The plots of the degenerate wave functions show that. The condition (2) speci es the initial shape of the string, I(x), and (3) expresses that the initial velocity of the string is zero. ( to 2D, the 2D wave equation may be written as The k-Wave toolbox is a powerful tool for general acoustic modelling. The wave equations may also be used to simulate large destructive waves Waves in fjords, lakes, or the ocean, generated by - slides - earthquakes - subsea volcanos - meteorittes Human activity, like nuclear detonations, or slides generated by oil drilling, may also generate tsunamis Propagation over large distances Wave amplitude increases near The wave equation is an important second-order linear partial differential equation for the description of waves—as they occur in classical physics—such as mechanical waves (e. The 2D wave equation Separation of variables Superposition Examples Solving the 2D wave equation Goal: Write down a solution to the wave equation (1) subject to the boundary conditions (2) and initial conditions (3). To unzip this file, use the unix command " unzip codes. 1. 9. Heat Transfer Matlab 2d Conduction Question Answers. Solving the 2D Wave Equation. For example in 1 dimension. Expanding these methods to 2 dimensions does not require significantly more work. docx. as_surface. Based on your location, we recommend that you select: . Both a second order or 5 point I am trying to code of rectangular wave equation but I got problem to code 2d wave simulation . . as_colormap. It turns out that the problem above has the following general solution A compact and fast Matlab code solving the incompressible Navier-Stokes equations on rectangular domains mit18086 navierstokes. It arises in fields like acoustics, electromagnetics, and fluid dynamics. Matlab Code Examples. These codes solve the advection equation using explicit upwinding. With such an indexing system, we will 3d plot of wave function. 2016): (On reply to the comments with the request for the code) It was long time ago. $\endgroup$ – Christian Clason Jan 14 '13 at 21:21 $\begingroup$ @ChristianClason it's can be either 2D or 3D in arbitrary domain. J. PROGRAMMING OF FINITE DIFFERENCE METHODS IN MATLAB LONG CHEN We discuss efﬁcient ways of implementing ﬁnite difference methods for solving Pois-son equation on rectangular domains in two and three dimensions. This method requires two initial conditions that are introduced using a simple Finite difference modeling of acoustic waves in Matlab Carrie F. Specializing Eq. Nov 30, 2018 an independent check, we test the traveling wave solutions from the steepest descent method using a . e. Although it will still work if you don't want to change your habits. Look at the next piece of code dx=0. . 1 CREWES Research Report — Volume 22 (2010) 3 Geological model definition (. I'm trying to figure out how to draw a wave equation progress in a 2D graph with Matlab. The Following is my Matlab code to simulate a 2D wave equation with a Gaussian source at center using FDM. 303 Linear Partial Diﬀerential Equations Matthew J. Select a Web Site. Youzwishen and Gary F. It basically consists of solving the 2D equations half-explicit and half-implicit along 1D proﬁles (what you do is the following: (1) discretize the heat equation implicitly in the x-direction and explicit in the z-direction. Today, I am sharing my experience on MATLAB plotting. The conclusion . 2d Unsteady Convection Diffusion Problem File Exchange Matlab. Therefore, we cannot generate a real continuous-time signal on it, rather we can generate a “continuous-like” signal by using a very very high sampling rate. matlab *. m Benjamin Seibold Applied Mathematics Massachusetts Institute of Technology www-math. Together with the heat conduction equation, they are sometimes referred to as the “evolution equations” because their solutions “evolve”, or change, with passing time. The Schrodinger equation is solved for the real and imaginary parts of the wavefunction < ( , )xt in the region from 0ddxL with the boundary conditions < (0, ) 0t and < ( , ) 0Lt. Then h satisﬁes the diﬀerential equation: ∂2h ∂t2 = c2 ∂2h ∂x2 (1) So I used the following code in Matlab (for a single inst Stack Overflow. Update (21. m % finite difference scheme for the 1D wave The second type of second order linear partial differential equations in 2 independent variables is the one-dimensional wave equation. Theory described in description. 0. mit. An interactive demo of the 2D wave equation. Showing our solver for the 2D Wave Equation. Lab 1 Solving A Heat Equation In Matlab. Traveling solutions, reflection, energy, plane waves, material interface; WKB, 2D wave equation and plane waves, 2D refraction, Matlab code for today's WKB plot. 9. Droplet put on the water surface to start Jan 27, 2016 This code aims to solve the wave equation on a 2d square plate and simulate the output in an user-friendly MATLAB-GUI You can find the Apr 18, 2013 An interactive demo of the 2D wave equation. There is a decay in wave equation. So if anyone could point out to me what causes the problem and how could I fix it that would be great. MATLAB sine wave plot is not correct. 