Improved cooling of photovoltaic panels by natural convection flow in a channel with adiabatic extensions.

In hot dry regions, photovoltaic modules are exposed to excessive temperatures, which leads to a drop in performance and the risk of overheating. The present numerical study aims to evaluate the natural air cooling of PV modules by an inclined chimney mounted at the back. The basic equations were solved using the finite volume method. The validity of the model is verified by comparison with the data available in the literature. Thermal and dynamic flow patterns are analyzed for a variety of parameters: Rayleigh numbers from 102 to 106, PV panel tilt angle from 15° to 90°, and channel aspect ratios from 1/20 to 1/5. A critical aspect ratio has been determined to minimize overheating of the PV module. According to the computational results, the tilt angle and modified Rayleigh number increase the mass flow rate and mean Nusselt number. The overheating zone with maximum temperatures is located in the upper part of the photovoltaic panel. The addition of an extension to both channel's inlet and outlet was found to improve the cooling of the photovoltaic panels; however, only the extensions downstream of the channel are truly effective. The critical lengths at which channel performance improves significantly were identified by examining the impact of longer extensions on channel performance. Increasing the extension length from 0 to 3H improves the mass flow rate by 65%, the average Nusselt number by 13.4%, and leads to an 11% decrease in maximum temperature when Ra* = 106. This cooling technique is particularly promising for hot dry regions where water is scarce.

Natural convection in the cytoplasm: Theoretical predictions of buoyancy-driven flows inside a cell.

The existence of temperature gradients within eukaryotic cells has been postulated as a source of natural convection in the cytoplasm, i.e. bulk fluid motion as a result of temperature-difference-induced density gradients. Recent computations have predicted that a temperature differential of ΔT ≈ 1 K between the cell nucleus and the cell membrane could be strong enough to drive significant intracellular material transport. We use numerical computations and theoretical calculations to revisit this problem in order to further understand the impact of temperature gradients on flow generation and advective transport within cells. Surprisingly, our computations yield flows that are an order of magnitude weaker than those obtained previously for the same relative size and position of the nucleus with respect to the cell membrane. To understand this discrepancy, we develop a semi-analytical solution of the convective flow inside a model cell using a bi-spherical coordinate framework, for the case of an axisymmetric cell geometry (i.e. when the displacement of the nucleus from the cell centre is aligned with gravity). We also calculate exact solutions for the flow when the nucleus is located concentrically inside the cell. The results from both theoretical analyses agree with our numerical results, thus providing a robust estimate of the strength of cytoplasmic natural convection and demonstrating that these are much weaker than previously predicted. Finally, we investigate the ability of the aforementioned flows to redistribute solute within a cell. Our calculations reveal that, in all but unrealistic cases, cytoplasmic convection has a negligible contribution toward enhancing the diffusion-dominated mass transfer of cellular material.

Electrokinetic convection-enhanced delivery for infusion into the brain from a hydrogel reservoir.

Electrokinetic convection-enhanced delivery (ECED) utilizes an external electric field to drive the delivery of molecules and bioactive substances to local regions of the brain through electroosmosis and electrophoresis, without the need for an applied pressure. We characterize the implementation of ECED to direct a neutrally charged fluorophore (3 kDa) from a doped biocompatible acrylic acid/acrylamide hydrogel placed on the cortical surface. We compare fluorophore infusion profiles using ECED (time = 30 min, current = 50 µA) and diffusion-only control trials, for ex vivo (N = 18) and in vivo (N = 12) experiments. The linear intensity profile of infusion to the brain is significantly higher in ECED compared to control trials, both for in vivo and ex vivo. The linear distance of infusion, area of infusion, and the displacement of peak fluorescence intensity along the direction of infusion in ECED trials compared to control trials are significantly larger for in vivo trials, but not for ex vivo trials. These results demonstrate the effectiveness of ECED to direct a solute from a surface hydrogel towards inside the brain parenchyma based predominantly on the electroosmotic vector.

The Influence of Size on the Intracranial Distribution of Biomedical Nanoparticles Administered by Convection-enhanced Delivery in Minipigs.

