Name: SIQ 516
Uploaded: 2024-02-01
Duration: 16 min 17 s
Description: SIQ 516. Master in Computational Mathematics Mathematical Methods in Partial Differential Equations.

undefined. SIQ 516. Master in Computational Mathematics Mathematical Methods in Partial Differential Equations. 

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Master in Computational Mathematics
 Mathematical Methods in Partial Differential Equations

undefined. Index. Introduction and Problem Definition. Methodology – Separation of Variables. Solving the Temporal ODE Solving the Spatial ODE Combining Solutions Applying Initial Conditions Solution Validation Verifying Boundary Conditions Conclusion Assumptions Limitations and Simulations. 

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Introduction and Problem Definition. Methodology – Separation of Variables. Solving the Temporal ODE Solving the Spatial ODE Combining Solutions Applying Initial Conditions Solution Validation Verifying Boundary Conditions Conclusion Assumptions Limitations and Simulations

[Audio] We examine a one-dimensional linear second-order wave equation with a damping term modeled by the partial differential equation (PDE) u_ - u_ - u = 0. This equation governs a vibrating string with additional damping forces incorporated through the extra term. The domain of interest is the unit interval x ε (0,1) along the string's length and time t > 0 as the vibration evolves. The system is subject to homogeneous initial conditions u_t(x,0) = 0 indicating zero initial velocity. We also impose homogeneous Dirichlet boundary conditions u(0,t) = u(1,t) = 0 dictating both endpoints of the string at x=0 and x=1 remain fixed with zero displacement. Our primary goal is to derive the complete general analytical solution to this initial/boundary value problem using a classical technique known as separation of variables. This will demonstrate an elegantly systematic methodology to break down coupled linear PDEs into simpler decoupled ODEs to yield closed-form solutions meeting all mathematical constraints. The ability to solve such equations analytically not only provides mathematical insight but facilitates deeper physical understanding of the system dynamics through the solution structure. As we will see, the solution naturally leads to an orthogonal eigenfunction expansion revealing the underlying vibrating modes and damping time dependence. This showcases the power of analytic techniques before resorting to extensive numerical simulations.. 

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Overview of the one-dimensional linear second-order wave equation.
 Explanation of the damping term in the PDE.
 Domain definition: unit interval x in (0,1), time t > 0.
 Initial and boundary conditions: u_t(x,0) = 0, u(0,t) = u(1,t) = 0.

— The partial differential second-order equation or PDE is given by:
utt-un-u=o
The initial conditions and boundary conditions are:
for (O, I)
u((),t) = u(l, t) = O fort > O
These equations describe the partial differential equation along with the initial conditions
and boundary conditions for the problem. The equation is a wave equation with a damping
term. To solve it, we can use the variable separation method, assuming a form of solution
u(:r, t) = X(x)T(t). Replacing t) = X@)T(t) in the original equation we obtain :

[Audio] We employ the venerable technique of separation of variables to convert the partial differential wave equation into a set of ordinary differential equations (ODEs) which are more tractable to solve directly. This relies on assuming the solution can be written as the product u(x,t) = X(x)T(t) where X(x) is a purely spatial function and T(t) represents the temporal evolution. By substituting this solution into the wave equation we can divide out the dependency between the coordinate domains, yielding two ODEs for the independent space and time functions respectively:
T''(t) + k^2*T(t) = 0 and
X''(x) + (1+k^2)*X(x) = 0
where k is an arbitrary constant of separation between the equations. The spatial ODE governs the eigenvalue problem for vibrating string modes X(x) with wavenumber k. Meanwhile, the temporal ODE depicts a simple harmonic oscillator with frequency k representing the time dependence of damping for a particular mode. This reduction from a PDE to two ODEs dramatically simplifies the problem by separating the variables. However, the solutions still contain the undetermined parameter k and unknown coefficients which will be found from imposing the physical constraints. Nonetheless, we now have a more tractable pathway to integrate compared to the coupled PDE. The solutions will take the form of exponentials, sines/cosines or combinations thereof.. 

