
 Under some suitable conditions, we study the existence and uniqueness of a solution to a new modification of a nonlinear fractional integro-differential equation (NFIDEq) in dual Banach space CE (E, [0, T]), which simulates several phenomena in mathematical physics, quantum mechanics, and other domains. The desired conclusions are demonstrated with the use of fixed-point theorems after applying the theory of fractional calculus. The validation of the provided strategy has been done by utilizing the Bernoulli matrix approach (BMA) method as a numerical method. The major motivation for selecting the BMA approach is that it combines Bernoulli polynomial approximation with Caputo fractional derivatives and numerical integral transformation to reduce the NFIDEq to an algebraic system and then derive the numerical solution; additionally, the convergence analysis indicated that the proposed strategy has more precision than other numerical methods. Finally, as a verification of the theoretical work, we apply two examples with numerical results by using [Matlab R2022b], illustrating the comparisons between the exact solutions and numerical solutions, as well as the absolute error in each case is computed.


Solvability of a nonlinear integro-differential equation with fractional order using the Bernoulli matrix approach

In this study, a fractional nonlinear mixed integro-differential equation (Fr-NMIDE) is presented and has a general discontinuous kernel based on position and time space. Conditions of the existence and uniqueness of the solution is provided through the principal form of the integral equation, based on the Banach fixed point theorem. After applying the properties of a fractional integral, the Fr-NMIDE conformed to the Volterra–Hammerstein integral equation (V-HIE) of the second kind, with a general discontinuous kernel in position with the Hammerstein integral term and a continuous kernel in time to the Volterra term. Then, using a technique of the separating method, we obtained HIE, where its physical coefficients were variable in time. The Toeplitz matrix method (TMM) and its schemes were used to obtain a nonlinear algebraic system by studying the convergence of the system. The Maple 18 program was implemented to present the numerical results, along with corresponding errors.

https://www.mdpi.com/2504-3110/7/2/173/pdf?version=1675942838

A Physical Phenomenon for the Fractional Nonlinear Mixed Integro-Differential Equation Using a General Discontinuous Kernel

Simple Summary According to cancer research statistics, there are 7500 new ovarian cancer diagnoses in the UK each year. An earlier detection of ovarian cancer leads to a better prognosis; however, there is currently no screening programme for ovarian cancer, and detection using ultrasound examinations remains challenging. The use of artificial intelligence in gynaecological ultrasound examinations aims to improve the diagnostic accuracy of ultrasound for ovarian cancer and improve outcomes for patients. This review aims to collate current research on AI in the ultrasound diagnosis of ovarian cancer and suggests the usefulness of incorporating this into clinical care. Abstract Ovarian cancer is the sixth most common malignancy, with a 35% survival rate across all stages at 10 years. Ultrasound is widely used for ovarian tumour diagnosis, and accurate pre-operative diagnosis is essential for appropriate patient management. Artificial intelligence is an emerging field within gynaecology and has been shown to aid in the ultrasound diagnosis of ovarian cancers. For this study, Embase and MEDLINE databases were searched, and all original clinical studies that used artificial intelligence in ultrasound examinations for the diagnosis of ovarian malignancies were screened. Studies using histopathological findings as the standard were included. The diagnostic performance of each study was analysed, and all the diagnostic performances were pooled and assessed. The initial search identified 3726 papers, of which 63 were suitable for abstract screening. Fourteen studies that used artificial intelligence in ultrasound diagnoses of ovarian malignancies and had histopathological findings as a standard were included in the final analysis, each of which had different sample sizes and used different methods; these studies examined a combined total of 15,358 ultrasound images. The overall sensitivity was 81% (95% CI, 0.80–0.82), and specificity was 92% (95% CI, 0.92–0.93), indicating that artificial intelligence demonstrates good performance in ultrasound diagnoses of ovarian cancer. Further prospective work is required to further validate AI for its use in clinical practice.

Artificial Intelligence in Ultrasound Diagnoses of Ovarian Cancer: A Systematic Review and Meta-Analysis

In this work, the existence and uniqueness solution of the fractional nonlinear mixed integro-differential equation (FrNMIoDE) is guaranteed with a general discontinuous kernel based on position and time-space L2Ω×C0,T, T&lt;1. The FrNMIoDE conformed to the Volterra-Hammerstein integral equation (V-HIE) of the second kind, after applying the characteristics of a fractional integral, with a general discontinuous kernel in position for the Hammerstein integral term and a continuous kernel in time to the Volterra integral (VI) term. Then, using a separation technique methodology, we developed HIE, whose physical coefficients were time-variable. By examining the system’s convergence, the product Nystrom technique (PNT) and associated schemes were employed to create a nonlinear algebraic system (NAS). 

