Research Interests: Mathematics and Diffusion
Matrix (MAT) Laboratory (LAB) is abbreviated as MATLAB®. It is a programming language that is designed to perform numerical calculations with matrix-based systems. It has a collection of built-in mathematical supports such as... more
Matrix (MAT) Laboratory (LAB) is abbreviated as MATLAB®. It is a programming language that is designed to perform numerical calculations with matrix-based systems. It has a collection of built-in mathematical supports such as diagonalization of matrix, optimization, and solving system of equations and differential equations. MATLAB® also works as an interpreter, so we can execute the program and get an intermediate result after each step. However, in computer languages, viz. Pascal or C, we need the whole program to be ready before we can get any result. As such, using MATLAB®, we can check the progress of our project continuously by building code in step by step. Hence, it works like a debugger ( Pratap, 2005 ; Chapman, 2007 ; Kattan, 2007 ).
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Research Interests: Engineering and Diffusion
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Diffusion is an important phenomenon in various fields of science and engineering. It may arise in a variety of problems viz. in heat transfer, fluid flow problem and atomic reactors etc. As such these diffusion equations are being solved... more
Diffusion is an important phenomenon in various fields of science and engineering. It may arise in a variety of problems viz. in heat transfer, fluid flow problem and atomic reactors etc. As such these diffusion equations are being solved throughout the globe by various methods. It has been seen from literature that researchers have investigated these problems when the material properties, geometry (domain) etc. are in crisp (exact) form which is easier to solve. But in real practice the parameters used in the modelled physical problems are not crisp because of the experimental error, mechanical defect, measurement error etc. In that case the problem has to be defined with uncertain parameters and this makes the problem complex. In this chapter related uncertain differential equation of various diffusion problems will be investigated using finite element method, which may be called fuzzy or interval finite element method.
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Abstract Epistemic uncertainties are one of the important class of uncertainties that occur in heat transfer problems. These uncertainties are caused due to experimental error, improper or insufficient knowledge of operating conditions,... more
Abstract Epistemic uncertainties are one of the important class of uncertainties that occur in heat transfer problems. These uncertainties are caused due to experimental error, improper or insufficient knowledge of operating conditions, and imprecise or vague information about the involved constants, coefficients, and parameters. As such, in this paper, a convection-diffusion heat transfer problem is considered with epistemic uncertainties viz. thermal conductivity, ambient temperature, convective heat transfer constant, and source. The uncertainties are taken as intervals with a 5% error to the exact value. To compute interval uncertainties effectively, a transformation-based technique is proposed that converts intervals into single variable linear functions. Then to solve the governing differential equation with uncertainties, the transformation-based method is used with Interval Finite Element Method (IFEM), and accordingly, the problem is modeled. Finally, the uncertain distribution of temperatures is quantified for a rectangular plate with nonhomogeneous boundary conditions, and the sensitiveness of the uncertainties are investigated.
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Research Interests: Mathematics and Elsevier
Uncertainties play a major role in stochastic mechanics problems. To study the trajectory involved in stochastic mechanics problems generally, probability distributions are considered. Accordingly, the stochastic mechanics problems govern... more
Uncertainties play a major role in stochastic mechanics problems. To study the trajectory involved in stochastic mechanics problems generally, probability distributions are considered. Accordingly, the stochastic mechanics problems govern by stochastic differential equations followed by Markov process. However, the observation still lacks some sort of uncertainties, which are important but ignored. These imprecise uncertainties involved in the various factors affecting the constants, coefficients, initial, and boundary conditions. Hence, there may be a possibility to model a more reliable strategy that will quantify the uncertainty with better confidence. In this context, a computational method for solving fuzzy stochastic Volterra-Fredholm integral equation, which is based on the Block Pulse Functions (BPFs) using fuzzy stochastic operational matrix, is presented. The developed model is used to investigate a test problem of fuzzy stochastic Volterra integral equation and the result...
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Diffusion is an important phenomenon in various fields of science and engineering. These problems depend on various parameters viz. diffusion coefficients, geometry, material properties, initial and boundary conditions etc. Governing... more
Diffusion is an important phenomenon in various fields of science and engineering. These problems depend on various parameters viz. diffusion coefficients, geometry, material properties, initial and boundary conditions etc. Governing differential equations with deterministic parameters have been well studied. But, in real practice these parameters may not be crisp (exact) rather it involves vague, imprecise and incomplete information about the system variables and parameters. Uncertainties occur due to error in measurements, observations, experiments, applying different operating conditions or it may be due to maintenance induced errors, etc. As such, it is an important concern to model these type of uncertainties. Traditionally uncertain problems are modelled through probabilistic approach. But probabilistic methods may not able to deliver reliable results at the required precision without sufficient data. In this context, interval and fuzzy theory may be used to manage such uncert...
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Research Interests: Mathematics and Elsevier
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This chapter describes the interval finite elements for space truss, space frame, and linear tetrahedral elements. A systematic formulation of interval stiffness matrices and the elemental equations is presented. Uncertain interval... more
This chapter describes the interval finite elements for space truss, space frame, and linear tetrahedral elements. A systematic formulation of interval stiffness matrices and the elemental equations is presented. Uncertain interval parameters and the boundary conditions are considered to formulate interval elemental equations by using interval finite element method. It may again be noted that due to the interval computations involved, the interval finite element method is bit more complex than the traditional finite element method. But the essence of uncertainty may be handled for actual practical problems. But direct application of interval parameters may be difficult. Hence, the interval parameters are transformed to crisp form (as done in previous chapters) by using parametric representation, and then the modified interval parameters are operated through the systematic procedure to obtain the interval stiffness matrices.
