A new regularisation for time-fractional backward heat conduction problem

It is well-known that the backward heat conduction problem of recovering the temperature $u(\cdot, t)$ at a time $t\geq 0$ from the knowledge of the temperature at a later time, namely $g:= u(\cdot, \tau)$ for $\tau>t$, is ill-posed, in the sense that small error in $g$ can lead to large deviation in $u(\cdot, t)$. However, in the case of a time fractional backward heat conduction problem (TFBHCP), the above problem is well-posed for $t>0$ and ill-posed for $t=0$. We use this observation to obtain stable approximate solutions for the TFBHCP for $t=0$, and derive error estimates under suitable source conditions. We shall also provide some numerical examples to illustrate the approximation properties of the regularized solutions

A source identification problem in a bi-parabolic equation: convergence rates and some optimal results

This paper is concerned with identification of a spatial source function from final time observation in a bi-parabolic equation, where the full source function is assumed to be a product of time dependent and a space dependent function. Due to the ill-posedness of the problem, recently some authors have employed different regularization method and analysed the convergence rates. But, to the best of our knowledge, the quasi-reversibility method is not explored yet, and thus we study that in this paper. As an important implication, the H{\"o}lder rates for the apriori and aposteriori error estimates obtained in this paper improve upon the rates obtained in earlier works. Also, in some cases we show that the rates obtained are of optimal order. Further, this work seems to be the first one that has broaden the applicability of the problem by allowing the time dependent component of the source function to change sign. To the best of our knowledge, the earlier known work assumed the fixed sign of the time dependent component by assuming some bounded below condition.Comment: Comments are welcome. Typos and some mistakes with sign in the PDE are rectified. Section 4 and 5 are majorly revise

Assimilating socio-economic perspective in designing crop sector technology interventions: A farmer participatory study on coconut sector in Kerala

The economic viability of coconut farming in the state has witnessed a steady decline due to a complex interplay of several socio-economic, environmental and institutional factors. But the crop sustains the livelihood of a significant share of the population in the state. Equitable growth in agricultural sector of the state cannot be attained unless the fortunes of coconut farming sector are revived. A critical understanding of the production environment is very important in crafting appropriate strategies for the sector. This study is based on a detailed analysis of socio-economic profile of 180 coconut farmers in Kerala across five major agro-ecological units, collected using pre-tested structured questionnaire. The study draws on trends in relevant socio-economic trends to examine the reasons for the vicious cycle of low investment-low profits -low productivity. The study identified structural agrarian changes like low dependence on farm income, High share of non-farm income, high cost and non-availability of skilled labour, etc., as contributing factors to the extant situation. Based on the socio-economic profile of the coconut farmer and technical studies, soil nutrient management centric strategy was identified as the key element in reorienting coconut farming. The intervention strategy was designed as an alternative approach for reviving the economic viability of coconut farming. The initial results on farmer perception on impact of technology intervention, with direct and indirect links to several biological and socio-economic limiting factors, indicate significant improvement across several parameters influencing crop productivity. Assimilating the lessons from the operation of the intervention strategy, the study also outlines a roadmap for multiple institutional involvements for scaling up this strategy across the state

Error estimates for Tikhonov regularization with unbounded regularizing operators

It is shown that Tikhonov regularization for ill- posed operator equation $Kx = y$ using a possibly unbounded regularizing operator $L$ yields an orderoptimal algorithm with respect to certain stability set when the regularization parameter is chosen according to the Morozov's discrepancy principle. A more realistic error estimate is derived when the operators $K$ and $L$ are related to a Hilbert scale in a suitable manner. The result includes known error estimates for ordininary Tikhonov regularization and also the estimates available under the Hilbert scale approach