diff --git a/work/week12/starting_point/CMakeLists.txt b/work/week12/starting_point/CMakeLists.txt
index 67e7652..94322be 100644
--- a/work/week12/starting_point/CMakeLists.txt
+++ b/work/week12/starting_point/CMakeLists.txt
@@ -1,94 +1,97 @@
 cmake_minimum_required (VERSION 3.1)
 project (Particles)
 cmake_policy(VERSION 3.3)
 
 set(CMAKE_CXX_STANDARD 14)
 set(CMAKE_MODULE_PATH ${CMAKE_MODULE_PATH} "${PROJECT_SOURCE_DIR}/cmake")
 
 ################################################################
 # libpart
 ################################################################
 
 add_library(part
   compute_boundary.cc
   compute_verlet_integration.cc
   particle.cc
   planet.cc
   compute_gravity.cc
   csv_reader.cc
   particles_factory_interface.cc
   planets_factory.cc
   compute_contact.cc
   compute_kinetic_energy.cc
   csv_writer.cc
   system.cc
   compute_energy.cc
   compute_potential_energy.cc
   ping_pong_ball.cc
   material_point.cc
   system_evolution.cc
   ping_pong_balls_factory.cc
   compute_interaction.cc
   compute_temperature.cc
   material_points_factory.cc
   )
 
 
 ################################################################
 # FFTW
 ################################################################
 
 
 set(FFTW_LIBRARY_PATH CACHE PATH "library where to search FFTW")
 find_library (FFTW_LIBRARY fftw3 /usr/include/ ${FFTW_LIBRARY_PATH})
 
 option (USE_FFTW "Use FFTW library" ON)
 
 if (USE_FFTW)
     target_link_libraries (part ${FFTW_LIBRARY})
 endif (USE_FFTW)
 
 ################################################################
 # Create executable
 ################################################################
 
 add_executable(particles main.cc)
 target_link_libraries(particles part)
 
 
 
 
 ################################################################
 # Google test
 ################################################################
 
 
 include(GoogleTest)
 enable_testing()
 find_package(GTest)
 if (GTEST_FOUND)
 include_directories(${GTEST_INCLUDE_DIRS})
 add_executable(test_kepler test_kepler.cc)
 add_executable(test_fft test_fft.cc)
+add_executable(test_heat_eq test_heat_equation_fft.cc)
 target_link_libraries(test_kepler part ${GTEST_BOTH_LIBRARIES} pthread)
+target_link_libraries(test_heat_eq part ${GTEST_BOTH_LIBRARIES} pthread)
 target_link_libraries(test_fft part ${GTEST_BOTH_LIBRARIES} ${FFTW_LIBRARIES} pthread)
 gtest_discover_tests(test_kepler)
 gtest_discover_tests(test_fft)
+gtest_discover_tests(test_heat_eq)
 endif()
 
 
 ################################################################
 # Doxygen
 ################################################################
 
 find_package(Doxygen)
 if (DOXYGEN_FOUND)
 
 # to set other options, read: https://cmake.org/cmake/help/v3.9/module/FindDoxygen.html
 doxygen_add_docs(
     doxygen
     ${PROJECT_SOURCE_DIR}
     COMMENT "Generate html pages"
 )
 add_custom_target(doc DEPENDS doxygen)
 endif(DOXYGEN_FOUND)
diff --git a/work/week12/starting_point/compute_temperature.cc b/work/week12/starting_point/compute_temperature.cc
index ca140a4..b7a4cbb 100644
--- a/work/week12/starting_point/compute_temperature.cc
+++ b/work/week12/starting_point/compute_temperature.cc
@@ -1,114 +1,114 @@
 #include <cmath>
 #include "compute_temperature.hh"
 #include "fft.hh"
 #include "material_point.hh"
 
 /* -------------------------------------------------------------------------- */
 
 ComputeTemperature::ComputeTemperature(Real dt) : dt(dt) {}
 
 
 void ComputeTemperature::compute(System& system) {
 
     // Get the number of material points
     auto N = system.getNbParticles();
 
     // Create matrix of temperature and heat source in the space domain
     Matrix<complex> M_theta(sqrt(N));
     Matrix<complex> M_hv(sqrt(N));
 
     // Fill in temperature and heat source matrices,
     // while computing min and max positions over the x-dimension
     int minI = std::numeric_limits<int>::infinity();
     int maxI = -std::numeric_limits<int>::infinity();
     for (auto&& entry : index(M_theta)) {
         int i = std::get<0>(entry);
         int j = std::get<1>(entry);
         auto& theta = std::get<2>(entry);
 
