bansarola Dist.Beed (Maharashtra) India. He completed his primary and Matriculation education from Maharashtra Vidyalaya Bansarola. He is graduated from Marathwada University an alma-meter of the same. After his postgraduation, he joined as a U.G.C Research Scholar in the university department as he was inclined towards research from the college life. He completed his Ph.D.degree in 1989 from the then Marathwada University, Aurangabad under the guidance of Professor D.Y. Kasture. In the year 1981 he joined as a lecturer at Pratishthan Mahavidyalaya, Paithan. In his thirteen years service at college, he remained popular among the students because of his cooperative nature and excellent teaching. He is also popular in many villages due to NSS activities for the common man in rural areas. He also worked as Incharge Principal of Pratishthan Mahavidyalaya, Paithan. In 1994, he joined the department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad and resumed the office of Professor in the same department in Professor Dhaigude worked as a Head of Department of Mathematics for two terms. As a result with his atmost efforts made the university felt to start Applied Mathematics in the department for the students. U.G.C. has sanctioned about 25 Laks to this course in 11 th plan. Because of his industrious attitude, capacity and keen interest in research, 10 students were conferred Ph.D. degree and 16 students were awarded M.Phil degree under his guidance. Now 6 students are working for their Ph.D. degree under his guidance. He completed four research projects viz. 1. Major Research Projects on Inequalities in Mechanics and Physics, Department of Atomic Energy (Through NBHM), Govt.of India.(with D.Y.Kasture) 2. Minor Research Project on Nonlinear Elliptic and Parabolic Systems, U.G.C. 3. Minor Research Project on Nonlinear Parabolic Systems, Marathwada University 4. Minor Research Project on Singular Perturbation Problems, Marathwada University He is a member of the following professional societies: 1. American Mathematical Society. 2. Indian Society of Theoretical and Applied Mechanics. 3. Life Member of Indian Academy of Mathematics. vi

2 4. Founder and Patron member of Marathwada Mathematical Society. 5. The Executive President of Marathwada Mathematical Society. Professor Dhaigude is associated with various academic journals, the details are as under: Executive Editor of Bulletin of the Marathwada Mathematical Society.[ISSN No ] Associate Editor of Dr. B. A. M. University Science Journal. Associate Editor of Global J.Appl. Math. Math. Sci.,Serial Pub. Associate Editor of J.Sci. Information. Reviewer of Mathematical Reviews. He attended 15 International conferences and 11 National conferences and various state level conferences, workshop and seminar. Furthermore Professor Dhaigude presented his research papers: 1. In the fourth international conference on Differential and Functional Differential Equations, Steklov Mathematical Institute, Moscow, Russia, August 14-21, Fifth international Conference on Dynamical Systems and Applications, Morehouse College, Atlanta, U. S. A, May 28-June 2, Major Areas of Interest of Prof.D.B.Dhaigude: 1. Differential and Integral Inequalities, 2. Monotone Iterative Technique, 3. Thermal Boundary Layer Theory, 4. Finite Difference Equations and Numerical Methods, 5. Fractional Differential Equations, 6. Delay Differential Equations, 7. Time Degenerate Parabolic Initial Boundary Value Problems, 8. Prefunctions and Differential Equations, 9. Maximum Principles and Applications, Professor Dhaigude is an external examiner for Ph.D. Thesis of various Indian Universities such as. Madras University, Chennai. Rashtrasant Tukadoji Maharaj Nagpur University, Nagpur vii

