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In this paper, a high-fidelity satellite orbit propagator, namely the Satellite Precise Orbit Propagator (SPOP), is developed for the Earth orbiting satellites. The SPOP numerically integrates the perturbed two-body differential equations of motion in the Earth Mean Equator and Equinox of 2000 (EME2000/J2000) reference frame using Cowell’s formulation and explicit Runge–Kutta class of integrator with the fixed time step size. Various perturbing forces are included in the numerical propagator such as a simple two-body problem up to $$J_{6}$$ terms for the academic purpose, full force gravitational field model using the spherical harmonics for the high precision orbit propagation in view of the commercial use, atmospheric drag, third-body gravity, solar radiation pressure, Earth radiation pressure due to albedo, and relativistic effect correction. In addition, SPOP also has the feature to predict the satellite ephemeris and orbit products using the NORAD compatible two-line element (TLE) through SGP4/SDP4 analytical propagator.

Chandrayaan-2, India’s second Moon mission, comprised of Orbiter, Lander and Rover modules was launched on July 22, 2019. The Chandrayaan-2 (CH2) Orbiter, which carries eight scientific payloads, is placed into a circular lunar orbit of 100 km altitude with an orbital period of around 120 minutes. After the successful injection of the spacecraft in the intended Earth parking orbit, it started facing eclipses during the Earth bound, Trans lunar, and lunar phases. This paper describes a high-fidelity orbit prediction propagator known as the Satellite Precise Orbit Propagator (SPOP), and an Earth shadow eclipse prediction model: the projection map method for the lunar orbiting spacecraft. The orbit and eclipse prediction models are simulated for the CH2 Orbiter. Results obtained by the models compare well with globally popular commercial package, Systems Tool Kit (STK) of Analytic Graphic Inc.

This paper concerns through the numerical treatment of Fisher’s reaction-diffusion equation by using a hybrid numerical method. In this method, the combination of cubic trigonometric B-spline (CTB) base functions and differential quadrature method is used. This reduces the problem to a system of first order ODEs which is solved by “an optimal five stage and fourth-order strong stability preserving Runge-Kutta (SSP-RK54)” scheme. Four examples are considered to compare the present results with exact solutions and the results obtained by existing methods. It is found that the present method is not only quite easy to implement, but also it gives better results than the ones already existing in the literature. Keywords: CTB functions, DQM, Fisher’s reaction-diffusion equation, SSP-RK54

This paper concerns with the modelling of a micro strip antenna with strip line feeding technique formulated using Ohm’s law. A partial differential equation is obtained from the model which is solved using “Reduced Differential Transformation Method (RDTM)”. A couple of numerical examples are considered to check the accuracy, efficiency and convergence of the method. The present method is a very powerful mathematical tool for solving wide range of problems arising in circuit theory, RF field, and science and communication system fields. Keywords: Micro strip, RDTM, Exact solution, Impedance

In the frame work of the perturbed restricted three-body problem, the solutions of halo orbits are developed up to fifth order approximation by using Lindstedt–Poincare technique. The effect of oblateness of the more massive primary on the size, location and period of halo orbits around L 1 and L 2 are studied by considering the Earth–Moon system. Due to oblateness of the Earth, halo orbits around L 1 and L 2 enlarge and move towards the Moon. Also, the period of halo orbits around L 1 and L 2 decreases. Numerical solution for halo orbits around L 1 and L 2 in the Sun–Earth system is obtained by using the differential correction method for different values of radiation pressure and oblateness. The separation between the orbits obtained using fourth and fifth order Lindstedt–Poincare method as well as differential correction method is found to be less than the separation between the orbits obtained using third and fourth order Lindstedt–Paincare as well as differential correction method. This indicates that as the order of the solution increases the separation between consecutive solution decreases leading to more accurate solution.

In this paper, the authors proposed a method based on exponential modified cubic B-spline differential quadrature method (Expo-MCB-DQM) for the numerical simulation of three dimensional (3D) nonlin...

In this paper, we construct a third-order analytic approximate solution using the Lindstedt-Poincare method in the photogravitational circular restricted three body problem considering the Sun as a radiating source and the Earth as an oblate spheroid for computing halo orbits around the collinear Lagrangian points L 1 and L 2 . Further, the well-known differential correction and continuation schemes are used to compute halo orbits and their families numerically. The effects of solar radiation pressure and oblateness on the orbit are studied around both Lagrangian points. From the study, it is noticed that time period of the halo orbit increases around L 1 and L 2 accounting oblateness of the Earth and solar radiation pressure of the Sun. It is also found that stability of halo orbits is a weak function of the out-of-plane amplitude and mass reduction factor.

