Numerical Scheme for the Computational Study of Two Dimensional Diffusion and Burgers’ Systems with Stability and Error Estimate

The objective of this study is to explore non-dyadic Haar wavelets for higher order integro-differential equations. In this research article, non-dyadic collocation method is introduced by using Haar wavelet for approximating the solution of higher order integrodifferential equations of Volterra and Fredholm type. The highest order derivatives in the integrodifferential equations are approximated by the finite series of non-dyadic Haar wavelet and then lower order derivatives are calculated by the process of integration. The integro-differential equations are reduced to a set of linear algebraic equations using the collocation approach. The Gauss - Jordan method is then used to solve the resulting system of equations. To demonstrate the efficiency and accuracy of the proposed method, numerous illustrative examples are given. Also, the approximated solution produced by the proposed wavelet technique have been compared with those of other approaches. The exact solution is also compared to the approximated solution and presented through tables and graphs. For various numbers of collocation points, different errors are calculated. The outcomes demonstrate the effectiveness of the Haar approach in resolving these equations.

Piecewise linear functions arc applied to solve the state equations of linear system science with the introduction of the methods of weighted residuals (MWR). There are four different weighted procedures of MWR to be derived: (a) collocation, (b) subdomain, (c) Galerkin's and (d) least-squares methods. Applying the MWR, a linear system can be represented by a set of simultaneous linear algebraic equations, and a recursive algorithm and/or an operational matrix are available. Analysis and parameter identification of stale space equations of linear systems can thus be converted into a set of linear algebraic equations and solved using a recursive algorithm.

Chapter 2 - Simultaneous linear algebraic equations

In this paper, a wavelet based collocation method is formulated for an approximate solution of (1 + 1)- and (1 + 2)-dimensional time fractional diffusion wave equations. The main objective of this study is to combine the finite difference method with Haar wavelets. One and two dimensional Haar wavelets are used for the discretization of a spatial operator while time fractional derivative is approximated using second order finite difference and quadrature rule. The scheme has an excellent feature that converts a time fractional partial differential equation to a system of algebraic equations which can be solved easily. The suggested technique is applied to solve some test problems. The obtained results have been compared with existing results in the literature. Also, the accuracy of the scheme has been checked by computing L 2 and L ∞ error norms. Computations validate that the proposed method produces good results, which are comparable with exact solutions and those presented before.

The traditional one-component transmission error of parallel-axis helical gears is generalized to a three-component transmission error which characterizes the composite displacement in the plane-of-contact resulting from arbitrary small deviations in the positions of both gears of a meshing pair from the positions of their rigid perfect involute counterparts. A set of linear algebraic equations is derived for the contribution to the three generalized transmission error components arising from elastic deformations of the teeth and gear bodies and deviations of the tooth running surfaces from equispaced perfect involute surfaces. It is shown how to combine this set of equations with the generalized transmission error definition and the equations of motion of a gear system to predict the dynamic response of gear elements in the system. For the case of negligible gearbody and bearing/bearing support inertial forces, an additional set of algebraic equations that includes the effects of bearing flexibility and misalignment is derived. Combining the solution of this set of equations with the above-mentioned generalized transmission error equations yields the three-component generalized static transmission error.

A compact kinematic synthesis procedure for serial robot's manipulator arms

This paper represents a portion of a study of decision analysis in large-scale systems in which an effort is made to attack problem complexity by taking advantage of any inherent structure in the problem. By arranging the elements in the problem in an appropriate graphical form, usually a tree or hierarchy, it may often be possible to decompose the problem into a number of loosely coupled subproblems which can be solved independently and then coordinated in such a way as to provide a solution to the overall problem. We first survey team decision theory in which a multiplicity of decision makers, each of whom has access to different information, is attempting to extremize the same cost function. Under the conditions of a linear system with quadratic costs and Gaussian noises, the solution of a static team problem, in which no member's decision affects any other member's information, reduces to the solution of a set of linear algebraic equations. A large class of dynamic team problems which possess certain information structures is found to reduce to a static problem and hence can also be solved as a set of linear algebraic equations. Team problems can often be represented as cooperative stochastic differential games. Finally a technique is proposed for reducing the computational burden by decomposing the problem into a set of subproblems which can be solved relatively independently employing the techniques of hierarchical systems theory.

It has been demonstrated that when estimating Lyapunov exponents using a time series, nonlinear mapping used for characterizing the evolution of the neighbors leads to more accurate negative exponents and is more robust to noise in the times series. However, the number of unknown elements of the matrices associated with nonlinear mapping increases significantly with the embedding dimensions of the state space where the dynamics is reconstructed. Such unknown coefficients are solved from a set of linear algebraic equations based on the least square-root fit method. Derivation of such linear equations and computer programming are tedious and error prone especially for the systems with high embedding dimensions. In this work, we develop a general form of the linear algebraic equations and the corresponding computer program in terms of arbitrary embedding dimensions. A stable robotic system with all negative Lyapunov exponents and the Lorenz system with positive, zero, and negative exponents are used to demonstrate the efficacy of the proposed method. The work can contribute significantly to estimating Lyapunov exponents for systems with large embedding dimensions.

