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Research Publication 35

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Extension of Picard’s Iteration Method to a Class of nth-Order Initial Value Problems
(Southeast Asian Bulletin of Mathematics, 2025-01-01) Jervin Zen Lobo; Sanket Tikare
This paper extends the method of successive approximations to an nth-order initial value problem without converting it to a first-order system. We obtain a bound in a closed form for the difference between two successive iterates. Finally, an error estimate for the solutions has been calculated and it is shown to be increasingly accurate as n increases. Suitable illustrations to support the result are provided.
Group analysis of the second order linear differential equation with variable delay
(Applied Mathematics E-Notes, 2022) Jervin Zen Lobo
We perform group analysis of the second order linear differential equation with the most general variable time delay by developing a Lie type invariance condition using Taylorís theorem for a function of more than one variable. This condition is then used to obtain the symmetry algebra and make a complete group classification of this delay differential equation. We deduce certain compatibility conditions for the infinitesimals, which lead to an extension of the symmetry algebra. Finally, we obtain a change of variables leading the differential equation with variable delay to be reduced to a differential equation with constant delay.
Cerror estimates in picard’s method of successive approximations for a particular second order initial value problem
(Vijnana Parishad of India (Jñānābha‎), 2022-06-01) Jervin Zen Lobo
In this paper, we extend the method of successive approximations to second order Initial Value Problems (IVPs) of the type y 00 = f(x, y), y(x0) = y0, y 0 (x0) = y1, without converting it to a system of first order differential equations. We obtain an upper bound in the closed form for the difference between two successive iterates. Further, we calculate an error bound for the solution and see that we get a tighter bound for the second order IVP as compared to its first order counterpart. 2020 Mathematical Sciences Classification: 34A12, 34A40.
Journal of Mathematical Extension
(Journal of Mathematical Extension, 2022-04-11) Jervin Zen Lobo; Y. S. Valaulikar
In this paper, we shall apply symmetry analysis to second order functional differential equations with constant coefficients. The determining equations of the admitted Lie group are constructed in a manner different from that of the existing literature for delay differential equations. We define the standard Lie bracket and make a complete classification of the second order linear functional differential equations with constant coefficients, to solvable Lie algebras. We also classify some second order non-linear functional differential equations with constant coefficients, to solvable Lie algebras.
Symmetry analysis of the korteweg-de vries Equation
(Palestine Journal of Mathematics, 2022-01-01) Jervin Zen Lobo
In this paper, we explain how symmetry analysis is applied to the Korteweg-devries equation. This equation under study is a nonlinear partial differential equation of third order. We develop the required Lie invariance condition using local inverse theorem, an approach different from the existing Lie-Bäcklund operators and Taylor series method. This condition is then used to obtain the determining equations of the admitted Lie group. The corresponding Lie algebra is found to be solvable. We obtain the invariant surface condition, symmetry algebra and the group classification admitted by this equation. Further, we obtain the exact and similarity solutions of this equation, comment on them and represent them graphically.