Because of the blood-brain barrier (BBB), successful drug delivery to the brain has long been a key objective for the medical community, calling for pioneering technologies to overcome this challenge. Convection-enhanced delivery (CED), a form of direct intraparenchymal microinfusion, shows promise but requires optimal infusate design and real-time distribution monitoring. The size of the infused substances appears to be especially critical, with current knowledge being limited. Herein, we examined the intracranial administration of polyethylene glycol (PEG)-coated nanoparticles (NPs) of various sizes using CED in groups of healthy minipigs (n = 3). We employed stealth liposomes (LIPs, 130 nm) and two gold nanoparticle designs (AuNPs) of different diameters (8 and 40 nm). All were labeled with copper-64 for quantitative and real-time monitoring of the infusion via positron emission tomography (PET). NPs were infused via two catheters inserted bilaterally in the putaminal regions of the animals. Our results suggest CED with NPs holds promise for precise brain drug delivery, with larger LIPs exhibiting superior distribution volumes and intracranial retention over smaller AuNPs. PET imaging alongside CED enabled dynamic visualization of the process, target coverage, timely detection of suboptimal infusion, and quantification of distribution volumes and concentration gradients. These findings may augment the therapeutic efficacy of the delivery procedure while mitigating unwarranted side effects associated with nonvisually monitored delivery approaches. This is of vital importance, especially for chronic intermittent infusions through implanted catheters, as this information enables informed decisions for modulating targeted infusion volumes on a catheter-by-catheter, patient-by-patient basis.

Enhancing low-concentration cell detection in single entity electrochemical systems through forced convection.

This study advances the detection of bacteria at low concentrations in single-entity electrochemistry (SEE) systems by integrating forced convection. Our results show that forced convection significantly improves the mass transfer rate of electrolyte, with the mass transfer coefficient demonstrating a proportional relationship to the flow rate to the power of 1.37. Notably, while the collision frequency of E. coli initially increases with the flow rate, a subsequent decrease is observed at higher rates. This pattern is attributed to the mechanics of cell collision under forced convection. Specifically, while forced convection propels cells towards the ultra-microelectrode (UME), it does not aid in their penetration through the boundary layer, leading to cells being driven away from the UME at higher flow rates. This hypothesis is supported by the statistical analysis of collision data, including signal heights and rise times. By optimizing the flow rate to 2 mL/min, we achieved enhanced detection of E. coli in concentrations ranging from 0.9 × 107 to 5.0 × 107 cells/mL. This approach significantly increased collision frequency by elevating the mass transfer of cells, with the mass transfer coefficient rising from 0.1 × 10-5 m/s to 0.9 × 10-5 m/s. It provides a viable solution to the challenges of detecting bacteria at low concentrations in SEE systems.

The effect of explicit convection on simulated malaria transmission across Africa.

Malaria transmission across sub-Saharan Africa is sensitive to rainfall and temperature. Whilst different malaria modelling techniques and climate simulations have been used to predict malaria transmission risk, most of these studies use coarse-resolution climate models. In these models convection, atmospheric vertical motion driven by instability gradients and responsible for heavy rainfall, is parameterised. Over the past decade enhanced computational capabilities have enabled the simulation of high-resolution continental-scale climates with an explicit representation of convection. In this study we use two malaria models, the Liverpool Malaria Model (LMM) and Vector-Borne Disease Community Model of the International Centre for Theoretical Physics (VECTRI), to investigate the effect of explicitly representing convection on simulated malaria transmission. The concluded impact of explicitly representing convection on simulated malaria transmission depends on the chosen malaria model and local climatic conditions. For instance, in the East African highlands, cooler temperatures when explicitly representing convection decreases LMM-predicted malaria transmission risk by approximately 55%, but has a negligible effect in VECTRI simulations. Even though explicitly representing convection improves rainfall characteristics, concluding that explicit convection improves simulated malaria transmission depends on the chosen metric and malaria model. For example, whilst we conclude improvements of 45% and 23% in root mean squared differences of the annual-mean reproduction number and entomological inoculation rate for VECTRI and the LMM respectively, bias-correcting mean climate conditions minimises these improvements. The projected impact of anthropogenic climate change on malaria incidence is also sensitive to the chosen malaria model and representation of convection. The LMM is relatively insensitive to future changes in precipitation intensity, whilst VECTRI predicts increased risk across the Sahel due to enhanced rainfall. We postulate that VECTRI's enhanced sensitivity to precipitation changes compared to the LMM is due to the inclusion of surface hydrology. Future research should continue assessing the effect of high-resolution climate modelling in impact-based forecasting.