         // Get corresponding system particle (material point cast)
         Particle& par = system.getParticle(i * sqrt(N) + j);
         auto& mp = static_cast<MaterialPoint&>(par);
 
         // Assign temperautre and heat distribution to corresponding matrices
         theta = mp.getTemperature();
         M_hv(i, j) = mp.getHeatDistribution();
-        std::cout << "particle(" << i*sqrt(N) + j << ")" << " ";
-        std::cout << "theta0: " << theta << std::endl;
+
+        // std::cout << "theta(" << i << ", " << j << ") = " << theta << std::endl;
 
         // Update space boundaries if needed
         auto pos = mp.getPosition();
         if (pos[0] < minI) { minI = pos[0];}
         if (pos[0] > maxI) { maxI = pos[0];}
     }
 
     // Get space dimension
-    float spaceDimension = maxI - minI;
+    float L = maxI - minI;
+    float norm = L / (2 * M_PI);
+    // std::cout << "L = " << L << ", norm  = " << norm << std::endl;
 
     // Apply FFT transformation to both matrices -> Fourier domain
     Matrix<complex> M_theta_hat = FFT::transform(M_theta);
     Matrix<complex> M_hv_hat = FFT::transform(M_hv);
 
     // Get coordinates in the Fourier domain
-    Matrix<std::complex<int>> M_q = FFT::computeFrequencies(N);
-    M_q /= spaceDimension/(2*M_PI);  // normalize by space dimension
+    Matrix<std::complex<int>> M_q = FFT::computeFrequencies(sqrt(N));
+    // M_q /= L;  // normalization
 
     // parameters for gold -> to be set outside of the method
     Real pho = 19.32;  // g/cm^3
     Real C = 0.129;  // J/g*°C
 
     // Compute time derivative of temperature distribution in the Fourier domain
     Matrix<complex> M_dtheta_hat_dt(sqrt(N));
     for (auto&& entry : index(M_theta_hat)) {
         int i = std::get<0>(entry);
         int j = std::get<1>(entry);
         auto& dtheta_hat_dt = std::get<2>(entry);
         auto theta_hat = M_theta_hat(i, j);
         auto hv_hat = M_hv_hat(i, j);
 
         if (i == 0 && j == 0){
             theta_hat = 0;
             hv_hat = 0;
         }
-        // std::cout << "theta_hat = " << theta_hat << ", hv_hat = " << hv_hat << std::endl;
+        // std::cout << "theta_hat(" << i << ", " << j << ") = " << theta_hat << std::endl;
+        // std::cout << "hv_hat(" << i << ", " << j << ") = " << hv_hat << std::endl;
 
         // Get particle heat rate
         Particle& par = system.getParticle(i * sqrt(N) + j);
         auto& mp = static_cast<MaterialPoint&>(par);
         Real hr = mp.getHeatRate();
 
-        Real laplacian_q = pow(M_q(i, j).real(), 2) + pow(M_q(j, i).real(), 2);
+        Real laplacian_q = pow((float) M_q(i, j).real() / norm, 2) + pow((float) M_q(j, i).real() / norm, 2);
         if (i == 0 && j == 0){
             laplacian_q = 1;
         }
 
         dtheta_hat_dt = (hv_hat - hr * theta_hat * laplacian_q) / (pho * C);
-        // std::cout << "dtheta_hat_dt = " << dtheta_hat_dt << std::endl;
+        // std::cout << "dtheta_hat_dt(" << i << ", " << j << ") = " << dtheta_hat_dt << std::endl;
     }
 
     // Compute inverse FFT to get time derivative of temperature distribution in the space domain
     Matrix<complex> M_dtheta_dt = FFT::itransform(M_dtheta_hat_dt);
 
     // Update the temperature of all the particles in our system
     for (auto&& entry2 : index(M_dtheta_dt)) {
         int i = std::get<0>(entry2);
         int j = std::get<1>(entry2);
         auto& dtheta_dt = std::get<2>(entry2);
-        // std::cout << " dtheta_dt = " << dtheta_dt << std::endl;
+        // std::cout << "dtheta_dt(" << i << ", " << j << ") = " << dtheta_dt << std::endl;
 
         Particle& par = system.getParticle(i*sqrt(N) + j);
         auto& mp = static_cast<MaterialPoint&>(par);
 