During the Mars solar conjunction, telecommunication and tracking between the spacecraft and the Earth degrades significantly. The radio signal degradation depends on the angular separation between the Sun, Earth and probe (SEP), the signal frequency band and the solar activity. All radiometric tracking data types display increased noise and signatures for smaller SEP angles. Due to scintillation, telemetry frame errors increase significantly when solar elongation becomes small enough. This degradation in telemetry data return starts at solar elongation angles of around 5° at S-band, around 2° at X-band and about 1° at Ka-band. This paper presents a mathematical model for predicting Mars superior solar conjunction for any Mars orbiting spacecraft. The described model is simulated for the Mars Orbiter Mission which experienced Mars solar conjunction during May–July 2015. Such a model may be useful to flight projects and design engineers in the planning of Mars solar conjunction operational scenarios.

In this paper, computation of the halo orbit for the KS-regularized photogravitational circular restricted three-body problem is carried out. This work extends the idea of Srivastava et al. (Astrophys. Space Sci. 362: 49, 2017) which only concentrated on the (i) regularization of the 3D-governing equations of motion, and (ii) validation of the modeling for small out-of-plane amplitude ( $$A_z =110000$$ km) assuming the third-order analytical approximation as an initial guess with and without differential correction. This motivated us to compute the halo orbits for the large out-of-plane amplitudes and to study their stability analysis for the regularized motion. The stability indices are described as a function of out-of-plane amplitude, mass reduction factor and oblateness coefficient. Three different Sun–planet systems: the Sun–Earth, Sun–Mars and the Sun–Jupiter are chosen in this study. Stable halo orbits do not exist around the $$L_{1}$$ point, however, around the $$L_{2}$$ point stable halo orbits are found for the considered systems.

In this paper, we study the invariant manifold and its application in transfer trajectory problem from a low Earth parking orbit to the Sun-Earth $L_{1}$ and $L_{2}$ -halo orbits with the inclusion of radiation pressure and oblateness. Invariant manifold of the halo orbit provides a natural entrance to travel the spacecraft in the solar system along some specific paths due to its strong hyperbolic character. In this regard, the halo orbits near both collinear Lagrangian points are computed first. The manifold’s approximation near the nominal halo orbit is computed using the eigenvectors of the monodromy matrix. The obtained local approximation provides globalization of the manifold by applying backward time propagation to the governing equations of motion. The desired transfer trajectory well suited for the transfer is explored by looking at a possible intersection between the Earth’s parking orbit of the spacecraft and the manifold.

Mars exploring spacecraft “Mars Orbiter Mission” is India’s first interplanetary mission. It is placed in a highly elliptical orbit around the planet Mars with an orbital period of 65 h and 27 min. There was no eclipse on the MOM spacecraft during its interplanetary transfer. During the Martian phase, it started to experience eclipse shadow of Mars from the beginning. In this paper, we discuss several conical shadow eclipse prediction models by accounting the effects of atmospheric dust of Mars considering both spherical and oblate shape of the red planet. A study is performed using the results obtained by different shadow models, Systems Tool Kit (STK), and the actual telemetry data. We notice that effects of the atmospheric dust of Mars cannot be neglected on the MOM spacecraft.

A solar eclipse occurs when the Sun, Moon and Earth are aligned in such a way that shadow of the Moon falls on the Earth. The Moon׳s shadow also falls on the Earth orbiting spacecraft. In this case, the alignment of the Sun, Moon, and spacecraft is similar to that of the Sun, Moon, and Earth but this phenomenon is often referred as a lunar eclipse falling on the spacecraft. Lunar eclipse is not as regular in terms of times of occurrence, duration, and depth as the Earth shadow eclipse and number of its occurrence per orbital location per year ranges from zero to four with an average of two per year; a spacecraft may experience two to three lunar eclipses within a twenty-four hour period [2] . These lunar eclipses can cause severe spacecraft operational problems. This paper describes two lunar shadow eclipse prediction models using a projection map approach and a line of intersection method by extending the Earth shadow eclipse models described by Srivastava et al. [10] , [11] for the Earth orbiting spacecraft. The attractive feature of both models is that they are much easier to implement. Both mathematical models have been simulated for two Indian low Earth orbiting spacecrafts: Oceansat-2, Saral-1, and two geostationary spacecrafts: GSAT-10, INSAT-4CR. Results obtained by the models compare well with lunar shadow model given by Escobal and Robertson [12] , and high fidelity commercial software package, Systems Tool Kit (STK) of AGI.

