Compact Mathematics for Undergraduates: Formulas and Identities -Part I | Book Publisher International

Mathematics formulae are fundamentally important because they express abstract relationships and principles that enable generalization, precise communication, and problem-solving across diverse contexts. Their abstraction fosters deeper understanding and broad applications, both within mathematics and in scientific disciplines. Mathematics formulae arise from abstraction—a process that extracts the underlying structures and patterns from concrete problems. By moving from specific examples to generalized forms, formulae allow for broader applicability and reveal connections between different areas of mathematics. mathematical formulae are indispensable tools for abstract reasoning, generalization, effective communication, and advancing knowledge in both mathematics and the applied sciences.

 

Author(s) Details

Dr. Thiruchinapalli Srinivas
Audisankara Deemed to be University, Gudur Bypass, Gudur, India.

 

Please see the book here :- https://doi.org/10.9734/bpi/mono/978-93-88417-25-9

vSimplification of Lie's Similarity Transformation Method of Solving Coupled Nonlinear Partial Differential Equations and Exact Dyon Solutions SU(2)Yang-Mills-Higg's Field Equations | Chapter 5 | Research Updates in Mathematics and Computer Science Vol. 9

 

For any \(\mathrm{n}\) number of coupled nonlinear partial differential equations for spherically symmetric field equations of the typer \({ }^2\left(\frac{\partial^2 \phi_j}{\partial r^2}-\frac{\partial^2 \phi_j}{\partial t^2}\right)=F \_i(\phi \mathrm{j})\), where \(\mathrm{j}=1,2, \ldots \mathrm{n}\), are the number of dependent variables and \(\mathrm{F} \mathrm{i}\left(\phi \_\mathrm{j}\right)\) are any functions of dependent variables \(\phi_j, \mathrm{j}=1,2, . . \mathrm{n}\). and free of independent variables \(\mathrm{r}\) and \(\mathrm{t}\) then a similarity variable is found as \(\mathrm{s}(\mathrm{r}, \mathrm{t})=\mathrm{r} /\left[\left(r^2-t^2\right)-\kappa t / \tau+\kappa^2 /\left(4 \tau^2\right)\right]\), where \(\kappa\) and \(\tau \neq 0\) are arbitrary integration constants. Using \(s(r, t)\) above coupled partial differential equations can be transformed into coupled ordinary differential equations. This result may reduce lengthy calculations for finding similarity transformations of coupled partial differential equations. Using this similarity variable two exact Dyon solutions of spherically symmetric Yang-Mills-Higg's field equations are found with 'circular functions.' For which known solutions are with hyperbolic functions.

 

Author(s) Details

B.V. Baby

3/88, Jadkal Post, Udupi District, Karnataka State -576 233, India.

 

Please see the link:- https://doi.org/10.9734/bpi/rumcs/v9/607

Discovering Two Novel Solutions: Circular and Hyperbolic Functions of SU(2) Yang-Mills Gauge Field Equations Using Null-Tetrad Formalism | Chapter 5 | Mathematics and Computer Science: Contemporary Developments Vol. 4

 

Two more new exact closed-form solutions of SU(2) Yang-Mills gauge field equations were found by the Carmeli-Charash-Kaye-Kh. Huleihil null-tetrad formalism. Unlike the reported first solutions with special functions of elliptic integrals, these new solutions are circular and hyperbolic trigonometric functions with two arbitrary integration constants. Electric parts corresponding to the solutions with circular functions cases are found as zero and nonzero magnetic parts so concluded as monopole solutions. In solutions with hyperbolic function cases, both electric and magnetic parts of Yang-Mills fields are zero.  Yang-Mills potentials and fields are also reported for all solutions.

 

Author (s) Details

B.V. Baby
3\88, Jadkal Post, Udupi District, Karnataka State, 576 233, India.

 

Please see the book here:- https://doi.org/10.9734/bpi/mcscd/v4/1847

Simplification of Lie's Similarity Transformation Method of Solving Coupled Nonlinear Partial Differential Equations and Exact Dyon Solutions SU(2)Yang-Mills-Higg's Field Equations | Chapter 5 | Research Updates in Mathematics and Computer Science Vol. 9

For any n number of coupled nonlinear partial differential equations for spherically symmetric field equations of the typer^2 ( (ð^2 ϕ_j)/(∂r^2 )-(∂^2 ϕ_j)/(∂t^2 ))=F_i (ϕ_j ), where j=1,2,…n, are the number of dependent variables and F_i (ϕ_j ) are any functions of dependent variables ϕ_j,j=1,2,..n. and free of independent variables r and t then a similarity variable is found as s(r,t)=r/[(r^2−t^2 )−κt/τ+κ^2/(4τ^2 )], where κ and τ≠0 are arbitrary integration constants. Using s(r,t) above coupled partial differential equations can be transformed into coupled ordinary differential equations. This result may reduce lengthy calculations for finding similarity transformations of coupled partial differential equations. Using this similarity variable two exact Dyon solutions of spherically symmetric Yang-Mills-Higg's field equations are found with ‘circular functions.' For which known solutions are with hyperbolic functions.

 

Author (s) Details

 

B.V. Baby
3/88, Jadkal Post, Udupi District, Karnataka State -576 233, India.

 

Please see the book here:- https://doi.org/10.9734/bpi/rumcs/v9/607