The improved residual power series method for a system of differential equations: a new semi-numerical method

ABSTRACTThis paper presents the extension of Improved Residual Power Series Method (IRPSM) towards the system of ordinary differential equations (ODEs). The present system of ODEs is based on the thin film flow over an inclined planar surface. The proposed system is highly nonlinear. Additionally, some embedded factors are taken into the flow analysis in order to investigate the impacts of these parameters on the flow profiles. The method is coded in MATHEMATICA 12.0 software. The results of the present analysis show that the IRPSM has a fast convergence. The impacts of embedded parameters have been successfully investigated and have comparable effects on the flow profiles. Comparison of the present results with the results present in the literature has confirmed the validity of IRPSM. The IRPSM is also applicable to the systems of both linear and highly nonlinear ordinary and partial differential equations.KEYWORDS: Thin film flowboundary value probleminclined planar surfaceIRPSM Nomenclature Symbol/Expression=Nameu,v,w=Velocity components ms−1x,y,z=Coordinates mΩ=Angular velocity rads−1T=Temperature KB0=Magnetic field strength Ω1/2 m−1 s−1/2kg1/2p=Pressure Paμ=Dynamic viscosity kgm−1s−1ρ=Density kgm−3ν=Kinematic viscosity m2 s−1k=Thermal conductivity Wm−1 K−1ρCp=Heat capacitance Jm−3 K−1σ=Electrical conductivity Ω−1 m−1W0=Spraying velocity ms−1α=Angle of inclination 0gˆ=Gravitational force m2 s−1L=Film thickness mδ=Normalized thickness factorM=Magnetic factorPr=Prandtl numberRex=Reynolds numberAbbreviations=ADM=Adomian decomposition methodVIM=Variational iteration methodHPM=Homotopy perturbation methodHAM=Homotopy analysis methodDTM=Differential transform methodOHAM=Optimal homotopy asymptotic methodLFRDTM=Local fractional reduced differential transformLFLVIM=Local fractional Laplace variational iteration methodKdV=Coupled Korteweg – De Vries equationHAM=Homotopy analysis methodOCM=Operational collocation methodFDM=Finite difference methodRPSM=Residual power series methodIRPSM=Improved residual power series methodIVPs=Initial value problemsBVPs=Boundary value problemsAcknowledgmentsThe authors express their cordial thanks to the respected Editor in chief and honorable reviewers for their valuable suggestions and comments to improve the presentation of this article.Disclosure statementThe authors declare that they have no known competing financial interest.Additional informationFundingNo funding was received for this work.Notes on contributorsAbdullah DawarAbdullah Dawar received his BS degree from Islamia College University, Peshawar, Pakistan and MS degree from Qurtuba University of Science and Information Technology, Peshawar, Pakistan. He is currently a PhD student at the Department of Mathematics, Abdul Wali Khan University, Mardan, Pakistan. He has published more than 80 research articles in well-reputed journals during his education career. His research proficiency includes magnetohydrodynamic, nanofluids, hybrid nanofluids, heat and mass transfer problems. Currently, he is working on a new method called the Improved Residual Power Series Method (IRPSM).Hamid KhanHamid Khan received his bachelor's degree from the Government postgraduate college Mardan and his MS degree from Kohat University of Science and Technology, Kohat. His MS research work was on Matrix Inversion and the Solution of a system of linear equations. During his MS, he wrote two research articles from his work and visited Japan to present his work in IPS Program at Saga University Japan. He discussed in detail his own research theme (Matrix Inversion and Solution of system of linear equations) with Prof Minamuto Teruyu (Department of Information Science Saga University Japan) and Prof Okazaki Yoshiaki (Kyushu Institute of Technology (KIT)). He gave two lectures on his own research theme to Professors and their research students at Saga University Japan. He is currently an Associate professor at the Department of Mathematics, Islamia College University Peshawar, Pakistan. Hamid Khan received his Ph.D. degree from Islamia college university Peshawar in 2013. He has published more than 43 research articles in well-reputed journals during his career. He also produced 7 Ph.D. and 16 MS students. Most of his research is on solving differential equations using the series solution techniques. He is working on a new method called the Improved Residual Power Series Method (IRPSM). He worked with serval MS and Ph.D. students on IRSPM. He published two research articles and submitted more than seven research articles based on the IRPSM technique.Saeed IslamSaeed Islam is an esteemed academician and mathematician who have made significant contributions to the field of Mathematics in Pakistan. With an impressive academic background and a remarkable career, he has not only excelled in research but also played a crucial role in shaping the educational landscape of the country. Professor Dr. Saeed Islam earned his PhD in Mathematics from a prestigious university in China (Harbin Institute of Technology), showcasing his dedication to pursuing higher education and