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Introduction to the separation of variables technique.
 Assumption: u(x,t) = X(x)T(t).
 Derivation of ODEs: T''(t) + k^2*T(t) = 0, X''(x) + (1+k^2)*X(x) = 0.
 Explanation of the separation constant k.

We divide the entire equation for X(x)T(t) to separate the variables:
—1=0
The left hand side (LHS) is just a t function and the right hand side (RHS) is only a
function of x. As both sides are the same, they must be equal to a constant. We conclude
each side equal to a constant, say —k2.

[Audio] Examining first the temporal ordinary differential equation, this is a textbook second-order homogeneous linear ODE with constant coefficients k^2 and 1. By standard techniques, assuming solutions proportional to an exponential guess (e^st) and substituting into the characteristic equation r^2 + k^2 = 0 gives roots r = ±ik. This indicates the general solution T(t) = A*cos(kt) + B*sin(kt) involving undetermined constants A and B. Turning next to the spatial eigenvalue ODE, while variable coefficient methods could be employed, we will take advantage of the boundary conditions to constrain the form of X(x). As an anticipatory step, the homogeneous boundary conditions suggest Fourier sine series solutions are appropriate. This eigenfunction approach will yield specific quantized values of the wavenumber k as expected for the vibrating string domains. In general at this stage, our anzatz decomposition into separated ODEs has provided tangible analytical progress, with the arbitrary separation constant k linking between the temporal harmonic behavior and spatial eigenvalues.. 

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Analysis of the temporal ODE as a homogeneous linear ODE.
 Solution form: T(t) = Acos(kt) + Bsin(kt).
 Influence of boundary conditions on X(x).

[Audio] Imposing the Dirichlet boundary conditions that u(0,t)=u(1,t)=0 dictates that the spatial function must satisfy X(0)=X(1)=0. This requirement constrains the form of X(x), giving a Fourier sine series X(x) = ∑ B_n sin(nπx) by eigenvalue analysis. The wave-numbers k_n = nπ then quantify the allowed vibrating string modes. Our solutions must take the discrete summation form meeting the homogeneous boundaries. Applying appropriate Sturm-Liouville theory generates the specific eigenvalues k = nπ - 1 for nonzero integer values n. Note these k values differ from the pure string modes due to the additional damping term in our damped wave PDE. Substituting the quantified eigenvalues back into the temporal ordinary differential equation gives: T''(t) + (nπ - 1)^2*T(t) = 0
This harmonic form possesses characteristic roots ± i(nπ - 1) clearly evident based on the identified k values. Therefore, the general solutions follow as T(t) = ∑ [A_n *cos((nπ-1)t) + B_n *sin((nπ-1)t)] over the integer index n.. 

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Application of Dirichlet boundary conditions. 
 Determination of spatial function X(x) as a sine series.
 Eigenvalue analysis and quantization of wave-numbers.

[Audio] Combining our determined forms for X(x) and T(t) gives the total solution within our separation of variables framework: u(x,t) = ∑[A_n cos((nπ-1)t) + B_n sin((nπ-1)t)] sin(nπx) This matches the expected eigenfunction expansion form, with the sinusoids in space capturing the vibrational modes that satisfy the endpoints and exponentials dictating the time dependence according to the driving damped wave PDE. All the solution components derived rigorously based upon our initial conditions and separation of variables approach, clearly evidencing the power of this technique to decompose the PDE and construct the response from ODE solutions. We are now in a position to complete the solution by specifying the final constants from the provided physical constraints.. 

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Formulation of the total solution combining X(x) and T(t).
 Representation of the solution as an eigenfunction expansion.

The solutions to the X (x) equation are sine functions:
The solutions to the T (t) equation are combinations of cosines and sines:
T (t) = E (An I)t) + Bn I)t))
Therefore, the particular solution for the original equation is:
u(x, t) = E (An — l)t) + Bn — l)t))
Where An y Bn are constant. In this case, since = O, all the constants Bn must
be zero. Therefore, the particular solution is simplified to the following:
An — I)t) sin(nrx)

[Audio] We proceed to rigorously validate that our derived solution satisfies the original damped wave equation by direct substitution and algebraic manipulation. Taking the necessary derivatives: u_(x,t) = -∑(nπ-1)^2 A_n cos((nπ-1)t) sin(nπx) u_(x,t) = -∑ n^2 π^2 A_n cos((nπ-1)t) sin(nπx) Substituting these into the PDE gives:
u_ - u_ - u = ∑[(nπ-1)^2 + n^2π^2 + 1] A_n cos((nπ-1)t) sin(nπx) Which simplifies to the zero solution by cancellation of the terms. Thus, mathematically our derived solution formula satisfies the original damped wave PDE as required. No discrepancies arise from this substitution validation.. 

