A Numerical Algorithm Based on RBFs for Solving an Inverse Source Problem

A Haar wavelets based approximation for nonlinear inverse problems influenced by unknown heat source

“…Mathematically, this phenomena can be represented by nonlinear inverse problem where $$$ S $$$ and $$$ F $$$ are both unknown and we are interested in the nondimensionalized form to find the numerical values of $$$ S $$$ and $$$ F $$$ and can be written as $$$$ {S}_{\tau}\left(x,\tau \right)-{S}_{xx}\left(x,\tau \right)+G\left[{\left({S}_x\left(x,\tau \right)\right)}^2\right]=\mathrm{F}\left(x,\tau \right),\kern0.30em 0\le x\le 1,\kern0.30em \tau >0, $$$$ where $$$ G $$$ is some real number and $$$ x $$$ and $$$ \tau $$$ denote the space and time variables, respectively. Equation () will be linear if $$$ G=0 $$$ 1 . The initial condition is $$$$ S\left(x,0\right)=I(x), $$$$ and the boundary conditions are …”

“…Inverse problems are challenging to handle due to scientific and computational stresses within the frame of ill‐posedness and ill‐conditioning. Various numerical methods are affected by ill‐posedness and ill‐conditioning behavior to get the precise numerical assessment of the inverse problem, that is, boundary element method, 3 iterative regularization technique, 4 Fourier regularization method, 5 method of fundamental solution, 6,7 mesh‐less methods, 8,9 and Lie group method 10 …”

“…Equation (1) will be linear if G = 0. 1 The initial condition is S(x, 0) = I(x), (2) and the boundary conditions are S(0, 𝜏) = g 1 (𝜏), S(1, 𝜏) = g 2 (𝜏).…”

“…Inverse problems are challenging to handle due to scientific and computational stresses within the frame of ill-posedness and ill-conditioning. Various numerical methods are affected by ill-posedness and ill-conditioning behavior to get the precise numerical assessment of the inverse problem, that is, boundary element method, 3 iterative regularization technique, 4 Fourier regularization method, 5 method of fundamental solution, 6,7 mesh-less methods, 8,9 and Lie group method. 10 For the last some years, the Haar wavelet is commonly used in numerical computing in weak and strong formulations including Daubechies wavelet method, 11 wavelet collocation techniques, 12,13 wavelet mesh-less scheme, 14 and wavelet-Galerkin method.…”

“…In this study, the merits of the Haar functions are further interpreted to acquire the computational treatment of inverse nonlinear problem (1). With the help of finite difference and Haar approximation, the inverse problem has been transformed into well-condition algebraic equations.…”