ISSLS prize in clinical/bioengineering science 2024: How standing and supine positions influence nutrient transport in human lumbar discs?-A serial post-contrast MRI study evaluating interplay between convection and diffusion.

PURPOSE: The intervertebral disc being avascular depends on diffusion and load-based convection for essential nutrient supply and waste removal. There are no reliable methods to simultaneously investigate them in humans under natural loads. For the first time, present study aims to investigate this by strategically employing positional MRI and post-contrast studies in three physiological positions: supine, standing and post-standing recovery. METHODS: A total of 100 healthy intervertebral discs from 20 volunteers were subjected to a serial post-contrast MR study after injecting 0.3 mmol/kg gadodiamide and T1-weighted MR images were obtained at 0, 2, 6, 12 and 24 h. At each time interval, images were obtained in three positions, i.e. supine, standing and post-standing recovery supine. The signal intensity values at endplate zone and nucleus pulposus were measured. Enhancement percentages were calculated and analysed comparing three positions. RESULTS: During unloaded supine position, there was slow gradual increase in enhancement reaching peak at 6 h. When the subjects assumed standing position, there was immediate loss of enhancement at nucleus pulposus which resulted in reciprocal increase in enhancement at endplate zone (washout phenomenon). Interestingly, when subjects assumed the post-standing recovery position, the nucleus pulposus regained the enhancement and endplate zone showed reciprocal loss (pumping-in phenomenon). CONCLUSIONS: For the first time, present study documented acute effects of physiological loading and unloading on nutrition of human discs in vivo. While during rest, solutes diffused gradually into disc, the diurnal short loading and unloading redistribute small solutes by convection. Standing caused rapid solute depletion but promptly regained by assuming resting supine position.

Convection-Enhanced Drug Delivery: Experimental and Analytical Studies of Infusion Behavior in an In Vitro Brain Surrogate.

Convection-enhanced drug delivery (CED) directly infuses drugs with a large molecular weight toward target cells as a therapeutic strategy for neurodegenerative diseases and brain cancers. Despite the success of many previous in vitro experiments on CED, challenges still remain. In particular, a theoretical predictive model is needed to form a basis for treatment planning, and developing such a model requires well-controlled injection tests that can rigorously capture the convective (advective) and diffusive transport of an infusate. For this purpose, we investigated the advection-diffusion transport of an infusate (bromophenol blue solution) in the brain surrogate (0.2% w/w agarose gel) at different injection rates, ranging from 0.25 to 4 µL/min, by closely monitoring changes in the color intensity, propagation distance, and injection pressures. One dimensional closed-form solution was examined with two variable sets, such as the mathematically calculated coefficient of molecular diffusion and average velocity, and the hydraulic dispersion coefficient and seepage velocity by the least squared method. As a result, the seepage velocity was greater than the average velocity to some extent, particularly for the later infusion times. The poroelastic deformation in the brain surrogate might lead to changes in porosity, and consequently, slight increases in the actual flow velocity as infusion continues. The limitation of efficiency of the single catheter was analyzed by dimensionless analysis. Lastly, this study suggests a simple but robust approach that can properly capture the convective (advective) and diffusive transport of an infusate in an in vitro brain surrogate via well-controlled injection tests.

Hemodiafiltration: a synergy yet to be convincing.

The desperate attempt to improve mortality, morbidity, quality of life and patient-reported outcomes in patients on hemodialysis has led to multiple attempts to improve the different modes, frequencies, and durations of dialysis sessions in the last few decades. Nothing has been more appealing than the combination of diffusion and convection in the form of hemodiafiltration. Despite the concrete evidence of better clearance of middle weight molecules and better hemodynamic stability, tangible evidence to support the universal adoption is still at a distance. Survival benefits seen in selected groups who are likely to tolerate hemodiafiltration with better vascular access and with lower comorbid burden, need to be extended to real life dialysis patients who are older than the population studied and have significantly higher comorbid burden. Technical demands of initiation hemodiafiltration, the associated costs, and the incremental benefits targeted, along with patient-reported outcomes, need to be explored further before recommending hemodiafiltration as the mode of choice.

Droplet microfluidics for time-resolved serial crystallography.