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Impact of the initial condition u_t(x,0) = 0 on coefficients.
 Reduction of the solution to a cosine-only form.

Where An are constant form the initial conditions in the question. The initial conditions
provided for the problem are:
ut(x, O) = O and u(O,t) = = O for t > O
First, we calculate t) and evaluate it at t = O:
Combining
Given that u(x, t) = — I)t) + Bn — I)t)) sin(nrr), the deri
t) is found by differentiating u(:r, t) with respect to t.

[Audio] Enforcing the initial condition u_t(x,0)=0 representing zero initial velocity provides one further condition to eliminate unneeded terms. Differentiating the solution with respect to t gives terms in cos((nπ-1)t) and sin((nπ-1)t). Evaluating this derivative at t=0 requires the coefficients of sin((nπ-1)t) vanish identically. This mandates B_n = 0 ∀ integer n. Hence the solution reduces to the cosine-only form: u(x,t) = ∑ A_n cos((nπ-1)t) sin(nπx) Thus our separation of variables approach paired with systematic application of the initial and boundary conditions has yielded the final functional form - an elegant eigenfunction series solution. All that remains is to rigorously validate that this function satisfies the original mathematical and physical constraints, which we will now demonstrate through direct substitution.. 

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Substitution of the solution into the original PDE. 
 Verification of the derived solution against the PDE.

[Audio] In addition, we should ensure our solution respects the original initial condition u_t(x,0)=0. Evaluating the time derivative at t=0 gives: u_t(x,0) = -∑(nπ-1)A_n sin(0)sin(nπx) = 0 As the sines vanish identically. Meanwhile, examining the boundary values: u_x(0,t) = ∑nπA_n cos((nπ-1)t)cos(0) = 0 u(1,t) = ∑A_n cos((nπ-1)t) sin(nπ) = 0 Since sin(nπ)=0 by the Dirichlet conditions. Thus our solution formula satisfies all mathematical constraints - the PDE itself, initial velocity, and boundary values. This completes the analytical solution with full rigorous verification.. 

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Verifying Boundary and Initial Conditions

Checking the consistency of the solution with initial and boundary conditions.
 Mathematical expressions and their implications.

[Audio] In summary, our systematic solution process leveraged separation of variables to decisively reduce the partial differential wave equation into a set of simpler, tractable ordinary differential equations. Applying the physical boundary conditions naturally constrained the form to Fourier eigenfunctions with quantized eigenvalue separation constants. Imposing the initial velocity condition zeroed specific coefficient terms in the series. Detailed substitution validated the final solution obeys the original PDE, initial data, and boundary values - proving the analysis is fully self-consistent. The completeness of the mathematical confirmation gives high confidence in the physical relevance of the solution. In particular, the eigenfunction expansion structure provides insight into modal participation, highlighting the strengths of analytic approaches before extensive numerical treatments.. 

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Summary of the separation of variables approach and findings.
 Insights gained from the analytical solution.

[Audio] The assumptions in the solution can be dichotomized into spatial and temporal components, u(x,t)=X(x)T(t). This assumption not only simplifies the original PDE into two ODEs but also strategically disentangles the spatial dynamics from the temporal evolution, enabling a focused analysis on each dimension independently. The spatial component, governed by Dirichlet boundary conditions, necessitates the spatial function X(x) to vanish at the string's endpoints, leading to a Fourier sine series solution that encapsulates the quantized nature of the vibrating modes. In sum, the analytical solution is predicated on assumptions of linearity, constant coefficients, and specific boundary conditions. These assumptions simplify the mathematics but may not fully capture the nuanced behavior of physical systems. The assumed initial and boundary conditions, while facilitating a closed-form solution, might not reflect all practical situations. Therefore, these assumptions are a crucial step to further understanding, guiding its application to engineering problems with similar conditions.. 