11 Finite difference methods for 2D and 3D wave equations . Observe in this M-ﬁle that the guess for fzero() depends on the value of x. The software is designed for time domain acoustic and ultrasound simulations in complex and tissue-realistic media. Solving the 2D heat equation in MATLAB. Matlab Code Examples The choice of Matlab as a programming environment definitely reflects the latter sensibility, though the use The 2D Wave Equation. Demonstrates the use of the Octave function "fsolve" to solve nonlinear equations in the form f ( x ) = 0 . Do you have any idea how to write the simulation of wave equation Finite difference methods for 2D and 3D wave equations¶. The key is the matrix indexing instead of the traditional linear indexing. edu March 31, 2008 1 Introduction On the following pages you ﬁnd a documentation for the Matlab Solutions to Problems for 2D & 3D Heat and Wave Equations 18. This was a project in Introduction to Multigrid Methods at Lund Institute of Technology (LTH), Sweden, 2014. Ask Question Asked 6 years, 2 2nd Order finite difference for 1D wave equation matlab issue. In particular, we examine questions about existence and This Matlab code implements a second order finite difference approximation to the 2D wave equation. It has been successfully applied to an extremely wide variety of problems, such as scattering from metal objects and Numerical solution to the wave equation - Explicit Method. Contents Simple wave modelling CREWES Research Report Š Volume 14 (2002) 1 Simplified wave modelling John C. (Update, much faster than the old version) Solution of 1D-collision-problem (wave equation) Oct 1, 2016 Python or Matlab program with a difference scheme. A natural next step is to consider extensions of the methods for various variants of the one-dimensional wave equation to two-dimensional (2D) and three-dimensional (3D) versions of the wave equation. Thus, one may obtain the function u(x) through knowledge of the Green's function in equation (1) and the source term on the right-hand side in equation (2). On one side, the grid is terminated with a Double Absorbing Boundary (DAB). 2 is the wave equation (you should check first that the coeffi- cients of the PDE give The Matlab PDE Toolbox uses the finite element method (FEM) to discretize in space. 6 A 2D Wave Equation with Variable Wave Velocity . Daley ABSTRACT Two subroutines have been added to the Matlab AFD (acoustic finite difference) package to permit acoustic wavefield modeling in variable density and variable velocity media. Plotting 2D equation with infinite sum. g. the wave doesn't bounce back, but simply continues to move outside the domain. The MATLAB tool distmesh can be used for generating a mesh of arbitrary shape that in turn can be used as input into the Finite Element Method. All can be viewed as prototypes for physical modeling sound synthesis. In this article I will give a step-by-step guide to implement the two dimensional wave equation in Octave. The 2D Wave Equation. function value = degwave(x) %DEGWAVE: MATLAB function M-ﬁle that takes a value x Finite difference modelling of the full acoustic wave equation in Matlab Hugh D. It uses central finite difference schemes to approximate derivatives to the scalar wave equation. The automated translation of this page is provided by a general purpose third party translator tool. As a starting point, Program 6 in Nick Trefethen's Spectral Methods in Matlab solves the wave equation for variable coefficients. The wave equation considered here is an extremely simplified model of the physics of waves. MATLAB's Parallel Computing Toolbox has direct support for Graphics Processing Units (GPUs or GPGPUs) for many different computations. The mathematics of PDEs and the wave equation Michael P. I used imagesc function to output the wave. 0 * pi. 3 2D hat function ˚ Applied Numerical Methods for Engineers using Matlab and C, R. png images Matlab script: nFrames = 1000; h Select a Web Site. Joshua Wall . Since this PDE contains a second-order derivative in time, we need two initial conditions . Daileda Trinity University Partial Diﬀerential Equations March 1, 2012 Daileda The 2D wave equation Octave - 2D Wave Equation. R I am going to write a program in MATLAB which will compare initial and final velocity profile for 1D Linear convection for different value of grid points. Why use Octave? The reasons for using Octave are: By wrinting the simulation in Octave we reduce the amount of auxiliary code needed to perform the calculations. The Green's tensor is a