In this article, we propose an Earth conical shadow model predicting umbra and penumbra states for the low Earth orbiting satellite considering the spherical shape of the Earth. The model is described using the umbra and penumbra cone geometries of the Earth’s shadow and the geometrical equations of these conical shadow regions into a Sun centered frame. The proposed model is simulated for three polar Sun-synchronous Indian Remote Sensing satellites: Cartosat-2A, Resourcesat-2 and Oceansat-2. The proposed model compares well with the existing spherical Earth conical shadow models such as those given by Vallado (2013), Wertz (2002), Hubaux et al. (2012), and Srivastava et al. (2013, 2014). An assessment is carried out of the existing Earth conical shadow models with Systems Tool Kit (STK), a high fidelity commercial software package of Analytic Graphic Inc., and the real time telemetry data.

In this paper, an exponential modified cubic B-spline differential quadrature algorithm is proposed for nonlinear one dimensional Fisher’s reaction–diffusion equation. The proposed method reduces the Fisher’s equation into a system of first order ordinary differential equations which is solved by using Runge–Kutta method. The accurateness and effectiveness of the method is tested by taking three numerical problems. The numerical results of the method are compared with the exact solutions and also compared with earlier published results. It is found that the proposed method produces more accurate results than results available in the literature. Straightforward and the economical implementation is main advantage of the method.

In this paper, a time- and space-coordinate transformation, commonly known as the Kustaanheimo–Stiefel (KS)-transformation, is applied to reduce the order of singularities arising due to the motion of an infinitesimal body in the vicinity of a smaller primary in the three-body system. In this work, the Sun–Earth system is considered assuming the Sun to be a radiating body and the Earth as an oblate spheroid. The study covers motion around collinear Lagrangian $L_{1}$ and $L_{2}$ points. Numerical computations are performed for both regularized and non-regularized equations of motion and results are compared for non-periodic as well as periodic motion. In the transformed space, time is also computed as a function of solar radiation pressure ( $q$ ) and oblateness of the Earth ( $A_{2}$ ). The two parameters ( $q, A_{2}$ ) have a significant impact on time in the transformed space. It is found that KS-regularization reduces the order of the pole from five to three at the point of singularity of the governing equations of motion.

A dynamical model of HIV infection of CD4+ T cells is solved numerically using an approximate analytical method so-called the differential transform method (DTM). The solution obtained by the method is an infinite power series for appropriate initial condition, without any discretization, transformation, perturbation, or restrictive conditions. A comparative study between the present method, the classical Euler’s and Runge–Kutta fourth order (RK4) methods is also carried out.

This article discusses the dynamic state analysis of underwater towed-cable when tow-ship changes its speed in a direction making parabolic profile path. A three-dimensional model of underwater towed system is studied. The established governing equations for the system have been solved using the central implicit finite-difference method. The obtained difference non-linear coupled equations are solved by Newton’s method and satisfactory results were achieved. The solution of this problem has practical importance in the estimation of dynamic loading and motion, and hence it is directly applicable to the enhancement of safety and the effectiveness of the offshore activities.

This article reports two conical shadow models: a spherical Earth conical shadow model (SECSM) and an oblate Earth conical shadow model (OECSM), and their comparative study to predict umbra and penumbra shadow regions for the low Earth orbiting (LEO) satellites. First model is described using a projection map technique considering the spherical shape of the Earth whereas the second model is illustrated using the line of intersection method for the oblate Earth. Both models have been implemented to four Indian Remote Sensing (IRS) satellites: Oceansat-2, Resourcesat-2, Cartosat-2A and Meghatropics-1. Computed results are compared well with the real time data and the commercial AGI package, Systems Tool Kit (STK).

The instability of a thin sheet of viscous and dielectric liquid moving in the same direction as an air stream in the presence of a uniform horizontal electric field has been carried out using viscous potential flow theory. It is observed that aerodynamic-enhanced instability occurs if the Weber number is much less than a critical value related to the ratio of the air and liquid stream velocities, viscosity ratio of two fluids, the electric field, and the dielectric constant values. Liquid viscosity has stabilizing effect in the stability analysis, while air viscosity has destabilizing effect.