Serial crystallography requires large numbers of microcrystals and robust strategies to rapidly apply substrates to initiate reactions in time-resolved studies. Here, we report the use of droplet miniaturization for the controlled production of uniform crystals, providing an avenue for controlled substrate addition and synchronous reaction initiation. The approach was evaluated using two enzymatic systems, yielding 3 µm crystals of lysozyme and 2 µm crystals of Pdx1, an Arabidopsis enzyme involved in vitamin B6 biosynthesis. A seeding strategy was used to overcome the improbability of Pdx1 nucleation occurring with diminishing droplet volumes. Convection within droplets was exploited for rapid crystal mixing with ligands. Mixing times of <2 ms were achieved. Droplet microfluidics for crystal size engineering and rapid micromixing can be utilized to advance time-resolved serial crystallography.

Manipulation of Convection Using Infrared Light Emitted from Human Hands.

Control of convection plays an important role in heat transfer regulation, bio/chemical sensing, phase separation, etc. Current convection controlling systems generally depend on engineered energy sources to drive and manipulate the convection, which brings additional energy consumption into the system. Here the use of human hand as a natural and sustainable infrared (IR) radiation source for the manipulation of liquid convection is demonstrated. The fluid can sense the change of the relative position or the shape of the hand with the formation of different convection patterns. Besides the generation of static complex patterns, dynamic manipulation of convections can also be realized via moving of hand or finger. The use of such sustainable convections to control the movement of a floating "boat" is further achieved. The use of human hands as the natural energy sources provides a promising approach for the manipulation of liquid convection without the need of extra external energy, which may be further utilized for low-cost and intelligent bio/chemical sensing and separation.

Comparison of HVT-ND recombinant and convection-based Newcastle disease vaccination programs in the protection against the genotype VII NDV challenges: an experimental study.

Newcastle disease virus (NDV) belongs to the Avulavirus genus and Paramyxoviridae family virus that causes acute, highly infectious Newcastle disease in poultry. The two proteins of haemagglutinin neuraminidase (HN) and fusion (F) are key virulence factors with an important role in its immunogenicity. Genotype VII NDV is still among the most serious viral hazards to the poultry industry worldwide. In this study, a commercial vector vaccine (HVT-NDV) was evaluated compared to the conventional vaccination strategy against Iranian genotype VII. This experiment showed that the group receiving the conventional vaccination strategy had higher antibodies, fewer clinical signs, and lower viral loads in tracheal swabs and feces. Also, two vaccine groups showed significant difference, which could have resulted from two extra vaccine doses in the conventional group. However, except for antibody levels in commercial chickens in the Iran new-generation vaccine, this difference was minor. Further, both groups showed 100% protection in the challenge study. Despite the phylogenetic gap between the NDV-F gene placed in the vector vaccine and the challenge virus (genotypes I and VII, respectively), the rHVT-NDV vaccine offered strong clinical protection and decreased challenge virus shedding considerably.

Adjuvant convection-enhanced delivery for the treatment of brain tumors.

BACKGROUND: Malignant gliomas are a therapeutic challenge and remain nearly uniformly fatal. While new targeted chemotherapeutic agentsagainst malignant glioma have been developed in vitro, these putative therapeutics have not been translated into successful clinical treatments. The lack of clinical effectiveness can be the result of ineffective biologic strategies, heterogeneous tumor targets and/or the result of poortherapeutic distribution to malignant glioma cells using conventional nervous system delivery modalities (intravascular, cerebrospinal fluid and/orpolymer implantation), and/or ineffective biologic strategies. METHODS: The authors performed a review of the literature for the terms "convection enhanced delivery", "glioblastoma", and "glioma". Selectclinical trials were summarized based on their various biological mechanisms and technological innovation, focusing on more recently publisheddata when possible. RESULTS: We describe the properties, features and landmark clinical trials associated with convection-enhanced delivery for malignant gliomas.We also discuss future trends that will be vital to CED innovation and improvement. CONCLUSION: Efficacy of CED for malignant glioma to date has been mixed, but improvements in technology and therapeutic agents arepromising.

Magnetic Resonance Imaging-Aided SmartFlow Convection Delivery of DNX-2401: A Pilot, Prospective Case Series.