For the inverse vibration problem, a Lie-group shooting method is proposed to simul- taneously estimate the time-dependent damping and stiffness functions by using two sets of dis- placement as inputs. First, we transform these two ODEs into two parabolic type PDEs. Sec- ond, we formulate the inverse vibration problem as a multi-dimensional two-point boundary value problem with unknown coefficients, allowing us to develop the Lie-group shooting method. For the semi-discretizations of PDEs we thus obtain two coupled sets of linear algebraic equations, from which the estimation of damping and stiff- ness coefficients can be writtenoutexplicitly. The present approach is very interesting,which result- ing to closed-form estimating equations without needing of any iteration and initial guess of co- efficient functions, and more importantly, it does not requireto assumea priori thefunctionalforms of unknown coefficients. The estimated results are rather accurate convicing that the new method can be employed in the vibrational engineering to identify viscoelastic property of time-aging mate-

In this Chapter, continuous Haar wavelet functions base and spline base have been discussed. Haar wavelet approximations are used for solving of differential equations (DEs). The numerical solutions of ordinary differential equations (ODEs) and fractional differential equations (FrDEs) using Haar wavelet base and spline base have been discussed. Also, Haar wavelet base and collocation techniques are used to approximate the solution of Lane-Emden equation of fractional-order showing that the applicability and efficacy of Haar wavelet method. The numerical results have clearly shown the advantage and the efficiency of the techniques in terms of accuracy and computational time. Wavelet transform studied as a mathematical approach and the applications of wavelet transform in signal processing field have been discussed. The frequency content extracted by wavelet transform (WT) has been effectively used in revealing important features of 1D and 2D signals. This property proved very useful in speech and image recognition. Wavelet transform has been used for signal and image compression.

A difficult matter for researchers is to develop numerical methods to analyze the behavior of nonlinear dynamical systems that arise in mathematical physics due to their nonlinearity and multi-dimensional nature. In this affection, a novel and effective coupling of spectral and one-step approaches is suggested and used to inspect the robust solutions to the mathematical problems. This proposed method is grounded on the one-step sixth-order Runge–Kutta (RK) method and the shifted Vieta–Fibonacci (sVF) polynomial-based spectral approach. First, use the sixth-order one-step technique for the temporal approximation, then utilize the sVF polynomials to estimate the spatial variables. The sVF polynomial-based new operational matrices are used to calculate the derivative terms of the mathematical models. Related theorems are presented in the paper to provide a mathematical validation of the technique. The suggested semi-spectral techniques transform the investigated nonlinear models into a set of linear algebraic equations, making them simpler to solve. We conduct theoretical and numerical research into the stability, convergence, and error-bound analysis to support the computational algorithm's mathematical formulation. Several mathematical models, including the Hartmann numbers ranging from 0 to 106, are taken into consideration to demonstrate the precision and efficacy of the proposed computational technique. A thorough comparison analysis illustrates the recommended computational approach's validity, correctness, and dependability. The suggested construction of sixth-order in the temporal direction and exponential order around 3.5545e0.1515M in the spatial direction is determined to be exceptional in handling nonlinear problems and examining the exact solutions.

By formulating the fundamental electromagnetic (EM) equations in terms of the stream potential of the surface current density, we can express the EM response of a rectangular thin plate as the solution of a single equation subject to simple boundary conditions. A finite difference approximation of this equation reduces the problem to that of solving a large set of linear algebraic equations. The solution of these equations by a modified Gauss‐Seidel iterative method yields the stream potential and thus permits visualization of the eddy currents circulating inside the plate conductors. The secondary field calculated from the stream potential compares well with that given by scale model measurements provided that the grid spacings used in the finite differences are small enough. Using a further approximation, we can also simulate inductively thick conductors if the conductors are not geometrically thick.

SIF statistics in micro cracked solid: Effect of crack density, orientation and clustering

This paper presents a detailed numerical analysis of a particular class of heat equations. To approximate the solutions, we develop a hybrid computational framework that combines the Haar wavelet collocation method with finite difference techniques. In the proposed approach, finite difference schemes are first applied to discretize the time variable, thereby reducing the original problem to a semi-discrete system. Subsequently, Haar wavelets defined on a uniform spatial grid are employed to approximate the spatial derivatives, allowing for efficient representation of both smooth and non-smooth solution profiles. Within the framework of Sobolev spaces, rigorous proofs of stability and convergence are provided, ensuring the reliability of the method from a theoretical standpoint. To validate the practical performance of the proposed scheme, extensive numerical experiments are conducted across a variety of test problems. The accuracy of the method is quantitatively assessed using both the <img src=image/13443061_01.gif> and <img src=image/13443061_02.gif> error norms. The results consistently demonstrate that the method delivers high accuracy while maintaining strong computational efficiency. Furthermore, the computed solutions exhibit excellent agreement with exact analytical solutions, thereby confirming the robustness and effectiveness of the hybrid scheme for solving heat equations. In addition, the findings of this research make several contributions to the field of numerical analysis of partial differential equations.

In this paper, a modified harmonic balance method is presented to solve nonlinear forced vibration problems. A set of nonlinear algebraic equations appears among the unknown coefficients of harmonic terms and the frequency of the forcing term. Usually a numerical method is used to solve them. In this article, a set of linear algebraic equations is solved together with a nonlinear one. The solution obtained by the proposed method has been compared to those obtained by variational and numerical methods. The results show good agreement with the results obtained by both methods mentioned above.