3 . Sri Padmavati Mahila Visvavidyalayam, Tirupati. Sri Venkateswara University, Tirupati. Anna University, Chennai. Periyar University, Salem. Shivaji University, Kolhapur After having seen his capacity and enthusiasm, involvement in a social activities, he was honoured by the university by appointing him as the Director of Academic Staff College of Dr. Babasaheb Ambedkar Marathwada University, Aurangabad. Beside this he worked as a Coordinator for 6 courses (2-Orientation and 4-Refresher) in the Academic Staff College. Professor Dhaigude occupied various positions in Dr. Babasaheb Ambedkar Marathwada University, Aurangabad such as Member of Management Council, Academic Council, Research and Recognition Committee, Faculty of Science and Chairman, Board of Studies in Mathematics. Also worked as Member of Board of Studies in almost all universities in Maharashtra and Gulbarga University, Gulbarga, as well. Extension Work/ Community Services: Professor Dhaigude not only worked in teaching and research but also worked in extension and community service which is a third dimension of higher education. He started his carrier in extension activities as a Programme Officer of NSS at collegiate level. He organized various youth camps which include Youth For Rural Development, Youth For Water Shade Management and Waste Land Development. He completed many projects for the society specially he worked for village Wadawali and village transformation concept is introduced in NSS for the first time. Details are in the following publications: 1. Success Achievements of NSS Unit of Pratishthan Mahavidyalaya, Paithan ( ), Rashtriya Seva, 8, 1(1985), New Delhi (With A.L.Deshmukh) 2. Role of NSS in Socio-Economic Development of Wadwali, Rashtriya Seva, 10, 2(1988), New Delhi (With A.L.Deshmukh) In real sense he has proved quality in education through community service and personality development through social service where ever possible. Therefore, Government of Maharashtra has honoured Pratishthan Mahavidyalaya, Paithan by awarding Best NSS Unit in Maharashtra in the year During he became NSS Programme Co-ordinator, NSS Cell, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad. He delivered more than 300 talks on various aspects of community development, environment enrichment, water shade management and waste land development at different special camp organized by various affiliated colleges of Dr. Babasaheb Ambedkar Marathwada University, Aurangabad. He developed a role model in water shade management and waste land development in Dr. Babasaheb Ambedkar Marathwada University campus as well as in different villages. Dr. Babasaheb Ambedkar Marathwada University, Aurangabad viii

4 awarded him the LIFE TIME ACHIEVEMENT AWARD for his valuable contribution in the field of extension work and community service. Main Contributions with his Students: Positivity Lemmas: The following lemmas commonly known as Positivity Lemmas, are wildly used by the researchers, and existence, uniqueness properties of the solutions are derived. They are stated below.. Lemma 0.1 (Positivity Lemma): Suppose that w C(D T ) C 2,1 (D T ) and satisfies the inequalities i d(x, t)w t D(x, t) 2 w + c(x, t)w 0; in D T ii α 0 (x) w ν + β 0(x)w 0; on S T iii w(x, 0) 0; in Ω where d(x, t) 0, c(x, t) 0 in D T Then w(x, t) 0 in D T. Lemma 0.2 (Positivity Lemma): Suppose that w i,n satisfies the following inequalix

5 ities d i,n kn 1 (w i,n w i,n 1 ) p D i,n (ν) w i,n + c i,n w i,n ν=1 α(x i, t n ) x i ˆx i 1 [w(x i, t n ) w(ˆx i, t n )] + β(x i, t n )w(x i, t n ) w i,0 0; (i, n) Λ p 0; (i, n) S p 0; i Ω p where c i,n 0, d i,n 0, ˆx i is a suitable point in Ω p and x i ˆx i is the distance between x i and ˆx i then w i,n 0 in Λ p. Maximum Principles: The following maximum principles are due to Dhaigude. They are widely used by the researchers to find the existence, uniqueness properties of the solutions. It is stated as follows. Theorem 0.1 Suppose that i) u(x) is a sufficiently smooth solution of 2 u + a(x, y) u + b(x, y)f(u) = 0. ii) a(x, y) and b(x, y) are bounded in Ω iii) a 0, b > 0 and b(x, y)u(x, y)f(u) + a(x, y) u 2 0 in Ω. Then the function P = u(x) 2 u u attain its non-negative maximum on Ω. Theorem 0.2 Suppose that i) Ω is plane domain with boundary Ω, ii) u(x) be a sufficiently smooth solution of 2 u + a(x, y) u + b(x, y)f(u) = 0., iii) a(x, y) and b(x, y) are bounded in Ω, iv) a(x, y) 0, b(x, y) > 0 in Ω and b(x, y)u(x, y)f(u) + a(x, y) u 2 0 in Ω. Then the function P = 1 ρ [ ] u(x) 2 u u attain its non-negative maximum on the boundary Ω. Monotone Technique: Monotone technique for time degenerate parabolic problems and problems in fractional differential equations are also developed and technique is successfully applied to study existence and uniqueness of solutions of the problems. Results due to Dhaigude et.al. for the following weakly coupled system of FPBVP for Caputo fractional differential equation of order q (0 < q < 1) c D q u 1 (t) = f 1 (t, u 1 (t), u 2 (t)), c D q u 2 (t) = f 2 (t, u 1 (t), u 2 (t)), u 1 (0) = u 1 (2π) u 2 (0) = u 2 (2π) (0.1) where f 1, f 2 C(J R 2, R), J = [0, 2π] are as under: x

