Explain how to solve the partial differential of a 1D heart transfer problem using finite element method from very basics to solving the pde

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The Finite Element Method transforms partial differential equations into algebraic equations. For one-dimensional heat transfer, we start with the heat equation that describes how temperature changes over time and space in a material. This equation includes density, specific heat, thermal conductivity, and heat sources. To derive the weak form, we multiply the PDE by a test function w of x and integrate over the domain. Then we apply integration by parts to reduce the order of derivatives. This naturally introduces boundary terms and transforms the strong form PDE into a weak form that is suitable for finite element approximation. Next, we discretize the domain by dividing it into finite elements connected at nodes. Within each element, we approximate the solution using simple shape functions. For linear elements, these are straight line functions that equal one at one node and zero at the other. The global solution is then constructed by combining these local approximations. For each element, we derive a matrix equation relating nodal temperatures to forces. The element stiffness matrix comes from integrating the shape function derivatives, while the force vector comes from the source term. These element equations are then assembled into a global system by summing contributions from elements that share nodes. Finally, boundary conditions are applied and the system is solved. To summarize, the Finite Element Method transforms partial differential equations into solvable algebraic systems. The weak form naturally handles boundary conditions while reducing derivative requirements. Domain discretization and shape functions provide local approximations that assemble into a global solution system.