         // Set null temperature at the domain boundaries
-        if ((i == 0 ) || (j == 0 ) || (j == (int)sqrt(N) - 1) || (i == (int)sqrt(N) - 1) ) {
-            mp.getTemperature() = 0;
-        } else {
-            mp.getTemperature() += dtheta_dt.real() * this->dt;
-        }
-        // std::cout << "particle(" << i * sqrt(N) + j << "): ";
-        // std::cout << "theta = " << mp.getTemperature() << std::endl;
+        // if ((i == 0 ) || (j == 0 ) || (j == (int)sqrt(N) - 1) || (i == (int)sqrt(N) - 1) ) {
+        //     mp.getTemperature() = 0;
+        // }
+        mp.getTemperature() += dtheta_dt.real() * this->dt;
+        // std::cout << "theta(" << i << ", " << j << ") = " << mp.getTemperature() << std::endl;
     }
-
 }
 
 /* -------------------------------------------------------------------------- */
diff --git a/work/week12/starting_point/fft.hh b/work/week12/starting_point/fft.hh
index 70a3797..ce4b524 100644
--- a/work/week12/starting_point/fft.hh
+++ b/work/week12/starting_point/fft.hh
@@ -1,132 +1,136 @@
 #ifndef FFT_HH
 #define FFT_HH
 /* ------------------------------------------------------ */
 #include "matrix.hh"
 #include "my_types.hh"
 #include <fftw3.h>
 /* ------------------------------------------------------ */
 
 /**
 
 This file defines an FFT structure that is a wrapper around the FFTW library.
 In particular, it defines 3 functions to work with signals of complex numbers:
 - the transform() function to compute the forward FFT of a complex signal
 - the itransform() function to compute the inverse FFT of a complex signal
 - the computeFrequencies() function to compute the sample frequencies of the
     forward FFT of a complex signal (i.e. the coordinates of the signal in the Fourier space).
     The Laplacian of these frequencies is then used to sovle the heat equation in the Fourier space.
 
 */
 struct FFT {
 
   static Matrix<complex> transform(Matrix<complex>& m);
   static Matrix<complex> itransform(Matrix<complex>& m);
 
   static Matrix<std::complex<int>> computeFrequencies(int size);
 
 };
 
 /* ------------------------------------------------------ */
 
 inline Matrix<complex> FFT::transform(Matrix<complex>& m_in) {
 
   // Check memory allocation of input matrix
   if (m_in.data() == nullptr) {
     throw std::runtime_error("no memory allocated for m_in");
   }
 
   // Get matrix size and initialize output matrix
   int N = m_in.size();
   Matrix<complex> m_out(N);
 
   // Initialize input and output 2D signals
   auto in = (fftw_complex*) m_in.data();
   auto out = (fftw_complex*) m_out.data();
 
   // Create, execute and destroy FFT plan
   fftw_plan p = fftw_plan_dft_2d(N, N, in, out, FFTW_FORWARD, FFTW_ESTIMATE);
   fftw_execute(p);
 
-  // Destroy plan and fftw signals
+  // Destroy plan
   fftw_destroy_plan(p);
-  fftw_free(in); fftw_free(out);
+  // fftw_free(in); fftw_free(out);
 
   // Return FFT matrix
   return m_out;
 }
 
 /* ------------------------------------------------------ */
 
 inline Matrix<complex> FFT::itransform(Matrix<complex>& m_in) {
 
   // Check memory allocation of input matrix
   if (m_in.data() == nullptr) {
     throw std::runtime_error("no memory allocated for m_in");
   }
 
   // Get matrix size and initialize output matrix
   int N = m_in.size();
   Matrix<complex> m_out(N);
 
   // Initialize input and output 2D signals
   auto in = (fftw_complex*) m_in.data();
   auto out = (fftw_complex*) m_out.data();
 
   // Create, execute and destroy inverse FFT plan
   fftw_plan p = fftw_plan_dft_2d(N, N, in, out, FFTW_BACKWARD, FFTW_ESTIMATE);
   fftw_execute(p);
 
-  // Destroy plan and fftw signals
+  // Destroy plan
   fftw_destroy_plan(p);
-  fftw_free(in); fftw_free(out);
+  // fftw_free(in); fftw_free(out);
+
+  // Divide output matrix by factor N2 (needed for 2D arrays)
+  for (int i = 0; i < N; ++i) {
+    for (int j = 0; j < N; ++j) {
+      m_out(i, j) /= (N * N);
+    }
+  }
 