BACKGROUND: The Combination Adenovirus + Pembrolizumab to Trigger Immune Virus Effects (CAPTIVE) study is a phase II clinical trial testing the efficacy of a recombinant adenovirus DNX-2401 combined with the immune checkpoint inhibitor pembrolizumab. Here, we report the first patients in this study who underwent viral delivery through real-time magnetic resonance imaging (MRI) stereotaxis-guided SmartFlow convection delivery of DNX-2401. METHODS: Patients who underwent real-time MRI-guided DNX-2401 delivery through the SmartFlow convection catheter were prospectively followed. RESULTS: Precise catheter placement was achieved in all patients treated, and no adverse events were noted. Average radial error from target was 0.9 mm. Average procedural time was 3 hours 16 minutes and was comparable to other convection-enhanced delivery techniques. In 2 patients, delivery of DNX-2401 was visualized as >1 cm maximal diameter of T1 hypointensity infusate on MRI obtained immediately after completion of viral infusion. These patients exhibited partial response based on Response Assessment in Neuro-Oncology assessment. The remaining patient showed <1 cm maximal diameter of infusate on immediate postinfusion MRI and showed disease progression on subsequent MRI. CONCLUSIONS: Our pilot case series supports compatibility of the SmartFlow system with oncolytic adenovirus delivery and provides the basis for future validation studies.

Chronic intermittent convection-enhanced delivery of vigabatrin to the bilateral subthalamic nucleus in an acute rat seizure model.

Targeted intracerebral drug delivery is an attractive experimental approach for the treatment of drug-resistant epilepsies. In this regard, the subthalamic nucleus (STN) represents a focus-independent target involved in the remote modulation and propagation of seizure activity. Indeed, acute and chronic pharmacological inhibition of the STN with vigabatrin (VGB), an irreversible inhibitor of GABA transaminase, has been shown to produce antiseizure effects. This effect, however, is lost over time as tolerance develops with chronic, continuous intracerebral pharmacotherapy. Here we investigated the antiseizure effects of chronic intermittent intra-STN convection-enhanced delivery of VGB in an acute rat seizure model focusing on circumventing tolerance development and preventing adverse effects. Timed intravenous pentylenetetrazol (PTZ) seizure threshold testing was conducted before and after implantation of subcutaneous drug pumps and bilateral intra-STN cannulas. Drug pumps infused vehicle or VGB twice daily (0.4 µg) or once weekly (2.5 µg, 5 µg) over three weeks. Putative adverse effects were evaluated and found to be prevented by intermittent compared to previous continuous VGB delivery. Clonic seizure thresholds were more clearly raised by intra-STN VGB compared to myoclonic twitch. Both twice daily and once weekly intra-STN VGB significantly elevated clonic seizure thresholds depending on dose and time point, with responder rates of up to 100% observed at tolerable doses. However, tolerance could not be completely avoided, as tolerance rates of 40-75% were observed with chronic VGB treatment. Results indicate that the extent of tolerance development after intermittent intra-STN VGB delivery varies depending on infusion dose and regimen.

Experimental investigation of an indirect solar dryer with PCM-integrated solar collector as a thermal energy storage medium.

An indirect-type forced convection solar dryer implementing a phase-changing material (PCM) as the energy-storing medium was designed, fabricated, and investigated in this study. The effects of changing the mass flow rate on the valuable energy and thermal efficiencies were studied. The experimental results showed that the instantaneous and daily efficiencies of the indirect solar dryer (ISD) increased with the initial increase in mass flow rate, beyond which the change is not prominent both with and without using the PCM. The system consisted of a solar energy accumulator (solar air collector with a PCM cavity), a drying compartment, and a blower. The charging and discharging characteristics of the thermal energy storage unit were evaluated experimentally. It was found that after using PCM, drying air temperature was higher than ambient air temperature by 9-12 ℃ after sunset for 4 h. Using PCM accelerated the process by which Cymbopogon citratus was effectively dried between 42 and 59 °C of drying air. Energy and exergy analysis of the drying process was performed. The daily energy efficiency of the solar energy accumulator reached 35.8%, while the daily exergy efficiency reached 13.84%. The exergy efficiency of the drying chamber was in the range of 47-97%. A free energy source, a large reduction in drying time, a higher drying capacity, a decrease in mass losses, and improved product quality all contributed to the proposed solar dryer's high potential.

Entropy generation in a ciliary flow of an Eyring-Powell ternary hybrid nanofluid through a channel with electroosmosis and mixed convection.

