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Brings mathematics to bear on your real-world, scientific problemsMathematical Methods in Interdisciplinary Sciences provides a practical and usable framework for bringing a mathematical approach to modelling real-life scientific and technological problems. The collection of chapters Dr. Snehashish Chakraverty has provided describe in detail how to bring mathematics, statistics, and computational methods to the fore to solve even the most stubborn problems involving the intersection of multiple fields of study. Graduate students, postgraduate students, researchers, and professors will all benefit significantly from the author's clear approach to applied mathematics.The book covers a wide range of interdisciplinary topics in which mathematics can be brought to bear on challenging problems requiring creative solutions. Subjects include:Structural static and vibration problemsHeat conduction and diffusion problemsFluid dynamics problemsThe book also covers topics as diverse as soft computing and machine intelligence. It concludes with examinations of various fields of application, like infectious diseases, autonomous car and monotone inclusion problems.

<p><b>Examines numerical and semi&#45;analytical methods for differential equations that can be used for solving practical ODEs and PDEs</b> <p>This student&#45;friendly book deals with various approaches for solving differential equations numerically or semi&#45;analytically depending on the type of equations and offers simple example problems to help readers along. <p>Featuring both traditional and recent methods, <i>Advanced Numerical and Semi&#45;Analytical Methods for Differential Equations</i> begins with a review of basic numerical methods. It then looks at Laplace, Fourier, and weighted residual methods for solving differential equations. A new challenging method of Boundary Characteristics Orthogonal Polynomials &#40;BCOPs&#41; is introduced next. The book then discusses Finite Difference Method &#40;FDM&#41;, Finite Element Method &#40;FEM&#41;, Finite Volume Method &#40;FVM&#41;, and Boundary Element Method &#40;BEM&#41;. Following that, analytical/semi<i>&#45;</i>analytic methods like Akbari Ganji&#39;s Method &#40;AGM&#41; and Exp&#45;function are used to solve nonlinear differential equations. Nonlinear differential equations using semi&#45;analytical methods are also addressed, namely Adomian Decomposition Method &#40;ADM&#41;, Homotopy Perturbation Method &#40;HPM&#41;, Variational Iteration Method &#40;VIM&#41;, and Homotopy Analysis Method &#40;HAM&#41;. Other topics covered include: emerging areas of research related to the solution of differential equations based on differential quadrature and wavelet approach; combined and hybrid methods for solving differential equations; as well as an overview of fractal differential equations. Further, uncertainty in term of intervals and fuzzy numbers have also been included, along with the interval finite element method. This book: <ul> <li>Discusses various methods for solving linear and nonlinear ODEs and PDEs</li> <li>Covers basic numerical techniques for solving differential equations along with various discretization methods</li> <li>Investigates nonlinear differential equations using semi&#45;analytical methods</li> <li>Examines differential equations in an uncertain environment</li> <li>Includes a new scenario in which uncertainty &#40;in term of intervals and fuzzy numbers&#41; has been included in differential equations</li> <li>Contains solved example problems, as well as some unsolved problems for self&#45;validation of the topics covered</li> </ul> <p><i>Advanced Numerical and Semi&#45;Analytical Methods for Differential Equations</i> is an excellent textbook for graduate as well as post graduate students and researchers studying various methods for solving differential equations, numerically and semi&#45;analytically.

In this manuscript, a new hybrid technique viz Sawi transform-based homotopy perturbation method is implemented to solve one-dimensional shallow water wave equations. In general, the quantities involved with such equations are commonly assumed to be crisp, but the parameters involved in the actual scenario may be imprecise/uncertain. Therefore, fuzzy uncertainty is introduced as an initial condition. The main focus of this study is to find the approximate solution of one-dimensional shallow water wave equations with crisp, as well as fuzzy, uncertain initial conditions. First, by taking the initial condition as crisp, the approximate series solutions are obtained. Then these solutions are compared graphically with existing solutions, showing the reliability of the present method. Further, by considering uncertain initial conditions in terms of Gaussian fuzzy number, the governing equation leads to fuzzy shallow water wave equations. Finally, the solutions obtained by the proposed method are presented in the form of Gaussian fuzzy number plots.