solution for a point source of the wave equation. ] [For solving this equation on an arbitrary region using the finite difference method, take a look at this post. (2) solve it for time n + 1/2, and (3) repeat the same but with an implicit discretization in the z-direction). of Partial Differential Equations in Finance with Matlab. In addition, PDEs need boundary conditions , give here as (4 pde matlab code for wave equation . The source code and files included in this project are listed in the project files section, please make sure whether the listed source code meet your needs there. I found this piece of code which effectively draw a 2D wave placing a droplet in the middle of the graph (I almost fully commented it to simplify things) and then letting it expanding till the border, then bouncing back (how can this code do that? Solution to the 2-D Wave Equation using finite difference method in Matlab. These programs are for the equation u_t + a u_x = 0 where a is a constant. A 2D PDE wave equation constrained in a square grid and then a rectangular grid. Next: Plate Reverberation Up: MATLAB Code Examples Previous: Vocal Tract Synthesis Contents Index The 2D Wave Equation. 198. Dynamics and Vibrations MATLAB tutorial School of Engineering Brown University This tutorial is intended to provide a crash-course on using a small subset of the features of MATLAB. Finite Difference Methods For Matlab or any other simulation softwares process everything in digital i. Learn more about pde, wave-equation MATLAB CODES Matlab is an integrated numerical analysis package that makes it very easy to implement computational modeling codes. The following Matlab project contains the source code and Matlab examples used for 2d wave equation. You may get the ENTIRE set of files by clicking here. Figure 1. FD1D_WAVE is a MATLAB library which applies the finite difference method to solve a version of the wave equation in one spatial dimension. MATLAB® allows you to develop mathematical models quickly, using powerful language constructs, and is used in almost every Engineering School on Earth. 001; % spatial increment WEEK 4: (Effect of Grid-Size on output for the solution of 1D linear wave equation ) 1. 0. ENTIRE set . Finite Difference Solution of Wave Equation [Program]. You can get them one at a time from the list below. surface waves in a bath). 3 – 2. The two dimensional wave equation 1. m — graph solutions to planar linear o. Ask Question Asked 3 years, 2 months ago. MATLAB, Simulink, Handle Graphics, and Real-Time Workshop are registered trademarks . k-Wave is an open source MATLAB toolbox designed for the time-domain simulation of propagating acoustic waves in 1D, 2D, or 3D [1]. A Simple Finite Volume Solver For Matlab File Exchange. Over 36 million developers use GitHub together to host and review code, project manage, and build software together across more than 100 million projects. Programs in Category Partial Differential Equations Computes the FE error measured in the L2- and H1-norm for 2D problems. In the case for 2D scattering the tensor de-. The Wave Equation . The simplest instance of the one A MATLAB-based frequency-domain finite-difference package for solving 2D visco-acoustic wave equation Article (PDF Available) in Waves in Random and Complex Media 21(1):161-183 · February 2011 Mar 29, 2017 This code solves the 2D Wave Equation on a square plate by finite differences method and plots an animation of the 2D movement and the http://simulations. [Edit: This is, in fact Poisson’s equation. I found this piece of code which effectively draw a 2D wave placing a droplet in the middle of the graph (I almost fully commented it to simplify things) and then letting it expanding till the border, then bouncing back (how can this code do that? 3. m files to solve the advection equation. The toolbox has a wide range of functionality, but at its heart is an advanced numerical model that can account for both linear and nonlinear wave propagation, an arbitrary distribution of heterogeneous material parameters, and power law acoustic absorption. You can gain valuable insight into quantum Unless specifically defined otherwise, Matlab by default cast any number to double, which is a 64 bit floating point type. 18 Finite di erences for the wave equation Similar to the numerical schemes for the heat equation, we can use approximation of derivatives by di erence quotients to arrive at a numerical scheme for the wave equation u tt = c2u xx. Dan Doherty Speed up MATLAB code with NVIDIA GPUs. We will follow the (hopefully!) familiar process of using separation of variables to produce simple solutions to (1) and (2), I am trying to code of rectangular wave equation but I got problem to code 2d wave simulation . 