The main goal of this paper is to present a new approximate series solution of the multi-dimensional (heat-like) diffusion equation with time-fractional derivative in Caputo form using a semi-analytical approach: fractional-order reduced differential transform method (FRDTM). The efficiency of FRDTM is confirmed by considering four test problems of the multi-dimensional time fractional-order diffusion equation. FRDTM is a very efficient, effective and powerful mathematical tool which provides exact or very close approximate solutions for a wide range of real-world problems arising in engineering and natural sciences, modelled in terms of differential equations.

The linear analysis of Rayleigh-Taylor instability of the interface between two viscous and dielectric fluids in the presence of a tangential electric field has been carried out when there is heat and mass transfer across the interface. In our earlier work, the viscous potential flow analysis of Rayleigh-Taylor instability in presence of tangential electric field was studied. Here, we use another irrotational theory in which the discontinuities in the irrotational tangential velocity and shear stress are eliminated in the global energy balance. Stability criterion is given by critical value of applied electric field as well as critical wave number. Various graphs have been drawn to show the effect of various physical parameters such as electric field, heat transfer coefficient, and vapour fraction on the stability of the system. It has been observed that heat transfer and electric field both have stabilizing effect on the stability of the system.

A fully implicit finite-difference method has been proposed for the numerical solutions of one dimensional coupled nonlinear Burgers’ equations on the uniform mesh points. The method forms a system of nonlinear difference equations which is to be solved at each iteration. Newton’s iterative method has been implemented to solve this nonlinear assembled system of equations. The linear system has been solved by Gauss elimination method with partial pivoting algorithm at each iteration of Newton’s method. Three test examples have been carried out to illustrate the accuracy of the method. Computed solutions obtained by proposed scheme have been compared with analytical solutions and those already available in the literature by finding L2 and L∞ errors., {"references":["S. E. Esipov, \"Coupled Burgers' equations: a model of polydispersive\nsedimentation\", Phys Rev E, Vol. 52 pp. 3711-3718, 1995.","J. M. Burgers, \"A mathematical model illustrating the theory of turbulence\",\nAdv. Appl. Mech., Vol. I, pp. 171-199, 1948.","J. D. Cole, \"On a quasilinear parabolic equations occurring in aerodynamics\",\nQuart. Appl. Math., Vol. 9, pp. 225-236, 1951.","J. Nee and J. Duan, \"Limit set of trajectories of the coupled viscous\nBurgers' equations\", Appl. Math. Lett, Vol. 11, no. 1, pp. 57-61, 1998.","D. Kaya, \"An explicit solution of coupled viscous Burgers' equations\nby the decomposition method\", J.J.M.M.S., Vol. 27, no. 11, pp. 675-680,\n2001.","A. A. Soliman, \"The modified extended tanh-function method for solving\nBurgers-type equations\", Physica A, Vol. 361, pp. 394-404, 2006.","M. A. Abdou and A. A. Soliman, \"Variational iteration method for solving\nBurgers and coupled Burgers equations\", J. Comput. Appl. Math., Vol.\n181, no. 2, pp. 245-251, 2005.","G. W. Wei and Y. Gu, \"Conjugate filter approach for solving Burgers'\nequation\", J. Comput. Appl. Math., Vol. 149, no. 2, pp. 439-456, 2002.","A. H. Khater, R. S. Temsah, and M. M. Hassan, \"A Chebyshev spectral\ncollocation method for solving Burgers-type equations\", J. Comput. Appl.\nMath., Vol. 222, no.2, pp. 333-350, 2008.\n[10] M. Deghan, H. Asgar and S. Mohammad, \"The solution of coupled\nBurgers' equations using Adomian-Pade technique\", Appl. Math. Comput.,\nVol. 189, pp. 1034-1047, 2007.\n[11] A. Rashid and A. I. B. Ismail, \"A fourierPseudospectral method for\nsolving coupled viscous Burgers' equations\", Comput. Methods Appl.\nMath., Vol. 9, no. 4, pp. 412-420, 2009.\n[12] R. C. Mittal and G. Arora, \"Numerical solution of the coupled viscous\nBurgers' equation\", Commun.Nonlinear Sci. Numer.Simulat., Vol. 16, pp.\n1304-1313, 2011.\n[13] R. Mokhtari, A. S. Toodar and N. G. Chegini, \"Application of the\ngeneralized differential quadrature method in solving Burgers' equations\",\nCommun. Theor. Phys., Vol. 56, no.6, pp. 1009-1015, 2011.\n[14] S. Kutley, A. R. Bahadir and A. Ozdes, \"Numerical solution of onedimensional\nBurgers' equation: explicit and exact-explicit finite difference\nmethods\", J. Compt. Appl. Math, Vol. 103, pp. 251-261, 1999.\n[15] M. K. Kadalbajoo and A. Awasthi, \"A numerical method based on\nCrank-Nicolson scheme for Burgers' equation\", Appl. Math. Compt., Vol.\n182, pp. 1430-1442, 2006.\n[16] P. C. Jain and D. N. Holla, \"Numerical solution of coupled Burgers'\nequations\", Int. J. Numer. Meth. Eng., Vol. 12,pp. 213-222, 1978.\n[17] C. A. J. Fletcher, \"A comparison of finite element and finite difference\nof the one and two-dimensional Burgers' equations\", J. Comput. Phys.,\nVol. 51, pp. 159-188, 1983.\n[18] F. W. Wubs and E. D. de Goede, \"An explicit-implicit method for a class\nof time-dependent partial differential equations\", Appl. Numer. Math., Vol.\n9, pp. 157-181, 1992.\n[19] O. Goyon, \"Multilevel schemes for solving unsteady equations\", Int. J.\nNumer. Meth. Fluids, Vol. 22, pp. 937-959, 1996.\n[20] A. R. Bahadir, \"A fully implicit finite-difference scheme for twodimensional\nBurgers' equation\", Applied Mathematics and Computation,\nVol. 137, pp. 131-137, 2003.\n[21] V. K. Srivastava, M. Tamsir, U. Bhardwaj and Y. V. S. S. Sanyasiraju,\n\"Crank-Nicolson scheme for numerical solutions of two dimensional\ncoupled Burgers' equations\", Int. J. Sci. Eng. Research, Vol. 2, no. 5,\npp. 1-6, 2011.\n[22] M. Tamsir and V. K. Srivastava, \"A semi-implicit finite-difference\napproach for two-dimensional coupled Burgers' equations\", Int. J. Sci.\nEng. Research, Vol. 2, no. 6, pp. 1-6, 2011.\n[23] V. K. Srivastava,Ashutosh and M. Tamsir, \"Generating exact solution\nof three-dimensional coupled unsteady nonlinear generalized viscous\nBurgers' equations\", Int. J. Mod. Math. Sci., Vol. 5, no. 1, pp. 1-13,\n2013."]}