The study of vibration equations of large membranes is crucial in various scientific and engineering fields. Analyzing the vibration equations of bridges, roofs, and spacecraft structures helps in designing structures that resist excessive oscillations and potential failures. Aircraft wings, parachutes, and satellite components often behave like large membranes. Understanding their vibration characteristics is essential for stability, efficiency, and durability. Studying large membrane vibration involves solving partial differential equations and eigenvalue problems, contributing to advancements in numerical methods and computational physics. In this paper, the Elzaki transformation decomposition method and the Shehu transformation decomposition method, along with inverse Elzaki and inverse Shehu transformations, are used to investigate the fractional vibration equation of a large membrane. The solutions are obtained in terms of Mittag–Leffler functions.

This paper presents a new approach for solving FFLP problems using a double parametric form (DPF), which is critical in decision-making scenarios characterized by uncertainty and imprecision. Traditional linear programming methods often fall short in handling the inherent vagueness in real-world problems. To address this gap, an innovative method has been proposed which incorporates fuzzy logic to model the uncertain parameters as TFNs, allowing for a more realistic and flexible representation of the problem space. The proposed method stands out due to its integration of fuzzy arithmetic into the optimization process, enabling the handling of fuzzy constraints and objectives directly. Unlike conventional techniques that rely on crisp approximations or the defuzzification process, the proposed approach maintains the fuzziness throughout the computation, ensuring that the solutions retain their fuzzy characteristics and better reflect the uncertainties present in the input data. In summary, the proposed method has the ability to directly incorporate fuzzy parameters into the optimization framework, providing a more comprehensive solution to FFLP problems. The main findings of this study underscore the method’s effectiveness and its potential for broader application in various fields where decision-making under uncertainty is crucial.

In this article, the Gershgorin disk theorem in complex interval matrices is proposed for enclosing interval eigenvalues. This is a non-iterative method for finding eigenvalue bounds for both real and imaginary parts. Moreover, we are able to find gaps between the clusters of interval eigenvalues and have compared the results with the previous theorems for interval eigenvalue bounds for complex interval matrices. These results can be decisive for checking Hurwitz and Schur stability of complex interval matrices that appear in uncertain dynamical systems. Further bounds obtained from the present formulae can be considered as the initial bounds for iterative methods.

The primary objective of this manuscript is to obtain the approximate analytical solution of Camassa–Holm (CH), modified Camassa–Holm (mCH), and Degasperis–Procesi (DP) equations with time-fractional derivatives labeled in the Caputo sense with the help of an iterative approach called fractional reduced differential transform method (FRDTM). The main benefits of using this technique are that linearization is not required for this method and therefore it reduces complex numerical computations significantly compared to the other existing methods such as the perturbation technique, differential transform method (DTM), and Adomian decomposition method (ADM). Small size computations over other techniques are the main advantages of the proposed method. Obtained results are compared with the solutions carried out by other technique which demonstrates that the proposed method is easy to implement and takes small size computation compared to other numerical techniques while dealing with complex physical problems of fractional order arising in science and engineering.

In the present investigation, the buckling behavior of Euler–Bernoulli nanobeam, which is placed in an electro-magnetic field, is investigated in the framework of Eringen’s nonlocal theory. Critical buckling load for all the classical boundary conditions such as “Pined–Pined (P-P), Clamped–Pined (C-P), Clamped–Clamped (C-C), and Clamped-Free (C-F)” are obtained using shifted Chebyshev polynomials-based Rayleigh-Ritz method. The main advantage of the shifted Chebyshev polynomials is that it does not make the system ill-conditioning with the higher number of terms in the approximation due to the orthogonality of the functions. Validation and convergence studies of the model have been carried out for different cases. Also, a closed-form solution has been obtained for the “Pined–Pined (P-P)” boundary condition using Navier’s technique, and the numerical results obtained for the “Pined–Pined (P-P)” boundary condition are validated with a closed-form solution. Further, the effects of various scaling parameters on the critical buckling load have been explored, and new results are presented as Figures and Tables. Finally, buckling mode shapes are also plotted to show the sensitiveness of the critical buckling load.

The present paper investigates the numerical solution of an imprecisely defined nonlinear coupled time-fractional dynamical model of marriage (FDMM). Uncertainties are assumed to exist in the dynamical system parameters, as well as in the initial conditions that are formulated by triangular normalized fuzzy sets. The corresponding fractional dynamical system has first been converted to an interval-based fuzzy nonlinear coupled system with the help of a single-parametric gamma-cut form. Further, the double-parametric form (DPF) of fuzzy numbers has been used to handle the uncertainty. The fractional reduced differential transform method (FRDTM) has been applied to this transformed DPF system for obtaining the approximate solution of the FDMM. Validation of this method was ensured by comparing it with other methods taking the gamma-cut as being equal to one.