2. zackg835 25,687 views. edu/~seibold seibold@math. Droplet put on the water surface to start waves. Overview. water waves, sound waves and seismic waves) or light waves. MathWorks does not warrant, and disclaims all liability for, the accuracy, suitability, or fitness for purpose of the translation. When plotted, such signals look like a continuous signal. GPUs. dial node. 1d Heat Transfer File Exchange Matlab Central. We have seen in other places how to use finite differences to solve PDEs. Fourier transform can be generalized to higher dimensions. 1 to 10. ▫ Use GPU Jun 17, 2014 A set of two-dimensional (2D) electromagnetic (EM) MATLAB codes, (Maxwell) equations and second-order decoupled (wave) equations, a grid of x and t values, solve the PDE and create a surface plot of its solution ( given in. Visualisation has been done with VirtuaDub 1. Since both time and space derivatives are of second order, we use centered di erences to approximate them. But you need at least two samples per cycle (2*pi) to depict your sine wave. Plotting wave equation. The boundary condition at x=0 generates the wave. Lamoureux ∗ University of Calgary Seismic Imaging Summer School August 7–11, 2006, Calgary Abstract Abstract: We look at the mathematical theory of partial diﬀerential equations as applied to the wave equation. Here are various simple code fragments, making use of the finite difference methods described in the text. conv2 function used for faster calculations. In this article, I am simply exaplain 'How to plot mathematical equation in MATLAB?'. zip ". Two-dimensional Fourier transform also has four different forms depending on whether the 2D signal is periodic and discrete. Ordinary Differential Matlab Program 3: Square-wave Test for the Explicit Method to solve. Solving the Wave Equation and Diffusion Equation in 2 dimensions. And it's required mutiple mathematical functions, code for the MATLAB program and writing format on the MATLAB command window. Skip navigation Sign in. The primary thing to notice here is that the DAB is essentially identical to the 1D case described in the 1D Klein-Gordon example. Partial Differential Equation Toolbox integrates with other MATLAB products, allowing you to build and share custom applications with MATLAB Compiler™, run design of experiments in parallel with Parallel Computing Toolbox™, and leverage high-fidelity simulation in Simulink ® and Simscape™. Particles can be represented by wavefunctions which obey a differential equation, the Schrodinger Wave Equation which relates spatial coordinates and time. Choose a web site to get translated content where available and see local events and offers. Equation: Finite Difference Scheme % matlab script waveeq1dfd. This process relies upon the linearity of the operator L. 39 % T: triangles (elements in 2D). Geiger and Pat F. m — phase portrait of 3D ordinary differential equation heat. Solve 2d wave equation with Finite Difference Method. 4. waveguide wave-equation 2d maxwell helmholtz-equation helmholtz fem dielectric physics physics-simulation optoelectronics photonics matlab wave guide MATLAB Updated May 21, 2019 howardjp / cmna Variational method, continuous wave equation, Fourier modes, and applet for playing with the wave equation (string under tension). The vast majority of students taking my classes have either little or rusty programming experience, and the minimal overhead and integrated graphics capabilities of Matlab makes it a good choice for beginners. The mfiles are grouped according to the chapter in which they are used. Bancroft ABSTRACT Wave motion is modelled using the acoustic wave equation and implemented using MATLAB. For example, many signals are functions of 2D space defined over an x-y plane. Otherwise you'll get an alias frequency, and in you special case the alias frequency is infinity as you produce a whole multiple of 2*pi as step size, thus your plot never gets its arse off (roundabout) zero. The coding style reflects something of a compromise between efficiency on the one hand, and brevity and intelligibility on the other. I have created a github repository containing matlab code that should help you GPU Computing with MATLAB. Uses MATLAB code with optional GPU acceleration for real-time performance. 3D Wave Equation and Plane Waves / 3D Differential Operators Overview and Motivation: We now extend the wave equation to three-dimensional space and look at some basic solutions to the 3D wave equation, which are known as plane waves. The Finite-Difference Time-Domain Method (FDTD) The Finite-Difference Time-Domain method (FDTD) is today’s one of the most popular technique for the solution of electromagnetic problems. 1). 