   // Return inverse FFT matrix
   return m_out;
 }
 
-/* ------------------------------------------------------ */
-
-
 /* ------------------------------------------------------ */
 
 inline Matrix<std::complex<int>> FFT::computeFrequencies(int size) {
 
   // Generate 1D frequency vector
   std::vector<float> vec(size);
   if (size % 2 == 0) {  // if size is even
     for (int i = 0; i < size / 2; ++i) {
-      vec[i] =(float)i;
+      vec[i] = (float)i;
     }
     for (int i = size / 2; i < size; ++i) {
       vec[i] = -size / 2  + ((float)i - size / 2);
     }
   } else { // if size is odd
     for (int i = 0; i < (size - 1) / 2; ++i) {
       vec[i] = (float)i;
     }
     for (int i = (size - 1) / 2; i < size; ++i) {
       vec[i] = -(size - 1) / 2  + ((float)i - (size - 1) / 2);
     }
   }
 
   // Check the 1D coordinate vecor on Fourier space
   // std::cout << "computeFrequencies, vect: " << std::endl;
   // for (int i = 0; i<vec.size(); ++i){
   //   std::cout << "vec(" << i << "): " << vec.at(i) << std::endl;
   // }
 
   // Populate 2D matrix with frequencies vector
   Matrix<std::complex<int>> m_out(size);
   for (int i = 0; i < size; ++i) {
     for (int j = 0; j < size; ++j) {
       m_out(i, j) = std::complex<int>(vec[j], 0);
       // std::cout << "(" << i << "," << j << "): " << vec.at(j) << " ";
     }
     // std::cout << std::endl;
   }
 
   return m_out;
 }
 
 
 #endif  // FFT_HH
diff --git a/work/week12/starting_point/generate_input.py b/work/week12/starting_point/generate_input.py
index 18b36f0..6ae1859 100644
--- a/work/week12/starting_point/generate_input.py
+++ b/work/week12/starting_point/generate_input.py
@@ -1,160 +1,160 @@
 import os
 import numpy as np
 import math
 import argparse
 import matplotlib.pyplot as plt
 
 
 def isSquare(n):
     ''' Check if an integer value is a perfect square or not. '''
     x = n // 2
     seen = set([x])
     while x * x != n:
         x = (x + (n // x)) // 2
         if x in seen:
             return False
         seen.add(x)
     return True
 
 
 def plotDistributions(hv2d, temp2d):
     ''' Plot 2D heat and temperature distributions. '''
     fig, axes = plt.subplots(1, 2, figsize=(12, 5))
 
     ax = axes[0]
     ax.set_title('Heat source distribution')
     ax.set_xlabel('Y')
     ax.set_ylabel('X')
     ax.imshow(hv2d, extent=[-1, 1, -1, 1])
 
     ax = axes[1]
     ax.set_title('Temperature distribution')
     ax.set_xlabel('Y')
     ax.set_ylabel('X')
     ax.imshow(temp2d, extent=[-1, 1, -1, 1])
 
     return fig
 
 
 def main():
 
     parser = argparse.ArgumentParser()
-    parser.add_argument('-n', '--number', type=int, default=32**2,
+    parser.add_argument('-n', '--number', type=int, default=4**2,
                         help='Number of particles')
     parser.add_argument('-o', '--outfile', type=str, default=None,
                         help='Name of output file')
     parser.add_argument('-H', '--heatfield', type=str, default='circular',
                         help='Heat distribution type ("null", "line" or "circular")')
     parser.add_argument('-T', '--tempfield', type=str, default=None,
                         help='Initial temperature field ("random" or "homogeneous"),\
                         only used for non-test heatfields')
     parser.add_argument('-t', '--test', default=False, action='store_true',
                         help='Test configuration (temperature field automtcially generated')
     parser.add_argument('-R', '--radius', type=float, default=1 / 3,
                         help='Heat source radius (if heatfield is set to "circular")')
     parser.add_argument('-p', '--plot', default=False, action='store_true',
                         help='Plot heat and temperature distribution')
     parser.add_argument('--hr', type=float, default=3,
                         help='Specific heat rate')
 
     args = parser.parse_args()
 
     # Get number of particles and square root of it
     N = int(args.number)
     if not(isSquare(N)):
         raise ValueError('number of particles must be square')
     sqrtN = int(math.sqrt(N))
 