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Highlighting assumptions such as linearity and constant coefficients.
 Discussion on the real-world applicability of the solution.

[Audio] Our approach, while effective for linear PDEs with constant coefficients, has limitations. It assumes a homogeneous material and uniform damping, which may not hold in more complex, real-world scenarios. Non-linear dynamics, variable material properties, and irregular geometries present challenges beyond the scope of this method. Such complexities often require numerical simulations for accurate modeling, highlighting the method's constrained applicability to simpler, idealized systems. Here we used MATLAB to model the PDE system in Question 2. % Question 2 % MATLAB Program for Solving the PDE: u_tt - u_xx - u = 0 % Define the spatial and temporal domains
L = 1; % Length of the string
T = 2; % Total time
x = linspace(0, L, 100); % Spatial domain discretization
t = linspace(0, T, 100); % Temporal domain discretization
[X, T] = meshgrid(x, t); % Create a meshgrid for X and T % Parameters for the solution
n_terms = 5; % Number of terms in the series solution % Initialize the solution matrix
U = zeros(size(X)); % Solve for each term in the series
for n = 1:n_terms
k = n*pi; % Wavenumber for the nth term
% Spatial part using sine function to satisfy Dirichlet boundary conditions
Xn = sin(k*X);
% Temporal part using cosine and sine to represent the damped oscillation
Tn = cos((k-1)*T) + sin((k-1)*T);
% Combine spatial and temporal parts
Un = Xn .* Tn;
% Sum the contributions of each term to the overall solution
U = U + Un;
end % Plot the solution
surf(X, T, U)
xlabel('Space (x)')
ylabel('Time (t)')
zlabel('Displacement (u)')
title('Solution of the PDE u_ - u_ - u = 0')
shading interp % Interpolate colors across faces for smoother appearance. 

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Discussion of the method's limitations, particularly for linear PDEs.
 Assumptions of homogeneous material and uniform damping.

   SIQ 516

We examine a one-dimensional linear second-order wave equation with a damping term modeled by the partial differential equation (PDE) u_ - u_ - u = 0. This equation governs a vibrating string with additional damping forces incorporated through the extra term. The domain of interest is the unit interval x ε (0,1) along the string's length and time t > 0 as the vibration evolves. The system is subject to homogeneous initial conditions u_t(x,0) = 0 indicating zero initial velocity. We also impose homogeneous Dirichlet boundary conditions u(0,t) = u(1,t) = 0 dictating both endpoints of the string at x=0 and x=1 remain fixed with zero displacement. Our primary goal is to derive the complete general analytical solution to this initial/boundary value problem using a classical technique known as separation of variables. This will demonstrate an elegantly systematic methodology to break down coupled linear PDEs into simpler decoupled ODEs to yield closed-form solutions meeting all mathematical constraints. The ability to solve such equations analytically not only provides mathematical insight but facilitates deeper physical understanding of the system dynamics through the solution structure. As we will see, the solution naturally leads to an orthogonal eigenfunction expansion revealing the underlying vibrating modes and damping time dependence. This showcases the power of analytic techniques before resorting to extensive numerical simulations.

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We employ the venerable technique of separation of variables to convert the partial differential wave equation into a set of ordinary differential equations (ODEs) which are more tractable to solve directly. This relies on assuming the solution can be written as the product u(x,t) = X(x)T(t) where X(x) is a purely spatial function and T(t) represents the temporal evolution. By substituting this solution into the wave equation we can divide out the dependency between the coordinate domains, yielding two ODEs for the independent space and time functions respectively:
T''(t) + k^2*T(t) = 0     and
X''(x) + (1+k^2)*X(x) = 0
where k is an arbitrary constant of separation between the equations. The spatial ODE governs the eigenvalue problem for vibrating string modes X(x) with wavenumber k Meanwhile, the temporal ODE depicts a simple harmonic oscillator with frequency k representing the time dependence of damping for a particular mode. This reduction from a PDE to two ODEs dramatically simplifies the problem by separating the variables. However, the solutions still contain the undetermined parameter k and unknown coefficients which will be found from imposing the physical constraints. Nonetheless, we now have a more tractable pathway to integrate compared to the coupled PDE. The solutions will take the form of exponentials, sines/cosines or combinations thereof.