specialization in Fluid mechanics and its applications in bio mathematics. Armed with a strong foundation in Mathematics, he embarked on a teaching career at COMSATS University Islamabad Pakistan, where he honed his pedagogical skills and passion for imparting knowledge to young minds. This marked the beginning of his journey towards becoming a prominent figure in the academic world. After his tenure at Comsat Islamabad, Professor Dr. Saeed Islam joined Abdul Wali Khan University (AWKUM), where he would leave an indelible mark on the institutions academic and research landscape. As a mathematician and educator, he has produced an outstanding number of 40 PhDs and over 120 MPhils, which reflects his commitment to nurturing and guiding the next generation of scholars in the field of Mathematics. Based on his research credentials, George Masson University already offered a position of Adjunct professor. Professor Dr. Saeed Islam's dedication to research is truly remarkable. With an impressive tally of 500 research articles published, having an impact factor of over 1000 and receiving more than 10200 citations (google scholar citations), 6700 citations (Web of Science Citations) & more than 7000 citations (Scopus citations), he has not only contributed significantly to the advancement of Mathematics but also left an indelible mark in the scientific community. His research work has been recognized and acknowledged both nationally and internationally. Professor Dr. Saeed Islam's contributions to Mathematics and research have been recognized and celebrated through various accolades and awards. His achievements have earned him the prestigious Quaid-E-Azam Gold Medal in 2013, a testament to his exceptional dedication and outstanding achievements in the field. Ministry of Science and Technology (Pakistan) awarded him the research productivity award in 2009, 2010, 2011, and 2014. The impact of Professor Dr. Saeed Islam's work has been acknowledged beyond Pakistan's borders. He was ranked 8th in the country by the web of sciences, placing him among the top academicians in Pakistan. Moreover, being in the top 2% of scientists globally in 2020 further highlights the international recognition of his research contributions. Professor Dr. Saeed Islam's commitment to research is evident not only in his publication record but also through his involvement in research projects as a PI and Co-PI. Having completed five research projects as a PI and Co-PI, he has actively contributed to advancing the frontiers of Mathematics and solving real-world problems. His dedication and productivity in research have been recognized by the Ministry of Science and Technology, which awarded him productivity awards in various years. Apart from his exemplary academic and research achievements, Professor Dr. Saeed Islam has also demonstrated exceptional leadership skills. He has held several administrative positions throughout his career, including serving as the Chairman of the Department of Mathematics from 2012 to 2020, Registrar in 2013, Director of the Office of Research, Innovation, and Commercialization (ORIC) in 2012, and Controller of Examinations. Currently, he is working as a Dean of the faculty (Physical & Numerical Sciences) and Director Academics & Research of Abdul Wali Khan University Mardan. Professor Dr. Saeed Islam's dedication to academia and his university's growth is evident through his involvement in various statutory bodies and committees. He has been an active member of Senate, Syndicate, Selection Boards, Academic Council, Affiliation Committee, ASRB (Academic Staff Review Board), Board of Studies, and other relevant bodies, contributing to the development and formulation of academic policies and guidelines. Professor Dr. Saeed Islam's journey in academia and Mathematics is an inspiration to aspiring researchers and educators alike. With an outstanding academic background, numerous publications, mentoring of numerous scholars, and a strong administrative presence, he has left an indelible mark on the academic and research landscape of Pakistan. His contributions to the field of Mathematics have not only earned him national and international recognition but have also advanced the frontiers of knowledge in the subject. Through his dedication to research, teaching, and administrative roles, he has played a vital role in shaping the future of education in Pakistan. Professor Dr. Saeed Islam's achievements stand as a testament to the transformative power of education and research, inspiring generations to come.Waris KhanWaris Khan received his M.Sc. degree from Quaid-e-Azam University, Islamabad, Pakistan, M.S. degree from COMSATS University Abbottabad, KP, Pakistan and Ph.D. degree from Islamia College University, Peshawar, Pakistan. He currently holds the position of Assistant Professor in Mathematics at Hazara University, Mansehra, KP, Pakistan. He has a prolific academic record with 80 articles published in high-impact factor journals, along with active participation in national conferences. His research expertise spans various areas including magnetohydrodynamics, nanofluids, heat transfer, hall effect, electrohydrodynamics, thin films, and heat exchangers.