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This paper will use Lagrange parameter in Adomain decomposition method to suggest new method for solving nonlinear differential equation. This method will be highly order convergent. Also, this method will be compared with old existence method. At last, some numerical examples will be given to illustrate the efficiency of newly developed method. [ABSTRACT FROM AUTHOR]

Osteoporosis is a fatal bone-wearing malady and a substantial reason behind the impermanence of human life and economic burden. Risedronate Sodium along with Ursolic acid has been studied to ameliorate osteoporosis. To bypass problems associated with bioavailability, we have developed a microneedle transdermal patch loaded with optimized formulation nanotransfersomes. It was optimized using three factor, three-level Central composite design with independent variables namely, the concentration of phospholipid, surfactant, and sonication time on dependent variables (vesicle size, entrapment efficiency and Polydispersity index). Vesicles of size 271.9 ± 8.45 nm with PDI 0.184 ± 0.01, having entrapment efficiency of 86.12 ± 5.20% and 85.65 ± 4.88% for RIS and UA respectively were observed. In vitro release study showed the sustained release pattern with 78.16 ± 1.12% and 75.72 ± 1.01% release of RIS and UA respectively. Dissolving MN patch prepared from gelatin was found to have good strength and folding endurance with uniform drug content (98.68 ± 0.004%). Ex vivo permeation study revealed that up to 80% of the drug can be permeated within 24 h. CLSM analysis was also performed to show penetration of RU-NTRs. From the results obtained, we can conclude that dissolving MN patch loaded with RU-NTRs has great potential than its conventional counterpart., Competing Interests: Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper., (Copyright © 2023 Elsevier B.V. All rights reserved.)