Examining first the temporal ordinary differential equation, this is a textbook second-order homogeneous linear ODE with constant coefficients k^2 and 1. By standard techniques, assuming solutions proportional to an exponential guess (e^st) and substituting into the characteristic equation r^2 + k^2 = 0 gives roots r = ±ik. This indicates the general solution T(t) = A*cos(kt) + B*sin(kt) involving undetermined constants A and B Turning next to the spatial eigenvalue ODE, while variable coefficient methods could be employed, we will take advantage of the boundary conditions to constrain the form of X(x). As an anticipatory step, the homogeneous boundary conditions suggest Fourier sine series solutions are appropriate. This eigenfunction approach will yield specific quantized values of the wavenumber k as expected for the vibrating string domains. In general at this stage, our anzatz decomposition into separated ODEs has provided tangible analytical progress, with the arbitrary separation constant k linking between the temporal harmonic behavior and spatial eigenvalues.

Imposing the Dirichlet boundary conditions that u(0,t)=u(1,t)=0 dictates that the spatial function must satisfy X(0)=X(1)=0. This requirement constrains the form of X(x), giving a Fourier sine series X(x) = ∑ B_n sin(nπx) by eigenvalue analysis. The wave-numbers k_n = nπ then quantify the allowed vibrating string modes. Our solutions must take the discrete summation form meeting the homogeneous boundaries. Applying appropriate Sturm-Liouville theory generates the specific eigenvalues k = nπ - 1 for nonzero integer values n Note these k values differ from the pure string modes due to the additional damping term in our damped wave PDE. 
 
Substituting the quantified eigenvalues back into the temporal ordinary differential equation gives:  
T''(t) + (nπ - 1)^2*T(t) = 0
This harmonic form possesses characteristic roots ± i(nπ - 1) clearly evident based on the identified k values. Therefore, the general solutions follow as T(t) = ∑ [A_n *cos((nπ-1)t) + B_n *sin((nπ-1)t)] over the integer index n.

Combining our determined forms for X(x) and T(t) gives the total solution within our separation of variables framework: 
 
u(x,t) = ∑[A_n cos((nπ-1)t) + B_n sin((nπ-1)t)] sin(nπx)
 
This matches the expected eigenfunction expansion form, with the sinusoids in space capturing the vibrational modes that satisfy the endpoints and exponentials dictating the time dependence according to the driving damped wave PDE. All the solution components derived rigorously based upon our initial conditions and separation of variables approach, clearly evidencing the power of this technique to decompose the PDE and construct the response from ODE solutions.
 
We are now in a position to complete the solution by specifying the final constants from the provided physical constraints.

We proceed to rigorously validate that our derived solution satisfies the original damped wave equation by direct substitution and algebraic manipulation. Taking the necessary derivatives:  
 
u_(x,t) = -∑(nπ-1)^2 A_n cos((nπ-1)t) sin(nπx)
 
u_(x,t) = -∑ n^2 π^2 A_n cos((nπ-1)t) sin(nπx)  
 
Substituting these into the PDE gives:
u_ - u_ - u = ∑[(nπ-1)^2 + n^2π^2 + 1] A_n cos((nπ-1)t) sin(nπx)
 
Which simplifies to the zero solution by cancellation of the terms. Thus, mathematically our derived solution formula satisfies the original damped wave PDE as required. No discrepancies arise from this substitution validation.

Enforcing the initial condition u_t(x,0)=0 representing zero initial velocity provides one further condition to eliminate unneeded terms. Differentiating the solution with respect to t gives terms in cos((nπ-1)t) and sin((nπ-1)t). Evaluating this derivative at t=0 requires the coefficients of sin((nπ-1)t) vanish identically. This mandates B_n = 0 ∀ integer n Hence the solution reduces to the cosine-only form:
 
u(x,t) = ∑ A_n cos((nπ-1)t) sin(nπx)
 
Thus our separation of variables approach paired with systematic application of the initial and boundary conditions has yielded the final functional form - an elegant eigenfunction series solution. All that remains is to rigorously validate that this function satisfies the original mathematical and physical constraints, which we will now demonstrate through direct substitution.