Drug delivery systems, where the nanofluid flow with electroosmosis and mixed convection can help in efficient and targeted drug delivery to specific cells or organs, could benefit from understanding the behavior of nanofluids in biological systems. In current work, authors have studied the theoretical model of two-dimensional ciliary flow of blood-based (Eyring-Powell) nanofluid model with the insertion of ternary hybrid nanoparticles along with the effects of electroosmosis, magnetohydrodynamics, thermal radiations, and mixed convection. Moreover, the features of entropy generation are also taken into consideration. The system is modeled in a wave frame with the approximations of large wave number and neglecting turbulence effects. The problem is solved numerically by using the shooting method with the assistance of computational software "Mathematica" for solving the governing equation. According to the temperature curves, the temperature will increase as the Hartman number, fluid factor, ohmic heating, and cilia length increase. It is also disclosed that ternary hybrid nanoparticles result in a change in flow rate when other problem parameters are varied, and the same is true for temperature graphs. Engineers and scientists can make better use of nanofluid-based cooling systems in electronics, automobiles, and industrial processes with the aid of the study's findings.

"Floating Catalytic Foam" with prominent heat-induced convection for the effective photocatalytic removal of antibiotics.

Immobilized photocatalysts represent a promising candidate for the wastewater treatments due to their good reusability, high stability and low eco-risk. Mass transfer within the immobilized catalytic bed is a crucial process that determines the contacting, adsorption, and degradation kinetics in the photodegradation. In this study, a floating catalytic foam (FCF) with a prominent pumping effect was designed to promote mass transfer. The polyurethane foam immobilized with rGO/TiO2/ultrathin-g-C3N4 photocatalyst (PRTCN) was prepared by a simple dip-coating and Uv-light aging process. It was found that the hydrophilic-hydrophobic interfaces could not only contribute to the floating of the catalyst but also establish a temperature gradient across the floating immobilized catalyst. In addition, the temperature gradient induced convection could serve as a built-in pump to effectively promote the diffusion and adsorption of target antibiotic molecules during the photocatalytic process. Therefore, the PRTCN demonstrated a high photodegradation and mineralization efficiency with excellent reusability and anti-interference capability. Moreover, the photodegradation mechanism and the intermediates' toxicity of norfloxacin were detailly investigated by ultra-high resolution electrospray time-of-flight mass spectrometry, density functional theory simulation and ECOSAR estimation. This work proposed a facile and sustainable strategy to enhance the mass transfer problem on immobilized photocatalysts, which could promote the application of the immobilized photocatalysts in the real water-treatment scenarios.

Design and Characterization of Pressure Monitoring and Insertion system for Intraparenchymal Convection Enhanced Delivery.

Neurological disorders are a significant societal and economic burden. Common pharmacological therapies often can only manage symptoms and have limited efficacy. Intraparenchymal convection enhanced delivery (IP CED) is a neurosurgical technique for direct brain delivery of therapeutics. Currently, the main applications of IP CED are targeted chemotherapy for glioblastoma and gene therapy. While IP CED has advantages over systemic approaches, its benefits can be drastically reduced by inadequate coverage as low as 21% of target anatomy, excessive infusion durations greater than 2 hours, and off-target effects. Addressing the limitations of IP CED requires thorough investigation and optimization of the relevant fluid dynamic and operational parameters. In this work, we present the design, fabrication, and characterization of low-cost, open-source, and fully automated CED cannula insertion control and pressure-monitoring systems. Using these automated CED control systems, we investigate the effects of pressure, insertion velocity, and flow rates on several outcome variables, including reflux, volume distribution, and infusion cloud morphology during CED infusions in brain phantoms.Clinical Relevance- CED pressure properties may be able to implicate reflux incidents and could provide clinicians with valuable, real-time information regarding ongoing infusions without the need for costly medical imaging modalities.

Uniformly convergent extended cubic B-spline collocation method for two parameters singularly perturbed time-delayed convection-diffusion problems.

This work proposes a uniformly convergent numerical scheme to solve singularly perturbed parabolic problems of large time delay with two small parameters. The approach uses implicit Euler and the exponentially fitted extended cubic B-spline for time and space derivatives respectively. Extended cubic B-splines have advantages over classical B-splines. This is because for a given value of the free parameter [Formula: see text] the solution obtained by the extended B-spline is better than the solution obtained by the classical B-spline. To confirm the correspondence of the numerical methods with the theoretical results, numerical examples are presented. The present numerical technique converges uniformly, leading to the current study of being more efficient.