In this paper, a modified cubic B-spline differential quadrature method (MCB-DQM) is employed for the numerical simulation of two-space dimensional nonlinear sine-Gordon equation with appropriate initial and boundary conditions. The modified cubic B-spline works as a basis function in the differential quadrature method to compute the weighting coefficients. Accordingly, two dimensional sine-Gordon equation is transformed into a system of second order ordinary differential equations (ODEs). The resultant system of ODEs is solved by employing an optimal five stage and fourth-order strong stability preserving Runge–Kutta scheme (SSP-RK54). Numerical simulation is discussed for both damped and undamped cases. Computational results are found to be in good agreement with the exact solution and other numerical results available in the literature.

In this paper, a numerical solution of the two dimensional nonlinear coupled viscous Burgers equation is discussed with the appropriate initial and boundary conditions using the modified cubic B spline differential quadrature method. In this method, the weighting coefficients are computed using the modified cubic B spline as a basis function in the differential quadrature method. Thus, the coupled Burgers equations are reduced into a system of ordinary differential equations (ODEs). An optimal five stage and fourth order strong stability preserving Runge Kutta scheme is applied to solve the resulting system of ODEs. The accuracy of the scheme is illustrated via two numerical examples. Computed results are compared with the exact solutions and other results available in the literature. Numerical results show that the MCB DQM is efficient and reliable scheme for solving the two dimensional coupled Burgers equation., Comment: 15 pages, 3 figures

In this article, an analytical solution based on the series expansion method is proposed to solve the time-fractional telegraph equation (TFTE) in two and three dimensions using a recent and reliable semi-approximate method, namely the reduced differential transformation method (RDTM) subjected to the appropriate initial condition. Using RDTM, it is possible to find exact solution or a closed approximate solution of a differential equation. The accuracy, efficiency, and convergence of the method are demonstrated through the four numerical examples.

This paper discusses a recently developed semi-analytic technique so called the reduced differential transform method (RDTM) for solving the (1 + n ) – dimensional Burgers' equation. The method considers the use of the appropriate initial or boundary conditions and finds the solution without any discretization, transformation, or restrictive assumptions. Four numerical examples are provided in order to validate the efficiency and reliability of the method and furthermore to compare its computational effectiveness with other analytical methods available in the literature.