Abstract This work presents a comprehensive efficiency and convergence analysis of wavelet-based methods within a multi-dimensional framework for detecting singularities in nonlocal weakly singular integro-partial differential equations in one and two dimensions. The proposed approach incorporates the multi-resolution properties of wavelets to accurately identify and localize singularities in solutions to such equations. Combinations of space-time wavelets with their advantages are very limited for higher-dimensional problems, and their convergence analysis on collocation points is not fully clear till the present day. For problems having time singularity, the present work shows that multi-resolution analysis through 2D/3D Haar wavelets requires a lower regularity assumption for the convergence of the proposed procedure than several approaches on finite-difference setup or other wavelets like Hermite, Chebyshev, or Bernoulli’s wavelets. In particular, we produce a higher-order convergence result (second-order accurate) in the $L^{2}$ L 2 norm, based on sufficient regularity assumptions on the solution. In addition to the higher-order estimate, we provide the wavelet-based truncation error estimate for several terms, such as the time-fractional derivative, Volterra & Fredholm integral operators, classical derivatives, and their effects on the regularity of the function for future researchers in this domain. Numerical tests are performed in the $L^{2}$ L 2 and $L^{\infty }$ L ∞ norms to compare the efficiency of this method over existing approaches for weakly singular nonlocal integro-partial differential equations. These experiments show the efficiency of the proposed approach in several kinds of regularity assumptions of the solution. This also guarantees the convergence of approximations to the functions having weak singularities in time and the higher-order accuracy for sufficiently smooth solutions.

Abstract The present manuscript studies a coupled thermoelastic-viscoelastic system, modelling the interaction between mechanical displacement and temperature in viscoelastic materials in a bounded interval. This topic is of interest in the fields of applied mathematics and continuum mechanics. The system under consideration reads $$\begin{aligned} \left \{ \textstyle\begin{array}{l} u_{tt} = (\gamma (\Theta ) u_{xt})_{x} + (\tilde {\gamma }(\Theta ) u_{x})_{x}+(f( \Theta ))_{x}, \\ \Theta _{t} = D\Theta _{xx} + \Gamma (\Theta ) u_{xt}^{2}+F(\Theta )u_{xt}, \end{array}\displaystyle \right . \end{aligned}$$ { u t t = ( γ ( Θ ) u x t ) x + ( γ ˜ ( Θ ) u x ) x + ( f ( Θ ) ) x , Θ t = D Θ x x + Γ ( Θ ) u x t 2 + F ( Θ ) u x t , in an open bounded interval, which with $\gamma \equiv \tilde {\gamma }\equiv \Gamma $ γ ≡ γ ˜ ≡ Γ as well as $f\equiv F$ f ≡ F reduces to the classical model for the evolution of strains and temperatures in thermoviscoelasticity. In contrast to the preceding related studies, the present study focuses on situations in which not only $f$ f and $F$ F , but also the core components $\gamma $ γ , $\tilde {\gamma }$ γ ˜ and $\Gamma $ Γ , are dependent on the temperature variable $\Theta $ Θ . Firstly, a statement regarding the local existence of classical solutions is derived for arbitrary $D > 0$ D > 0 , $0 <\gamma $ 0 < γ , $\tilde {\gamma }\in C^{2}([0,\infty ))$ γ ˜ ∈ C 2 ( [ 0 , ∞ ) ) , and $0 \le \Gamma \in C^{1}([0, \infty ))$ 0 ≤ Γ ∈ C 1 ( [ 0 , ∞ ) ) , for functions $f\in C^{2}([0,\infty );\mathbb{R})$ f ∈ C 2 ( [ 0 , ∞ ) ; R ) and $F\in C^{1}([0,\infty );\mathbb{R})$ F ∈ C 1 ( [ 0 , ∞ ) ; R ) with $F(0)=0$ F ( 0 ) = 0 , and for suitably regular initial data of arbitrary size. Secondly, if $\tilde{\gamma } = a\cdot \gamma +\mu $ γ ˜ = a ⋅ γ + μ , with $a>0$ a > 0 and arbitrary $\mu >0$ μ > 0 , there exists $\delta >0$ δ > 0 with the property that whenever in addition to the above we have $$ \frac{a}{\gamma (\Theta _{\star })}\le \delta \qquad \text{and}\qquad \frac{|f'(\Theta _{\star })|\cdot |F(\Theta _{\star })|}{D\cdot \gamma (\Theta _{\star })} \le \delta , $$ a γ ( Θ ⋆ ) ≤ δ and | f ′ ( Θ ⋆ ) | ⋅ | F ( Θ ⋆ ) | D ⋅ γ ( Θ ⋆ ) ≤ δ , for initial data close to the constant level given by $u = 0$ u = 0 and $\Theta =\Theta _{\star }$ Θ = Θ ⋆ , with any fixed $\Theta _{\star }\ge 0$ Θ ⋆ ≥ 0 , it is demonstrated that these solutions are indeed global in time and possess the property that $u_{xt}$ u x t , $u_{x}$ u x , $u_{xx}$ u x x and $\Theta _{x}$ Θ x decay exponentially fast in $L^{2}$ L 2 . In this context, the parameter $\mu $ μ captures the weak inclusion of the electric field within the system. This aspect constitutes the primary novel contribution of the present analysis. The aforementioned results are obtained by detecting suitable dissipative properties of functionals involving norms of these gradients in $L^{2}$ L 2 spaces.

Abstract Joint inverse problems occur in many practical situations, where different modalities are used to image the same object. Structural similarity is a way to regularize such joint inverse problems by imposing similarity between the images. While structural similarity has found widespread use in many practical settings, its theoretical foundations remain underexplored. This study develops an over-arching formulation for these types of problems and studies their well-posedness via the Direct Method from the calculus of variations. We focus in particular on lower semi-continuity and coerciveness as essential properties for the well-posedness of the variational problem in $W^{m,p}$ W m , p and $SBV$ S B V . Here quasiconvexity and growth properties of the structural similarity quantifier turns out to be essential. We find that the use of gradient-difference, cross-gradient or Schatten norms as structural similarity quantifiers is theoretically justified. A generalized form of the cross-gradient that inherently works on $N$ N coupled problems is introduced.