The differential equations, solved numerically, are often implemented on simulation and computational tools. This requires programming skills. Excel spreadsheet has been used frequently for statistical analysis, however it is rarely used for computation in mathematics, instead of having great usability and usefulness. It also easily enables a non-programmer to perform programming-like tasks in a visual tabular pen and paper approach. Many physical phenomena are modelled in terms of odes and PDEs, so solving these types of equations, by cooperative iterative methods, are very important there are several iterative methods. The Euler and Runge-Kutta are the most famous ones among the numerical methods for solving the differential equations as well as system(s) of the differential equation(s). Euler's method has slow convergence; therefore, methods of a higher order of accuracy are often needed. Since the iterative methods manually with pen-paper are quite tedious and laboring because it involves numerous repetitive calculations. The spreadsheet technique in Excel is in cooperated to solve the system of an ode by applying the RK5 methods, but one of the pitfalls of an excel spreadsheet is that it is limited containing less than 256 variables (columns) and 65536 records (rows).
<abstract><p>It's undeniably true that fractional calculus has been the focus point for numerous researchers in recent couple of years. The writing of the Caputo-Fabrizio fractional operator has been on many demonstrating and real-life issues. The main objective of our article is to improve integral inequalities of Hermite-Hadamard and Pachpatte type incorporating the concept of preinvexity with the Caputo-Fabrizio fractional integral operator. To further enhance the recently presented notion, we establish a new fractional equality for differentiable preinvex functions. Then employing this as an auxiliary result, some refinements of the Hermite-Hadamard type inequality are presented. Also, some applications to special means of our main findings are presented.</p></abstract>
The Helmholtz differential equation is often used for efficient and dynamic modeling of wave scattering realworld problems. The time harmonic wave scattering phenomena find applications in several scientific areas such as acoustic, electromagnetism, sensors, seismic, radar, and solar technology. Lately, it has been vital in modeling medical imaging problems; an emerging need of humans. Helmholtz type’s differential equation(s) also arises from a physical phenomenon, therefore, it is important to solve Helmholtz types of equation(s) for several purposes. Analytically, it is difficult, in some cases impossible, to find the solution of such equation(s). The numerical method can be used to find the solutions of such equations. Therefore, for the numerical method, the finite difference method is simple and is widely used as compared to other methods. The second order central finite difference is usually used in solving the differential equation(s), but sometimes in many scientific applications we do desire to have higher order approximations and that desire stems from two issues first the better accuracy. Suppose we have done computations of a problem with moderate size mesh, but if the error that we are getting is not acceptable for some unknown reasons, at the same time it is difficult to solve it with a further finer mesh that can happen at times and that is where we want a higher order approximation which gives us the same accuracy as a finer mesh but with fewer mesh points. This is the basic idea and memory storage is another important issue. The discretization of the space differential in the finite difference method is usually derived using the Taylor series expansion; however, if we use a method that adopts algebraic polynomial interpolations in the calculation around near-wall elements, all the calculations over irregular domains reduce to those over regular domains. However, if we use the polynomial interpolation systematically, exceptional advantages are gained in deriving high-order differences which is explained in detail in this paper with examples.
This study explored the convergence rate of the hybrid numerical iterative technique (HNIT) for the solution of nonlinear problems (NLPs) of one variable ( f (x) = 0) . It is sightseen that convergence rate is two for the HNIT. By the HNIT, several algebraic and transcendental NLPs of one variable have been illustrated as an approximate real root for efficient performance. In many instances, HNIT is more vigorous and attractive than well-known conventional iterative techniques (CITs). The computational tool MATLAB has been used for the solution of iterative techniques.
The finite difference technique is oldest numerical method to solve differential equations. Like many differential equations, Helmholtz differential equation which is used to describe many physical phenomena, has long been solved using finite difference method. can be described by Helmholtz Differential equations. The solution of the Helmholtz type differential equations is very important. The information that it belongs together because it tells one coherent story just knowing a little bit about finite differences through to how to solve differential equations an especial technique is used, how to implement finite difference method and the tool which is used as generic enough that will immediately be given a whole new differential equation. The analysis of small to moderate sized presented with the help of a few examples. The improved finite difference method is presented with examples, the method is simple, clear, and short the MatLab code is available, the improved finite difference method is suitable and easy to implement, manually as well as computationally.
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