56. narod. % % Discussion: % % FEM_50 is a set of MATLAB routines to apply the finite % element method to solving Laplace's equation in an arbitrary % region, using about 50 lines of MATLAB code. I am trying to code of rectangular wave equation but I got problem to code 2d wave simulation . Either runs Use command-line functions to solve a wave equation. In mathematics, a partial differential equation (PDE) is a differential equation that contains Common examples of linear PDEs include the heat equation, the wave Differential Equations in Cleve Moler: Numerical Computing with MATLAB; Partial Differential Equations at nag. Either runs interactively, click anywhere to poke the surface and generate a new wave, or let the program do it by itself. wavelet pseudo spectral method in wave % matlab function to initialize and call the solve_sps2D % function, which implements a 2-D solution of % the nonlinear wave equation using a psuedospectral % / staggered A-B method, including attenuation modeled % with two relaxation mechanisms. Sean's pick this week is 2D Wave Equation by Daniel Armyr. So I used the following code in Matlab (for a single inst Stack Overflow. I've tried running the same code on a problem with known analytic solution like 2D Quantum Harmonic Oscillator, I still get the same result and the eigenvalues are wrong as well. Followed by this: y = 3*sin(x(2*pi/4))+2; I have multiplied x by 2*pi/4 in order to resize the period to a quarter of its size, but I have errors regarding indexes being positive. Taking a In this section we do a partial derivation of the wave equation which can be used to find the one dimensional displacement of a vibrating string. Matt Kawski's personal MATLAB resources: From calculus, thru nonlinear dynamical combinations of a vibrating membrane (2d-wave equation on rectangle). 2D Wave Equation MATLAB Animation - Duration: 1:15. For more information about the algorithm Scientiﬁc Programming Wave Equation 1 The wave equation The wave equation describes how waves propagate: light waves, sound waves, oscillating strings, wave in a pond, Suppose that the function h(x,t) gives the the height of the wave at position x and time t. The Finite Element Method is a popular technique for computing an approximate solution to a partial differential equation. A Matlab Figure window gives a summary of the 2D finite-difference modelling in Matlab, v. geo files) The geological definition file (. 11. Do you have any idea how to write the simulation of wave equation All your code in one place. Solving Laplace’s Equation With MATLAB Using the Method of Relaxation By Matt Guthrie Submitted on December 8th, 2010 Abstract Programs were written which solve Laplace’s equation for potential in a 100 by 100 Doing Physics with Matlab Quantum Mechanics Bound States 3 SCHRODINGER EQUATION On an atomic scale, all particles exhibit a wavelike behavior. 4x speedup in Solve 2D wave equation in MATLAB. 2 Dimensional Wave Equation Analytical and Numerical Solution This project aims to solve the wave equation on a 2d square plate and simulate the output in an user-friendly MATLAB-GUI you can find the gui in mathworks file-exchange here Implementing Explicit formulation of 1D wave equation in Matlab. This code aims to solve the wave equation on a 2d square plate and simulate the output in an user-friendly MATLAB-GUI You can find the solution derivations here MathWorks Machine Translation. MATLAB® function fem_50 ( ) %% FEM_50 applies the finite element method to Laplace's equation. Problem Setup: Given Partial Differential Equation: `(delu)/(delt) + If m = 0, then Eq. The first data line of the model gives the Coincidentally, I had started to use MATLAB® for teaching several other subjects around this time. m - visualization of waves as colormap. The general equation is given as. The boundary condition at x=5 refers to Mur boundary condition, i. I found this piece of code which effectively draw a 2D wave placing a droplet in the middle of the graph (I almost fully commented it to simplify things) and then letting it expanding till the border, then bouncing back (how can this code do that? Solving The Wave Equation And Diffusion In 2 Dimensions. ru/ Solve 2d wave equation with Finite Difference Method . (Kac drum problem links are below). The name of the zip file is "codes. If you complete the whole of this tutorial, you will be able to use MATLAB to integrate equations of motion MATLAB Codes for Introduction to PDE with MATLAB . This is because we only need to use the pwe fem microcavity maxwell maxwell-equations-solver helmholtz-equation helmholtz dielectric physics physics-simulation optoelectronics photonics photonic-mode-solver 2d resonant-cavity wave wave-equation matlab cavity-simulators The heat and wave equations in 2D and 3D 18. zip" . The Heat Equation: A Heated Metal Block . Schilling step size governed by Courant condition for wave equation. 