A Physical Phenomenon for the Fractional Nonlinear Mixed Integro-Differential Equation Using a Quadrature Nystrom Method

Fertility-sparing surgery in ovarian cancer is an increasing need in gynecology-oncology clinical practice because of the frequent childbearing delay in developed countries. As the evidence in literature is based on observational studies, this review focuses on summarizing the most recent and relevant evidence for the conservative management of young patients with ovarian cancer.Staging surgery is mandatory in epithelial ovarian tumors. In sex cord-stromal tumors, fertility-sparing surgery is only recommended in FIGO stage IA. Due to its good prognosis, conservative treatment is possible in some cases of advanced disease of germ cell tumors and borderline tumors. Cystectomy is a reasonable option in serous borderline tumors but is not recommended in other tumor subtypes. Successful conception rates after fertility-sparing surgery ranges from 30 to 60%.Fertility-sparing surgery is a well tolerated option in young women with early-stage ovarian cancer with acceptable reproductive outcomes. 

Current limits of conservative treatment in ovarian cancer

Abstract In this paper, the semi-group method is used to discuss the existence and uniqueness of solutions for fractional and partial integro differential equations (F-PIDEs) of heat type in Banach space E. In addition, the stability of the solutions for F-PIDEs are discussed. Moreover, the Adomian decomposition method (ADM) is used to obtain the solutions numerically. Finally, numerical results of each case are obtained, and the difference of results between the fractional partial integro differential equation and partial integro differential equation are explained. Furthermore, the error is computed in each case.

On the behaviour solutions of fractional and partial integro differential heat equations and its numerical solutions

Fertility Sparing Surgery (FSS) appears to be a safe means of treating early-stage ovarian cancer based on relatively limited evidence. However, there is currently insufficient evidence to aid women in counselling about their potential fertility outcomes. The aim of this study was to assess the reproductive outcomes and prognosis of women who have undergone FSS for ovarian malignancy. Between 1 June 2008 and 1 June 2018, a retrospective review of a clinical database was conducted to identify all consecutive patients who underwent FSS in a central London gynaecological oncology centre. All patients with a histological diagnosis of ovarian malignancy (excluding borderline ovarian tumours) were eligible. All identified patients were then prospectively called into a follow up and asked to complete a questionnaire about their fertility outcomes. A total of 47 women underwent FSS; 36 were included in this study. The mean age was 30.3 years (95% Confidence Interval [CI]: 27.6 to 33.0 years). During the study period, 17/36 (47.2%) of the women had attempted to conceive following surgery, with a successful live birth rate of 52.9% (9/17). The mean time of recurrence was 125.3 months (95% CI: 106.5–144.1 months). The mean time to death was 139.5 months (95% CI: 124.3–154.8). The cancer grade, tumour stage and use of Assisted Reproductive Technology (ART) were the main factors significantly associated with the risk of recurrence and death. In conclusion, this study suggests that a large proportion of women will not attempt to conceive following FSS. For those who do attempt to conceive, the likelihood of achieving a live birth is high. However, careful counselling about the higher risk of recurrence and worse survival for women with high grade cancer, disease Stage > IA and potentially those who undergo ART is essential before contemplating FFS.

/pdf/fertility-preserving-surgery-outcomes-for-ovarian-malignancy-1g00gw20.pdf

Fertility Preserving Surgery Outcomes for Ovarian Malignancy: Data from a Tertiary Cancer Centre in Central London

High grade endometrioid endometrial cancer (HGEEC) is a heterogeneous group of tumors with unclear prognostic features. The aim of the present study is to evaluate the independent risk factors for recurrence and mortality and to describe the recurrence patterns of HGEEC. Ninety-six consecutive cases of HGEEC treated with primary surgery in a single Tertiary Center were retrospectively reviewed. Clinicopathological and treatment details were recorded, and all patients were closely followed up. Disease-free, overall and cancer-specific survival rates were 83.8%, 77.8% and 83.6%, respectively. Cervical stromal involvement was independently related to recurrence (HR = 25.67; 95%CI 2.95–223.30; p = 0.003) and cancer-related death (HR = 15.39; 95%CI 1.29–183.43; p = 0.031) after adjusting for other pathological and treatment variables. Recurrence rate was 16%, with 60% of these cases having lung metastases and only one case with single vaginal vault recurrence. 81.81% of the recurrences presented with symptoms and not a single recurrence was diagnosed in routine follow-up clinical examination. In conclusion, the recurrence pattern may suggest that patient-initiated follow-up (PIFU) could be considered a potential alternative to clinical-based follow-up for HGEEC survivors, especially for patients without cervical involvement and after two years from treatment. Additional caution is needed in patients with cervical stromal involvement.

The Prognostic Characteristics and Recurrence Patterns of High Grade Endometrioid Endometrial Cancer: A Large Retrospective Analysis of a Tertiary Center

Kasr Alainy simplified uterine preserving surgery for conservative management of placenta accreta spectrum (PAS): A modified surgical approach.

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On the behaviour solutions of fractional and partial integro differential heat equations and its numerical solutions

Fertility Preserving Surgery Outcomes for Ovarian Malignancy: Data from a Tertiary Cancer Centre in Central London

The Prognostic Characteristics and Recurrence Patterns of High Grade Endometrioid Endometrial Cancer: A Large Retrospective Analysis of a Tertiary Center

Kasr Alainy simplified uterine preserving surgery for conservative management of placenta accreta spectrum (PAS): A modified surgical approach.