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We established several novel inequalities of Gronwall–Pachpatte type on time scales. Our results can be used as handy tools to study the qualitative and quantitative properties of the solutions of the initial boundary value problem for a partial delay dynamic equation. The Leibniz integral rule on time scales has been used in the technique of our proof. Symmetry plays an essential role in determining the correct methods to solve dynamic inequalities. [ABSTRACT FROM AUTHOR]

In this article, with the help of Leibniz integral rule on time scales, we prove some new dynamic inequalities of Gronwall–Bellman–Pachpatte-type on time scales. These inequalities can be used as handy tools to study the qualitative and quantitative properties of solutions of the initial boundary value problem for partial delay dynamic equation. [ABSTRACT FROM AUTHOR]

In engineering disciplines, many important problems are to be formed as boundary value problems (BVPs) that have conditions that are specified at the extremes. To handle such problems, the conventional shooting method that transforms BVPs into initial value problems has been extensively used, but it does not always guarantee solving the problem due to the possible failure of finding a proper initial guess. This paper proposes a universal initial guess finder that is composed of Bézier functions. Various dimensional problems that include Lambert's problem for several orbits around the spherically symmetric Earth are studied to validate the efficacy of the proposed approach, and the results are compared. [ABSTRACT FROM AUTHOR]