^{1,2}, Martin Calasan

^{2}, Drasko Kovac

^{1}and Ivana Tosic

^{3}

### Abstract

The problem of finding an analytical solution of some families of Kepler transcendental equation is studied in some detail, by the Special Trans Functions Theory – STFT. Thus, the STFT mathematical approach in the form of STFT iterative methods with a novel analytical solutions are presented. Structure of the STFT solutions, numerical results and graphical simulations confirm the validity of the basic principle of the STFT. In addition, the obtained analytical results are compared with the calculated values of other analytical methods for alternative proving its significance. Undoubtedly, the proposed novel analytical approach implies qualitative improvement in comparison with conventional numerical and analytical methods.

The paper is a part of the research done within the project No.01/2337/14, supported and financed by Ministry of Science of Montenegro. The authors would like to thank to this continuous interest and support.

I. INTRODUCTION II. REVIEW OF THE LITERATURE KNOWN METHODS FOR KEPLER’S EQUATION SOLVING III. OBTAINING AN ANALYTICAL SOLUTION TO THE KEPLER’S TRANSCENDENTAL EQUATION (1) BY USING A SIMPLE ITERATIVE METHOD (SIM) WITHIN THE SPECIAL TRANS FUNCTIONS THEORY A. Concerning an exact analytical closed-form solution of transcendental equations (11) and (12) B. Determining the planet position P(r, v) at time t by using the Simple iterative method within the Special trans functions theory IV. OBTAINING A NOVEL ANALYTICAL SOLUTION TO THE KEPLER’S TRANSCENDENTAL EQUATION BY USING AN ADVANCED ITERATIVE PROCEDURE WITHIN THE SPECIAL TRANS FUNCTIONS THEORY A. Concerning an approximate STFT solution obtained within Eqs. (53) and (54) V. NUMERICAL RESULTS A. Numerical results analysis based on formulae (20) (or, (23)) and (57) (or, (57a)) VI. CONCLUSIONS

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### Abstract

The problem of finding an analytical solution of some families of Kepler transcendental equation is studied in some detail, by the Special Trans Functions Theory – STFT. Thus, the STFT mathematical approach in the form of STFT iterative methods with a novel analytical solutions are presented. Structure of the STFT solutions, numerical results and graphical simulations confirm the validity of the basic principle of the STFT. In addition, the obtained analytical results are compared with the calculated values of other analytical methods for alternative proving its significance. Undoubtedly, the proposed novel analytical approach implies qualitative improvement in comparison with conventional numerical and analytical methods.

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