6 Theorem 0.3 Suppose that (i) v 0 (t) = (v1 0(t), v0 2 (t)) and w0 (t) = (w1 0(t), w0 2 (t)) in C1 (J, R) are ordered lower and upper solutions of FPBVP (0.1) such that vi 0(0) w0 i (0) on J (ii) f i = f i (t, u 1 (t), u 2 (t)) in C[J R 2, R] is quasimonotone nondecreasing (iii) f i f i (t, u 1 (t), u 2 (t)) satisfies one-sided Lipschitz condition f i (t, u 1, u 2 ) f i (t, u 1, u 2 ) M i (u i u i ) for i = 1, 2 whenever v 0 i u i u i w 0 i, M i 0. Then there exist monotone sequences {v n (t)} = (v1 n(t), vn 2 (t)) and {wn (t)} = (w1 n(t), wn 2 (t)) such that {v n (t)} v(t) = (v 1 (t), v 2 (t)) and {w n (t)} w(t) = (w 1 (t), w 2 (t)) as n and v(t), w(t) are extremal solutions of FPBVP (0.1). Theorem 0.4 Suppose that assumptions (i)-(iii) in Theorem 0.3 holds and if, for 0 < N i < M i f i (t, u 1, u 2 ) f i (t, u 1, u 2 ) N i (u i u i ) v 0 i u i u i w 0 i, i = 1, 2, then the FPBVP (0.1) has a unique solution. He is the author of two graduate books on Mathematics viz. (1) Calculus and Algebra, Anand Prakashan, Aurangabad. (2) Real Analysis and Differential Equations, Anand Prakashan, Aurangabad. Research Publications: (1) A Note on Comparison Theorem, J. Indian Acad. Math. 5,(1983), (IN- DIA) (2) Comparison Theorems for Nonlinear Elliptic System with Applications, J. Math. Anal. Appl. 112, 1(1985), (with D.Y.Kasture) MR 87c: 35008(U.S.A) (3) Comparison Theorems for a Class of Elliptic Equations of Order 4m, Chinese J. Math 15(1987), MR 88m: (CHINA) (4) Sturmian Theorems for Time Degenerate Parabolic Inequalities, Ann. St.Uni Al.l.Cuza Iasi, 36, (1990), MR 92j: (RUMANIA) xi

7 (5) Nonlinear Time Degenerate Parabolic Inequalities and Applications, J. Math. Anal. Appl.175, 2(1993), (with D.Y.Kasture) MR 94a: (U.S.A) (6) Squaring of Numbers, Bull. Marathwada Math. Soc. 1 (2000), (with C.R.Bembalkar) (INDIA) (7) Maximum Principles for Nonlinear Time Degenerate Parabolic Equations, Proc. International Conf. On Nonlinear Systems: Modelling, Simulation & Applications, (2000), (with M.R.Gosavi) (INDIA) (8) Maximum Principles in Ordinary Differential Equations, Bull.Marathwada Math. Soc. 1 (2000), (INDIA) (9) Salient Contributions of Prof.B.G.Pachpatte, Bull.Marathwada Math. Soc. 4 (2003), 1-16 (with E.Thandapani & M.B.Dhakne) (INDIA) (10) Maximum Principles for Fourth Order Semi linear Elliptic Equations with Applications, Differential Equations and Dynamical Systems, 12, 3&4, (2004), (with M.R.Gosavi) (INDIA) (11) Hermitian Type Operators of Second Order Differential Equations, J. Natural & Physical Sciences, 20, 1,(2006),77-85 (with A.P.Bhadane) (INDIA) (12) Upper-Lower Solutions for System of Finite Difference Reaction-Diffusion Equations, Proc. Second Internat. Conf. On Nonlinear Systems: Bull. Marathwada Math. Soc., 2,8(2007), 1-11 ( with S.B.Kiwne & R.M.Dhaigude) (13) Monotone Iterative Scheme for Weakly Coupled System of Finite Difference Reaction-Diffusion Equations, Communications in Applied Analysis 12,(2008), No.2, (with S.B.Kiwne & R.M.Dhaigude), (U.S.A.) (14) Monotone Technique for Nonlinear Time Degenerate Reaction -Diffusion Problems, Recent Advances in Mathematical Sciences and Applications (2009), 46-53, (with R.M.Dhaigude & Chandradeepa D. Dhaigude) (15) A Two Level Explicit Finite Difference Scheme for One Dimensional Pennes Bioheat Equation, Recent Advances in Mathematical Sciences and Applications (2009), , (with Takale K.C) (16) Preexponential and Pretrigonometric Functions, Communications in Applied Analysis 14(2010), no , (with R.B.Khandeparkar & S.G.Deo), Zbl (U.S.A.) (17) Fuzzy Programming Technique to Solve Multi objective Solid Transport Problems with Nonlinear Membership Function, Advances in Computational Research, 2, 1(2010), (with Bodkhe S.G. & Bajaj V.H.) (18) A Weighted Average Finite Difference Scheme for One Dimensional Penne s Bioheat Equation, Proc. Neural, Parallel and Scientific Computations 4, (2010), (with Takale K.C.), Zbl (U.S.A.) xii