     # Set initial positions on a 2D rectilinear grid (z = 0)
     x = np.linspace(-1, 1, sqrtN)
     pos2d = np.array(np.meshgrid(x, x)).T
 
     # Check for input validity
     test_error_str = 'cannot be specified for test configurations'
     if args.test:
         if args.outfile is not None:
             raise ValueError('output file {}'.format(test_error_str))
         if args.tempfield is not None:
             raise ValueError('temperature field {}'.format(test_error_str))
 
     # Set heat volumetric source according to user input
     hv2d = np.zeros((sqrtN, sqrtN))
     if args.heatfield == 'null':
         if args.test:
             # Test configuration: set constant temperature field
-            theta2d = np.zeros((sqrtN, sqrtN))
+            theta2d = np.ones((sqrtN, sqrtN))
             outfile = 'testnull.csv'
 
     elif args.heatfield == 'line':
         # Set punctual (dirac) heat distribution
         hv2d[int(sqrtN / 4), :] = -sqrtN
         hv2d[int(3 * sqrtN / 4), :] = sqrtN
 
         # Test configuration: set corresponding temperature field from analytical solution
         if args.test:
             outfile = 'testline.csv'
             theta1d = np.piecewise(x, [x < -0.5, (x >= -0.5) & (x < 0.5), x >= 0.5],
                                    [lambda x: -x - 1, lambda x: x, lambda x: -x + 1])
             theta2d = np.tile(theta1d, (sqrtN, 1)).T
 
     elif args.heatfield == 'circular':
         if args.test:
             raise ValueError('no test configuration available for "circular" heat field')
 
         # Set circular heat distribution
         R = args.radius
         if R < 0:
             raise ValueError('radius must be positive')
         for i in range(sqrtN):
             for j in range(sqrtN):
                 if (x[i]**2 + x[j]**2 < R**2):
                     hv2d[i, j] = N
 
     else:
         raise ValueError('heatfield must be "null", "line" or "circular"')
 
     if not args.test:
         # Set temperature field according to user input
         if args.tempfield == 'random' or args.tempfield is None:
             # Random field (between zero and 2)
             theta2d = np.random.random((sqrtN, sqrtN)) * 2
         elif args.tempfield == 'homogeneous':
             # Homogeneous field (all equal to one)
             theta2d = np.ones((sqrtN, sqrtN))
         else:
             raise ValueError('tempfield must be "random" or "homogeneous"')
 
         # Set output file according to user input
         outfile = args.outfile if args.outfile is not None else 'step-00000.csv'
 
     # Reshape matrices into serialized vectors
     positions = np.hstack((pos2d.reshape(N, 2), np.zeros((N, 1))))
     hv = np.reshape(hv2d, (hv2d.size, 1))
     theta = np.reshape(theta2d, (theta2d.size, 1))
 
     # Generate serialized vectors for other, constant quantities
     hr = args.hr * np.ones((N, 1))
     velocities = np.zeros((N, 3))
     forces = np.zeros((N, 3))
-    masses = np.ones((N, 3))
+    masses = np.ones((N, 1))
 
     # Save data into csv file
     file_data = np.hstack((positions, velocities, forces, masses, theta, hr, hv))
     outdir = os.path.join(os.getcwd(), 'build', 'dumps')
     if not os.path.isdir(outdir):
         os.makedirs(outdir)
     filepath = os.path.join(os.getcwd(), 'build', 'dumps', outfile)
     np.savetxt(filepath, file_data, delimiter=" ")
 