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Recently, Wavelet Method (WM) is becoming a powerful tool to solve various types of differential equations. In this method orthogonal wavelet functions are used as shape functions which are easier to compute. Till date WM has been used... more
Recently, Wavelet Method (WM) is becoming a powerful tool to solve various types of differential equations. In this method orthogonal wavelet functions are used as shape functions which are easier to compute. Till date WM has been used for crisp problems that is where the variables and parameters in the differential equations are considered as crisp. But generally in real world problems, every system contains uncertainty and this makes the corresponding mathematical model as uncertain. The uncertain and imprecise parameters make the system complex. Here the uncertainty has been managed by considering the parameters as interval. Accordingly, WM method is modified and Interval Wavelet Method (IWM) has been proposed to solve ODEs.
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In this chapter, we will develop the MATLAB® codes for three-dimensional interval finite element, viz. that of space truss, space frame, and linear tetrahedral elements. A systematic procedure is again followed to develop the MATLAB®... more
In this chapter, we will develop the MATLAB® codes for three-dimensional interval finite element, viz. that of space truss, space frame, and linear tetrahedral elements. A systematic procedure is again followed to develop the MATLAB® codes. A set of MATLAB® functions are created first, and then these are executed to investigate few example problems.
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Stochastic differential equations (SDEs) with impreciseness and vagueness form uncertain stochastic differential equations (SDE), which may be the more generalized differential equation for handling uncertainties. In this chapter, two... more
Stochastic differential equations (SDEs) with impreciseness and vagueness form uncertain stochastic differential equations (SDE), which may be the more generalized differential equation for handling uncertainties. In this chapter, two different approaches for solving uncertain stochastic differential equations (SDE) are discussed. Uncertainties are taken in the initial conditions as well as the associated parameters in terms of triangular fuzzy numbers (TFN). The limit method for fuzzy arithmetic has been used as a tool to handle the fuzzy stochastic differential equation (FSDE). In particular, a system of Ito SDEs has been analyzed with fuzzy parameters. Further, the fuzzy Euler-Maruyama method (FEMM) and fuzzy Milstein method (FMM) are demonstrated through an example problem for different cases.
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Research Interests: Mathematics and Diffusion
In general, theories related to the solution of various problems involved in different fields of science and engineering have been well established. Usually, the problems of mechanics, heat transfer, and electromagnetics are modeled by... more
In general, theories related to the solution of various problems involved in different fields of science and engineering have been well established. Usually, the problems of mechanics, heat transfer, and electromagnetics are modeled by partial differential equations. These problems possess complexity due to the presence of involved parameters, surrounding temporal, and/or spatial scales or the discontinuities in the domain. Hence, the modeling and computer simulations for these problems are often challenging. There exist various well-known numerical methods to handle these problems. In particular, finite element method (FEM) is found to be a versatile method. But FEM has been developed and used where no uncertainties are usually modeled and/or introduced. To handle the uncertainties mentioned in the previous chapter, suitable numerical methods should be developed to understand. The efficiency and accuracy of the solution depends on data such as material properties, geometry, and bou...
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Research Interests: Mathematics and Elsevier
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In this paper an alternative approach to solve uncertain Stochastic Differential Equation (SDE) is proposed. This uncertainty occurs due to the involved parameters in system and these are considered as Triangular Fuzzy Numbers (TFN). Here... more
In this paper an alternative approach to solve uncertain Stochastic Differential Equation (SDE) is proposed. This uncertainty occurs due to the involved parameters in system and these are considered as Triangular Fuzzy Numbers (TFN). Here the proposed fuzzy arithmetic in [2] is used as a tool to handle Fuzzy Stochastic Differential Equation (FSDE). In particular, a system of Ito stochastic differential equations is analysed with fuzzy parameters. Further exact and Euler Maruyama approximation methods with fuzzy values are demonstrated and solved some standard SDE.
Scattering of neutron collision inside a reactor depends upon geometry of the reactor, diffusion coefficient and absorption coefficient etc. In general these parameters are not crisp and hence we may get uncertain neutron diffusion... more
Scattering of neutron collision inside a reactor depends upon geometry of the reactor, diffusion coefficient and absorption coefficient etc. In general these parameters are not crisp and hence we may get uncertain neutron diffusion equation. In this paper we have investigated the above problem for a bare triangular homogeneous reactor. Here the uncertain governing differential equation is modelled by a modified fuzzy finite element method using newly proposed interval arithmetic. Obtained eigenvalues by the proposed method are studied in detail. Further the eigenvalues are compared with the classical finite element method in special cases and various uncertain results have been discussed.
Here a novel idea to handle imprecise or vague set viz. Pseudo fuzzy set has been proposed. Pseudo fuzzy set is a triplet of element and its two membership functions. Both the membership functions may or may not be dependent. The... more
Here a novel idea to handle imprecise or vague set viz. Pseudo fuzzy set has been proposed. Pseudo fuzzy set is a triplet of element and its two membership functions. Both the membership functions may or may not be dependent. The hypothesis is that every positive sense has some negative sense. So, one membership function has been considered as positive and another as negative. Considering this concept, here the development of Pseudo fuzzy set and its property along with Pseudo fuzzy numbers has been discussed.