8 (19) Monotone Technique for Nonlinear Degenerate Weakly Coupled System of Parabolic Problems, Communications in Applied Analysis 15, (2011), N0 1, (with B.R. Sontakke & Chandradeepa D. Dhaigude), Zbl (U.S.A.) (20) Douglas Higher Order Finite Difference Scheme for One Dimensional Pennes Bioheat Equation, International Journal of Advanced Engineering & Application, 3,(2 011), (with K.C.Takale, V.R.Nikam ) (21) Bounded and Asymptotic Behavior of Second Order Integro- differential Equations with Retarded Argument, International Journal of Mathematical Archive (11), Nov.2011, 1-8 (with J. V. Patil) (22) Monotone Iterative Scheme for System of Riemann-Liouville Fractional Differential Equations with Integral Boundary Condition, Mathematical Modelling and Scientific Computation, Vol.283, (2012), , Springer Verlag, (U.S.A.) (with J.A. Nanaware ) (23) Adomian Decomposition Method for Fractional Benjamin-Bona-Mahony-Burger s Equations, International Journal of Applied Mathematics and Mechanics 8 (12), (2012), 42-51, (with Gunvant A. Birajdar V. R. Nikam) (24) Maximum Principles for Fourth Order Uniformly Elliptic Equations with Applications, International Journal of Applied Mathematical Research 1 (3), (2012), , ( with R.M. Dhaigude, G.C. Lomte) (25) Numerical Solution of System of Fractional Partial Differential Equations by Discrete Adomian Decomposition Method, Journal of Fractional Calculus and Applications,3. July 2012, No.12, pp (EGYPT) ISSN: (with G. A. Birajdar) (26) Monotone Technique for System of Caputo Fractional Differential Equations with Periodic Boundary Conditions, Dynamics of Continuous, Discrete and Impulsive Systems, Series A: Mathematical Analysis 19 (2012), ( CANADA) (with J.A. Nanaware and V.R.Nikam) (27) Existence and Uniqueness of Solution of Riemann-Liouville Fractional Differential Equations with Integral Boundary Conditions, International Journal of Nonlinear Sciences, Vol.14, (2012) No.4, (U.K.) (with J.A. Nanaware ) (28) Linear Initial Value Problems for Fractional Partial Differential Equations, Bull.Marathwada Math.Soc.13, 2(2012), (with Chandradeepa D. Dhaigude) (INDIA) (29) Maximum Principles for Fourth Order Nonlinear Elliptic Equations with Applications, Bol. Soc. Paran. Mat. (3serie.) Vol. 32, 1 (2014)(In press), (BRASIL)(with R.M. Dhaigude, G.C. Lomte)(Bulletin of Parana s Mathematical Society named as Boletim de Sociedade Paranaense de Matematica (BSPM) Brasil) (30) Maximum Principles for Nonlinear Fourth Order Differential Equations, J. Indian Acad.Math. (Accepted), (With C.R.Bembalkar) xiii

9 Summary: Professor Dhaigude s commitment, dedication to a work assigned to him, zeal and enthusiasm for the developments of Mathematics in particular and community development in general and his unconditional services are valuable to both researchers and lovers of Mathematics as well as downtrodden peoples of the society. On this occasion, we extend our sincere gratitude towards his contribution to Mathematics and extension activities as well. We wish him a grand success in his future life. xiv