     # Plot the heat and temperature fields
     if args.plot:
         plotDistributions(hv2d, theta2d)
         plt.show()
 
 
 if __name__ == '__main__':
     try:
         main()
     except ValueError as e:
         print('Error:', e)
         quit()
diff --git a/work/week12/starting_point/step.csv b/work/week12/starting_point/step.csv
deleted file mode 100644
index 9d3c2b0..0000000
--- a/work/week12/starting_point/step.csv
+++ /dev/null
@@ -1,9 +0,0 @@
--1.000000000000000000e+00 -1.000000000000000000e+00 0.000000000000000000e+00 0.000000000000000000e+00 0.000000000000000000e+00 0.000000000000000000e+00 0.000000000000000000e+00 0.000000000000000000e+00 0.000000000000000000e+00 1.105296388216345349e+00 2.000000000000000000e+00 3.000000000000000000e+00 0.000000000000000000e+00
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--1.000000000000000000e+00 1.000000000000000000e+00 0.000000000000000000e+00 0.000000000000000000e+00 0.000000000000000000e+00 0.000000000000000000e+00 0.000000000000000000e+00 0.000000000000000000e+00 0.000000000000000000e+00 1.820369108671012093e+00 2.000000000000000000e+00 3.000000000000000000e+00 0.000000000000000000e+00
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-0.000000000000000000e+00 0.000000000000000000e+00 0.000000000000000000e+00 0.000000000000000000e+00 0.000000000000000000e+00 0.000000000000000000e+00 0.000000000000000000e+00 0.000000000000000000e+00 0.000000000000000000e+00 1.790316306983258610e+00 2.000000000000000000e+00 3.000000000000000000e+00 9.000000000000000000e+00
-0.000000000000000000e+00 1.000000000000000000e+00 0.000000000000000000e+00 0.000000000000000000e+00 0.000000000000000000e+00 0.000000000000000000e+00 0.000000000000000000e+00 0.000000000000000000e+00 0.000000000000000000e+00 1.945211012159552944e+00 2.000000000000000000e+00 3.000000000000000000e+00 0.000000000000000000e+00
-1.000000000000000000e+00 -1.000000000000000000e+00 0.000000000000000000e+00 0.000000000000000000e+00 0.000000000000000000e+00 0.000000000000000000e+00 0.000000000000000000e+00 0.000000000000000000e+00 0.000000000000000000e+00 1.094870165133711826e+00 2.000000000000000000e+00 3.000000000000000000e+00 0.000000000000000000e+00
-1.000000000000000000e+00 0.000000000000000000e+00 0.000000000000000000e+00 0.000000000000000000e+00 0.000000000000000000e+00 0.000000000000000000e+00 0.000000000000000000e+00 0.000000000000000000e+00 0.000000000000000000e+00 1.147579217696644616e+00 2.000000000000000000e+00 3.000000000000000000e+00 0.000000000000000000e+00
-1.000000000000000000e+00 1.000000000000000000e+00 0.000000000000000000e+00 0.000000000000000000e+00 0.000000000000000000e+00 0.000000000000000000e+00 0.000000000000000000e+00 0.000000000000000000e+00 0.000000000000000000e+00 1.344093002773754097e+00 2.000000000000000000e+00 3.000000000000000000e+00 0.000000000000000000e+00
diff --git a/work/week12/starting_point/stepter b/work/week12/starting_point/stepter
deleted file mode 100644
index 0bed52a..0000000
--- a/work/week12/starting_point/stepter
+++ /dev/null
@@ -1,256 +0,0 @@
--1.000000000000000000e+00 -1.000000000000000000e+00 0.000000000000000000e+00 0.000000000000000000e+00 0.000000000000000000e+00 0.000000000000000000e+00 0.000000000000000000e+00 0.000000000000000000e+00 0.000000000000000000e+00 1.139871358819202873e+00 2.000000000000000000e+00 3.000000000000000000e+00 0.000000000000000000e+00
--1.000000000000000000e+00 -8.666666666666666963e-01 0.000000000000000000e+00 0.000000000000000000e+00 0.000000000000000000e+00 0.000000000000000000e+00 0.000000000000000000e+00 0.000000000000000000e+00 0.000000000000000000e+00 1.558855164112928460e+00 2.000000000000000000e+00 3.000000000000000000e+00 0.000000000000000000e+00
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diff --git a/work/week12/starting_point/stepter.csv b/work/week12/starting_point/stepter.csv
deleted file mode 100644
index 4846bc8..0000000
--- a/work/week12/starting_point/stepter.csv
+++ /dev/null
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diff --git a/work/week12/starting_point/test_fft.cc b/work/week12/starting_point/test_fft.cc
index dc340a9..f506ba0 100644
--- a/work/week12/starting_point/test_fft.cc
+++ b/work/week12/starting_point/test_fft.cc
@@ -1,126 +1,100 @@
-#include "my_types.hh"
-#include "fft.hh"
-#include "compute_temperature.hh"
 #include <gtest/gtest.h>
 #include <cstdlib>
+#include <fstream>
+#include <sstream>
+#include "my_types.hh"
+#include "fft.hh"
+
+
 /*****************************************************************/
 TEST(FFT, transform) {
   UInt N = 512;
   Matrix<complex> m(N);
 
   Real k = 2 * M_PI / N;
   for (auto&& entry : index(m)) {
     int i = std::get<0>(entry);
     int j = std::get<1>(entry);
     auto& val = std::get<2>(entry);
     val = cos(k * i);
   }
 
