In this work I have presented the exact solution of some non-Newtonian fluids in different situat... more In this work I have presented the exact solution of some non-Newtonian fluids in different situation after some preliminaries regarding continuity equation, constitutive equation, equation of motion and integral transforms, the newly exact solutions for second grade fluid with fractional derivatives, Maxwell fluid with fractional derivatives and ordinary Oldroyd-B fluid have been found in which we have calculated the velocity and shear stress. In the name of Allah, the most Gracious and the most Merciful. All praise and glory to Allah for His blessing upon me to finish my thesis. I am thankful to almighty Allah who gave me the courage and patience for doing research. I would like to thank Prof. Constantin Fetecau, my supervisor, who show me the path of research in which I am doing my best. Special thanks to Prof. Corina Fetecau and Prof. Dumitru Vieru for their help and guidance. I would like to thank Prof. Alla Ditta Raza Choudary for his invaluable suggestions. I am especially thankful to the HEC (Higher Education Commission of Pakistan) and Government of Punjab for financial support and providing me an excellent opportunity to complete my Ph.D. My thanks are also for all colleagues from Abdus Salam School of Mathematical Sciences for giving me a chance for their love. I am also grateful to colleagues in Batch IV for the wonderful environment they created in ASSMS. I am very much thankful to my seniors Mudassar, Waseem, Amir, and specially my research fellow Muhammad Kamran and also to juniors Imran, Aziz Ullah and Jamil. Of course, I am grateful to my parents, brothers and sisters for their prayers and love. Finally, I wish to thank the following: All the foreign professors at ASSMS ; Malik, Peer, Don, Chhota and ... (for their friendship and for all the good time we had together); and also Nauman, Shaukat, Shahbaz and Awais (for so many things) and my beloved life partner and children (Shoaib and Warishah) for their love.
Lie symmetry analysis of differential equations proves to be a powerful tool to solve or atleast ... more Lie symmetry analysis of differential equations proves to be a powerful tool to solve or atleast to reduce the order and non-linearity of the equation. The present article focuses on the solution of Generalized Equal Width wave (GEW) equation using Lie group theory. Over the years, different solution methods have been tried for GEW but Lie symmetry analysis has not been done yet. At first, we obtain the infinitesimal generators, commutation table and adjoint table of Generalized Equal Width wave (GEW) equation. After this, we find the one dimensional optimal system. Then we reduce GEW equation into non-linear ordinary differential equation (ODE) by using the Lie symmetry method. This transformed equation can take us to the solution of GEW equation by different methods. After this, we get the travelling wave solution of GEW equation by using the Sine-cosine method. We also give graphs of some solutions of this equation.
Sierpinski networks are networks of fractal nature having several applications in computer scienc... more Sierpinski networks are networks of fractal nature having several applications in computer science, music, chemistry, and mathematics. These networks are commonly used in chaos, fractals, recursive sequences, and complex systems. In this article, we compute various connectivity polynomials such as M -polynomial, Zagreb polynomials, and forgotten polynomial of generalized Sierpinski networks S k n and recover some well-known degree-based topological indices from these. We also compute the most general Zagreb index known as α , β -Zagreb index and several other general indices of similar nature for this network. Our results are the natural generalizations of already available results for particular classes of such type of networks.
This paper aims to explore the possible solutions for the movement of an Oldroyd-B fluid placed u... more This paper aims to explore the possible solutions for the movement of an Oldroyd-B fluid placed under certain conditions, i.e. the fluid is present within two cylinders, which are coaxial and oscillating within. Having said that the governing model will be an Oldroyd-B fluid, we wish to achieve our goal of finding the velocity and shear stress by using some common transformations, namely the Laplace transformation and the Hankel transformation. The final results, for the sake of simplicity, will be expressed in the form of generalized G-function and they satisfy all imposed initial and boundary conditions.
In this work I have presented the exact solution of some non-Newtonian fluids in different situat... more In this work I have presented the exact solution of some non-Newtonian fluids in different situation after some preliminaries regarding continuity equation, constitutive equation, equation of motion and integral transforms, the newly exact solutions for second grade fluid with fractional derivatives, Maxwell fluid with fractional derivatives and ordinary Oldroyd-B fluid have been found in which we have calculated the velocity and shear stress. In the name of Allah, the most Gracious and the most Merciful. All praise and glory to Allah for His blessing upon me to finish my thesis. I am thankful to almighty Allah who gave me the courage and patience for doing research. I would like to thank Prof. Constantin Fetecau, my supervisor, who show me the path of research in which I am doing my best. Special thanks to Prof. Corina Fetecau and Prof. Dumitru Vieru for their help and guidance. I would like to thank Prof. Alla Ditta Raza Choudary for his invaluable suggestions. I am especially thankful to the HEC (Higher Education Commission of Pakistan) and Government of Punjab for financial support and providing me an excellent opportunity to complete my Ph.D. My thanks are also for all colleagues from Abdus Salam School of Mathematical Sciences for giving me a chance for their love. I am also grateful to colleagues in Batch IV for the wonderful environment they created in ASSMS. I am very much thankful to my seniors Mudassar, Waseem, Amir, and specially my research fellow Muhammad Kamran and also to juniors Imran, Aziz Ullah and Jamil. Of course, I am grateful to my parents, brothers and sisters for their prayers and love. Finally, I wish to thank the following: All the foreign professors at ASSMS ; Malik, Peer, Don, Chhota and ... (for their friendship and for all the good time we had together); and also Nauman, Shaukat, Shahbaz and Awais (for so many things) and my beloved life partner and children (Shoaib and Warishah) for their love.
Lie symmetry analysis of differential equations proves to be a powerful tool to solve or atleast ... more Lie symmetry analysis of differential equations proves to be a powerful tool to solve or atleast to reduce the order and non-linearity of the equation. The present article focuses on the solution of Generalized Equal Width wave (GEW) equation using Lie group theory. Over the years, different solution methods have been tried for GEW but Lie symmetry analysis has not been done yet. At first, we obtain the infinitesimal generators, commutation table and adjoint table of Generalized Equal Width wave (GEW) equation. After this, we find the one dimensional optimal system. Then we reduce GEW equation into non-linear ordinary differential equation (ODE) by using the Lie symmetry method. This transformed equation can take us to the solution of GEW equation by different methods. After this, we get the travelling wave solution of GEW equation by using the Sine-cosine method. We also give graphs of some solutions of this equation.
Sierpinski networks are networks of fractal nature having several applications in computer scienc... more Sierpinski networks are networks of fractal nature having several applications in computer science, music, chemistry, and mathematics. These networks are commonly used in chaos, fractals, recursive sequences, and complex systems. In this article, we compute various connectivity polynomials such as M -polynomial, Zagreb polynomials, and forgotten polynomial of generalized Sierpinski networks S k n and recover some well-known degree-based topological indices from these. We also compute the most general Zagreb index known as α , β -Zagreb index and several other general indices of similar nature for this network. Our results are the natural generalizations of already available results for particular classes of such type of networks.
This paper aims to explore the possible solutions for the movement of an Oldroyd-B fluid placed u... more This paper aims to explore the possible solutions for the movement of an Oldroyd-B fluid placed under certain conditions, i.e. the fluid is present within two cylinders, which are coaxial and oscillating within. Having said that the governing model will be an Oldroyd-B fluid, we wish to achieve our goal of finding the velocity and shear stress by using some common transformations, namely the Laplace transformation and the Hankel transformation. The final results, for the sake of simplicity, will be expressed in the form of generalized G-function and they satisfy all imposed initial and boundary conditions.
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