's Solve 2d wave equation with Finite Difference Method. Apr 28, 2014 Solving Differential Equations on 2-D Geometries with Matlab. m — numerical solution of 1D heat equation (Crank—Nicholson method) wave. m - visualization of waves as surface. com; Sanderson, Grant (April 21, 2019). Do you have any idea how to write the simulation of wave equation Wave modelling by shallow water equations. Although we will not discuss it, plane waves can be used as a basis for k-Wave is an open source acoustics toolbox for MATLAB and C++ developed by Bradley Treeby and Ben Cox (University College London) and Jiri Jaros (Brno University of Technology). damped wave equation with 2 sources and neumann boundary conditions. 2D wave equation matlab code Search and download 2D wave equation matlab code open source project / source codes from CodeForge. Based on its page rank, this paper has apparently helped a lot of students. Since solving the wave equation in 2D has all the essential features of the 3D case, we will look at the 2D case in this section. %PDE1: %and returns values for a standing wave solution to 2D (which should be the default) and then double-click on Classical PDEs. I guess that the variable dx is the relevant here. In Chapter 4 we study the traveling wave equations of (1. Matlab 2D wave equation using FDM. But you are never using it. The Mur boundary condition can be expressed mathematically as degenerate viscous shock wave. In-class demo script: February 5. 1) in 2D satisfying zero Dirichlet we solve using MATLAB. The 2D wave equation Separation of variables Superposition Examples The two dimensional wave equation Ryan C. Margrave ABSTRACT A Matlab toolkit, called the AFD package, has been written to model waves using acoustic finite differences. However, this doesn't mean it's the best tool for every purpose! There is a diverse range of other acoustics-related software available, both commercially and open-source. Matlab Programs for Math 5458 Main routines phase3. Hancock Fall 2006 1 2D and 3D Heat Equation Ref: Myint-U & Debnath §2. MATLAB code for solving Laplace's equation using the Jacobi method - Duration: 12:06. ] I will present here how to solve the Laplace equation using finite differences 2-dimensional case: Pick a step , where is a positive integer. 11 software from series of *. PDE Toolbox. The wave seems to spread out from the center, but Solution to the 2-D Wave Equation using finite difference method (Matlab). So writing 2 * pi is enough to get the maximum precision (Matlab won't cast pi as an integer ;-)), you do not need to write 2. In addition, we also give the two and three dimensional version of the wave equation. 5 [Nov 2, 2006] Consider an arbitrary 3D subregion V of R3 (V ⊆ R3), with temperature u(x,t) deﬁned at all points x = (x,y,z) ∈ V. The initial values for the wavefunction < ( ,0)x must be specified to describe the wave packet representing a particle. 2014/15 Numerical Methods for Partial Differential Equations 55,826 views MathWorks Machine Translation. In other words, the solution of equation (2), u(x), can be determined by the I'm trying to create a sine wave audio signal within MatLab based on this function: So far I have created a vector x that starts at 0, increments in 0. The 1D Wave Equation: Finite Difference Scheme. com The equation above is a partial differential equation (PDE) called the wave equation and can be used to model different phenomena such as vibrating strings and propagating waves. Two-Dimensional Fourier Transform. geo file) must be used to describe the geology of the section of the earth to be modelled. May 3, 2011 small public domain programs via Matlab to full visualization systems like . Shallow water equations can be used to model Rossby and Kelvin waves in the atmosphere, rivers, lakes and oceans as well as gravity waves in a smaller domain (e. The constant term C has dimensions of m/s and can be interpreted as the wave speed. m — numerical solution of 1D wave equation (finite difference method) go2. d. Hancock 1 Problem 1 A rectangular metal plate with sides of lengths L, H and insulated faces is heated to a paraheat_pwc_sample, a MATLAB program which repeatedly calls paraheat_pwc() to set up and solve a parameterized steady heat equation in a 2D spatial domain, with a piecewise constant diffusivity, saving values of the finite element solution at selected points, and writing the saved data to a file. ) Since the equilibrium solution ¯u(x) is deﬁned implicitly in this case, we ﬁrst write a MATLAB M-ﬁle that takes values of x and returns values ¯u(x). e, discrete in time. It models an initial drop of water at several different positions and the resulting ripples in a body of water. 40 % T is size License: Berkeley Software Distribution (BSD) Description: Finite-difference time domain (FDTD) solution of coupled first-order acoustic equations in 2D using a May 29, 2002 tensor technique in Matlab. Equation (1) is known as the one-dimensional wave equation . 2d wave equation matlab

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