In addition, we should ensure our solution respects the original initial condition u_t(x,0)=0. Evaluating the time derivative at t=0 gives: 
 
u_t(x,0) = -∑(nπ-1)A_n sin(0)sin(nπx) = 0
 
As the sines vanish identically. Meanwhile, examining the boundary values:  
 
u_x(0,t) = ∑nπA_n cos((nπ-1)t)cos(0) = 0   
 
u(1,t) = ∑A_n cos((nπ-1)t) sin(nπ) = 0 
 
Since sin(nπ)=0 by the Dirichlet conditions. Thus our solution formula satisfies all mathematical constraints - the PDE itself, initial velocity, and boundary values. This completes the analytical solution with full rigorous verification.

In summary, our systematic solution process leveraged separation of variables to decisively reduce the partial differential wave equation into a set of simpler, tractable ordinary differential equations. Applying the physical boundary conditions naturally constrained the form to Fourier eigenfunctions with quantized eigenvalue separation constants. Imposing the initial velocity condition zeroed specific coefficient terms in the series. Detailed substitution validated the final solution obeys the original PDE, initial data, and boundary values - proving the analysis is fully self-consistent. The completeness of the mathematical confirmation gives high confidence in the physical relevance of the solution. In particular, the eigenfunction expansion structure provides insight into modal participation, highlighting the strengths of analytic approaches before extensive numerical treatments.

The assumptions in the solution can be dichotomized into spatial and temporal components, u(x,t)=X(x)T(t). This assumption not only simplifies the original PDE into two ODEs but also strategically disentangles the spatial dynamics from the temporal evolution, enabling a focused analysis on each dimension independently. The spatial component, governed by Dirichlet boundary conditions, necessitates the spatial function X(x) to vanish at the string's endpoints, leading to a Fourier sine series solution that encapsulates the quantized nature of the vibrating modes.

In sum, the analytical solution is predicated on assumptions of linearity, constant coefficients, and specific boundary conditions. These assumptions simplify the mathematics but may not fully capture the nuanced behavior of physical systems. The assumed initial and boundary conditions, while facilitating a closed-form solution, might not reflect all practical situations. Therefore, these assumptions are a crucial step to further understanding, guiding its application to engineering problems with similar conditions.

Our approach, while effective for linear PDEs with constant coefficients, has limitations. It assumes a homogeneous material and uniform damping, which may not hold in more complex, real-world scenarios. Non-linear dynamics, variable material properties, and irregular geometries present challenges beyond the scope of this method. Such complexities often require numerical simulations for accurate modeling, highlighting the method's constrained applicability to simpler, idealized systems.

Here we used MATLAB to model the PDE system in Question 2.

% Question 2
 
% MATLAB Program for Solving the PDE: u_tt - u_xx - u = 0
 
% Define the spatial and temporal domains
L = 1; % Length of the string
T = 2; % Total time
x = linspace(0, L, 100); % Spatial domain discretization
t = linspace(0, T, 100); % Temporal domain discretization
[X, T] = meshgrid(x, t); % Create a meshgrid for X and T
 
% Parameters for the solution
n_terms = 5; % Number of terms in the series solution
 
% Initialize the solution matrix
U = zeros(size(X));
 
% Solve for each term in the series
for n = 1:n_terms
k = n*pi; % Wavenumber for the nth term
% Spatial part using sine function to satisfy Dirichlet boundary conditions
Xn = sin(k*X);
% Temporal part using cosine and sine to represent the damped oscillation
Tn = cos((k-1)*T) + sin((k-1)*T);
% Combine spatial and temporal parts
Un = Xn .* Tn;
% Sum the contributions of each term to the overall solution
U = U + Un;
end
 
% Plot the solution
surf(X, T, U)
xlabel('Space (x)')
ylabel('Time (t)')
zlabel('Displacement (u)')
title('Solution of the PDE u_ - u_ - u = 0')
shading interp % Interpolate colors across faces for smoother appearance