   Matrix<complex> res = FFT::transform(m);
-  // Matrix<std::complex<int>> a = FFT::computeFrequencies(10);
 
   for (auto&& entry : index(res)) {
     int i = std::get<0>(entry);
     int j = std::get<1>(entry);
     auto& val = std::get<2>(entry);
     if (std::abs(val) > 1e-10)
       std::cout << i << "," << j << " = " << val << std::endl;
 
     if (i == 1 && j == 0)
       ASSERT_NEAR(std::abs(val), N * N / 2, 1e-10);
     else if (i == N - 1 && j == 0)
       ASSERT_NEAR(std::abs(val), N * N / 2, 1e-10);
     else
       ASSERT_NEAR(std::abs(val), 0, 1e-10);
   }
 }
 /*****************************************************************/
 
 TEST(FFT, inverse_transform) {
   UInt N = 512;
   Matrix<complex> m(N);
 
   Real k = 2 * M_PI / N;
   for (auto&& entry : index(m)) {
     int i = std::get<0>(entry);
     int j = std::get<1>(entry);
     auto& val = std::get<2>(entry);
     if ((i == 1 && j == 0) || (i == N - 1 && j == 0)) {
       val = N * N / 2;
     } else {
       val = 0;
     }
   }
 
   Matrix<complex> res = FFT::itransform(m);
 
   for (auto&& entry : index(res)) {
     int i = std::get<0>(entry);
     int j = std::get<1>(entry);
     auto& val = std::get<2>(entry);
     ASSERT_NEAR(std::abs(val), std::abs(cos(k * i)), 1e-10);
   }
 }
-/*****************************************************************/
 
-/*****************************************************************/
+TEST(FFT, compute_frequencies) {
+  UInt N = 8;
 
-TEST(FFT, Homogeneous) {
-  UInt N = 512;
-  Matrix<complex> m(N);
-
-
-  //Create a homogeneous matrix of material points
-  for (auto&& entry : index(m)) {
-    int i = std::get<0>(entry);
-    int j = std::get<1>(entry);
-    auto& val = std::get<2>(entry);
-    val = 5;
+  // Populate reference frequency matrix from numpy generated file
+  Matrix<std::complex<int>> freqs_ref(N);
+  std::ifstream is("test_computeFrequencies.txt");
+  if (is.is_open() == false) {
+    std::cerr << "cannot open file" << std::endl;
+    throw;
   }
+  std::string line;
+  double tmp;
+  for (int i=0; i < N; ++i) {
+    getline(is, line);
+    std::stringstream sstr(line);
+    for (int j = 0; j < N; ++j) {
+        sstr >> tmp;
+        freqs_ref(i, j) = std::complex<int>(tmp, 0);
+    }
+  }
+  is.close();
 
-  // Apply FFT transformation
-  Matrix<complex> tempMtx_fft = FFT::transform(m);
-
-  // Get FFT coordinates
-  Matrix<std::complex<int>> fft_coord = FFT::computeFrequencies(N);
-
-  // Heat flux
-  Real hv = 0;
-
-  for (auto&& entry : index(tempMtx_fft)) {
+  // Create frequency matrix with our own implementation
+  Matrix<std::complex<int>> freqs = FFT::computeFrequencies(N);
 
+  // Compare the two matrices
+  for (auto&& entry : index(freqs)) {
     int i = std::get<0>(entry);
     int j = std::get<1>(entry);
     auto& val = std::get<2>(entry);
-
-    // Generates random particle heat rate
-    Real hr = rand() % 1;
-
-    Real q_squared;
-    if (i == 0 && j == 0){
-      val = 0;
-      q_squared = 1;
-    }else{
-      q_squared = pow(2*M_PI*fft_coord(i,j).real(),2) + pow(2*M_PI*fft_coord(j,i).real(),2);
-    }
-    // std::cout << "Before val is: " << val << ", " << hr << ", " << q_squared;
-    val = hv - val * hr * q_squared;
-    // std::cout << " After val is: " << val << std::endl;
+    auto& ref = freqs_ref(i, j);
+    ASSERT_NEAR(std::abs(val), std::abs(ref), 1e-10);
   }
-
-    // Inverse FFT transformation
-  Matrix<complex> tempMtx_ifft = FFT::itransform(tempMtx_fft);
-  tempMtx_ifft *= 0.1;
-
-  for (auto&& entry2 : index(tempMtx_ifft)) {
-    int i = std::get<0>(entry2);
-    int j = std::get<1>(entry2);
-    auto& val = std::get<2>(entry2);
-    // std::cout << " IFFT val is: " << val << std::endl;
-    Real new_m = std::abs(m(i,j)) + val.real();
-/*    std::cout << m(i,j)<< std::endl;
-    Real new_m=5;*/
-    // std::cout << " New temp is: " << mp.getTemperature() << std::endl;
-    ASSERT_NEAR(new_m,std::abs(m(i,j)), 1e-10);
-  }
-}
\ No newline at end of file
+}
diff --git a/work/week12/starting_point/test_fftfreq.py b/work/week12/starting_point/test_fftfreq.py
index 3416662..a5b023b 100644
--- a/work/week12/starting_point/test_fftfreq.py
+++ b/work/week12/starting_point/test_fftfreq.py
@@ -1,18 +1,13 @@
 import numpy as np
 
