The report aims to explore the application of differential equations in modeling the motion of planets and stars within our universe, serving as an introduction to the captivating realm of celestial mechanics. We utilize differential equations to represent the movement and positions of celestial bodies within a gravitational field, grounding our analysis in Newton's laws of motion and gravitation. Moreover, we employ Kepler's laws of planetary motion to elucidate the orbits of planets around the sun. It is important to note that this report offers a simplified perspective, designed for educational purposes. In reality, celestial mechanics can be exceedingly intricate, involving n-body problems, relativistic effects, and a multitude of other factors.
Further, the RD Sharma Solutions for Class 12 are curated in a stepwise manner for every question in simple language. The primary aim is to provide ease of understanding to students. Referring to these RD Sharma Solutions while solving can increase and strengthen the conceptual knowledge of students. To maximise the possibilities of scoring high marks, solutions are designed according to the latest CBSE guidelines and marking schemes.
Differential Equation Dr Bd Sharma Pdf Download
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\(\begin{array}{l}\boldsymbol{P_{0}\left ( \frac{d^{n}y}{dx^{n}} \right )\;+\;P_{1}\left ( \frac{d^{n-1}y}{dx^{n-1}} \right )\;+\;P_{2}\left ( \frac{d^{n-2}y}{dx^{n-2}} \right )\;+ \;. \;.\; . \;+ \;P_{n-1}\frac{dy}{dx}+P_{n}y\;=\;Q }\end{array} \)
Examples: The Equation \(\begin{array}{l}\frac{d^{4}y}{dx^{4}}-7y= 10 \left ( \frac{dy}{dx} \right )^{8}\end{array} \) is of degree 1 because, the power of the highest order derivative in this differential equation is 1.
The Equation \(\begin{array}{l}\left ( \frac{d^{3}y}{dx^{3}} \right )^{5}= 21y-10 \left ( \frac{dy}{dx} \right )^{2}\end{array} \) is of degree 5 because, the power of the highest order derivative in this differential equation is 5.
In this study, a delay differential equation model of gene expression for both retroviruses and normal cell is proposed to study the dynamics of functional gene products. The model is categorised into two sub-models to understand the characteristics of a cell by incorporating time delays in the processes of gene expression. The first model which is for retroviruses, involves time delay in replication, transcription, reverse transcription and translation processes taking place in the cell, while in the second model which is for normal cell, the time delay in transcription and translation processes are incorporated. A numerical solution is obtained using semi-temporal data set. The impact of time delays on temporal concentration profile of DNA, mRNA and proteins have been analysed which gives better insight into the normal cell as well as retroviruses. Further, sensitivity analysis has been performed for both models to study the behaviour of gene expression in the cell. The results obtained from such models can be useful for biomedical applications.
In this article, we demonstrated the study of the time-fractional nonlinear Sharma-Tasso-Olever (STO) equation with different initial conditions. The novel technique, which is the mixture of the q-homotopy analysis method and the new integral transform known as Elzaki transform called, q-homotopy analysis Elzaki transform method (q-HAETM) implemented to find the adequate approximated solution of the considered problems. The wave solutions of the STO equation play a vital role in the nonlinear wave model for coastal and harbor designs. The demonstration of the considered scheme is done by carrying out some examples of time-fractional STO equations with different initial approximations. q-HAETM offers us to modulate the range of convergence of the series solution using , called the auxiliary parameter or convergence control parameter. By performing appropriate numerical simulations, the effectiveness and reliability of the considered technique are validated. The implementation of the new integral transform called the Elzaki transform along with the reliable analytical technique called the q-homotopy analysis method to examine the time-fractional nonlinear STO equation displays the novelty of the presented work. The obtained findings show that the proposed method is very gratifying and examines the complex nonlinear challenges that arise in science and innovation.
Naveen Sanju Malagi is a research scholar, working under the guidance of Dr. D. G. Prakasha Associate Professor, Department of Mathematics, Davangere University, Davangere. He completed his Master's Degree from Rani Channamma University, Belagavi. His areas of interest are Fractional Calculus, Applications of Fractional differential equations, Applied Mathematics and Mathematical Modeling.
Pundikala Veeresha is currently an Assistant Professor in the Department of Mathematics, CHRIST (Deemed to be University), Bangalore and received the Ph.D. degree in 2020 from Karnatak University, Dharwad and Master Degree from Davangere University, Davangere. His area of research interests are Fractional Calculus, Applied Mathematics, Mathematical Physics, Mathematical Methods and Models for Complex Systems. He has published more than fifty (50) research articles in various reputed international journals.
Ballajja Chandrappa Prasannakumara obtained Master degree in Mathematics in 2000, and Doctoral Degree in Applied Mathematics in 2007 from Kuvempu University. At present he is serving as an Associate Professor, Department of Mathematics, Davangere University. His research focuses on semi analytical and numerical solutions to heat and mass transfer of Newtonian/non-Newtonian fluids. He has developed mathematical model and simulation pertaining to thermodynamic performance of nanofluid. His work centers around the study of heat and mass transfer through fins, micro and nano channel and over a stretched surface.
Doddabhadrappla Gowda Prakasha received his M.Sc., (2005) and Ph.D., (2008) from Kuvempu University. He has started his teaching career in 2008 from Karnatak University, Dharwad. Later, joined to Department of Mathematics, Davangere University, Davangere as an Associate Professor of Mathematics in the year 2019. His area of research specialization is Differential Geometry of manifolds, Fractional Calculus, Graph theory, and General Theory of relativity witnessed by more than 125 research papers in reputed journals. Presently, seven students got Ph.D. degree and four more are working under his supervision. He is a Referee / Reviewer for more than 65 research papers for various reputed journals.
Yavuz, M., (2020). Europian option pricing models described by fractional operators with classical an generalized Mittag-Leffler kernels. Numerical Methods for Partial Differential Equations, 1-23. DOI:
Prakasha, D. G., Malagi, N. S., & Veeresha, P. (2020). New approach for fractional Schrdinger?Boussinesq equations with Mittag?Leffler kernel. Mathematical Methods in the Applied Sciences, 43(17), 9654-9670. DOI:
Prakasha, D. G., Malagi, N. S., Veeresha, P., & Prasannakumara, B. C. (2021). An efficient computational technique for time?fractional Kaup?Kupershmidt equation. Numerical Methods for Partial Differential Equations, 37(2), 1299-1316. DOI:
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Natasha Shilla Sharma was born and raised in India where she obtained her Bachelors and Masters in Mathematics from the University of Delhi, New Delhi. She earned a doctorate in numerical methods for differential equations in 2011 from the University of Houston, Houston, Texas. Between 2012 and 2014, she was a post doctoral fellow at Heidelberg University, Heidelberg, Germany where she became more involved in the computational aspects of numerical methods to solve differential equations. 152ee80cbc
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