+# size of matrix dimensions
 n = 8
-timestep = 1
 
 #  Generate 1D frequency coordinates
-freq_1D = np.fft.fftfreq(n, d=timestep) * 2 * np.pi * n
-print(freq_1D)
+freq1d = np.fft.fftfreq(n) * n
+
 #  Generate 2D frequency coordinates
-freq_2D = np.einsum('i,j->ij', np.ones(n), freq_1D)
-print(freq_2D)
+freq2d = np.einsum('i,j->ij', np.ones(n), freq1d)
 
-#  Store 1D frequency coordinates
-np.savetxt("test_computeFrequency_1D.txt", freq_1D, delimiter=" ",
-           fmt='%.2f')
 #  Store 2D frequency coordinates
-np.savetxt("test_computeFrequency_2D.txt", freq_2D, delimiter=" ",
-           fmt='%.2f')
+np.savetxt("build/test_computeFrequencies.txt", freq2d, delimiter=" ", fmt='%.2f')
diff --git a/work/week12/starting_point/test_heat_equation_fft.cc b/work/week12/starting_point/test_heat_equation_fft.cc
new file mode 100644
index 0000000..d436655
--- /dev/null
+++ b/work/week12/starting_point/test_heat_equation_fft.cc
@@ -0,0 +1,57 @@
+#include <gtest/gtest.h>
+#include <cstdlib>
+#include "my_types.hh"
+#include "fft.hh"
+#include "compute_temperature.hh"
+#include "system.hh"
+#include "csv_reader.hh"
+#include "material_point.hh"
+#include "material_points_factory.hh"
+
+/*****************************************************************/
+
+void updateMatrix(System& system, Matrix<complex>& M_theta) {
+    auto N = system.getNbParticles();
+    for (auto&& entry : index(M_theta)) {
+        int i = std::get<0>(entry);
+        int j = std::get<1>(entry);
+        auto& theta = std::get<2>(entry);
+        Particle& par = system.getParticle(i * sqrt(N) + j);
+        auto& mp = static_cast<MaterialPoint&>(par);
+        theta = mp.getTemperature();
+    }
+}
+
+TEST(HEAT_EQ, Homogeneous) {
+    // Get factory instance
+    MaterialPointsFactory::getInstance();
+    ParticlesFactoryInterface& factory = ParticlesFactoryInterface::getInstance();
+
+    // Create system of particles and compute object
+    auto system = std::make_unique<System>();
+
+    // Parse input file to populate system
+    CsvReader reader("dumps/testnull.csv");
+    reader.read(*system);
+    auto N = system->getNbParticles();
+    auto ct = ComputeTemperature(0.1);
+
+    // Create and fill temperature matrices
+    Matrix<complex> M_theta(sqrt(N));
+    Matrix<complex> M_theta0(sqrt(N));
+    updateMatrix(*system, M_theta);
+    updateMatrix(*system, M_theta0);
+
+    // Integrate the system for a small number of steps,
+    // and check stability of temperature distribution at every iteration
+    for (int i=0; i < 10; ++i) {
+        for (auto&& entry : index(M_theta)) {
+            int i = std::get<0>(entry);
+            int j = std::get<1>(entry);
+            auto& theta = std::get<2>(entry);
+            ASSERT_NEAR(std::abs(theta), std::abs(M_theta0(i, j)), 1e-10);
+        }
+        ct.compute(*system);
+        updateMatrix(*system, M_theta);
+    }
+}