Pub Date : 2025-10-23DOI: 10.1016/j.rinam.2025.100659
Saman Bagherbana, Jafar Biazar, Hossein Aminikhah
We present a reliable numerical method for solving multidimensional partial Volterra integro-differential equations (PVIDEs). This comprehensive approach integrates techniques from product integration, the Nyström method, and spectral collocation, all founded on ultraspherical polynomials. The primary objective of our methodology is to employ variable and function transformations to reformulate the equations into a novel class of PVIDEs. Subsequently, the ultraspherical product integration-spectral collocation approach is applied to derive equivalent algebraic equations. Newton’s iterative method is then utilized to simultaneously compute the numerical solution and the first-order partial derivative. We rigorously analyze the error bounds of the proposed method in both - and -norms. Our results demonstrate that the errors in the numerical solution, as well as in the numerical first-order partial derivative, decay exponentially. Numerical examples are provided to validate reliability and efficiency of the ultraspherical product integration-spectral collocation approach.
{"title":"An ultraspherical product integration-spectral collocation method for multidimensional partial Volterra integro-differential equations and its convergence analysis","authors":"Saman Bagherbana, Jafar Biazar, Hossein Aminikhah","doi":"10.1016/j.rinam.2025.100659","DOIUrl":"10.1016/j.rinam.2025.100659","url":null,"abstract":"<div><div>We present a reliable numerical method for solving multidimensional partial Volterra integro-differential equations (PVIDEs). This comprehensive approach integrates techniques from product integration, the Nyström method, and spectral collocation, all founded on ultraspherical polynomials. The primary objective of our methodology is to employ variable and function transformations to reformulate the equations into a novel class of PVIDEs. Subsequently, the ultraspherical product integration-spectral collocation approach is applied to derive equivalent algebraic equations. Newton’s iterative method is then utilized to simultaneously compute the numerical solution and the first-order partial derivative. We rigorously analyze the error bounds of the proposed method in both <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>∞</mi></mrow></msup></math></span>- and <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-norms. Our results demonstrate that the errors in the numerical solution, as well as in the numerical first-order partial derivative, decay exponentially. Numerical examples are provided to validate reliability and efficiency of the ultraspherical product integration-spectral collocation approach.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"28 ","pages":"Article 100659"},"PeriodicalIF":1.3,"publicationDate":"2025-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145363082","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper presents the valuation of commodity options within the context of a Wishart stochastic volatility model that is equipped with a jump process. To achieve this, we propose a semi-analytical solution for pricing European options on commodity futures by introducing the characteristic function of the proposed model. The unique challenges posed by this model underscore the necessity for effective calibration techniques. To address this, we utilize an Artificial Neural Network (ANN) designed to improve the precision and efficiency of the calibration process. To optimize the presented ANN model, we use the flower pollination (FP) algorithm. Empirical studies suggest that the Wishart stochastic volatility model incorporating a jump factor enhances calibration accuracy compared to common models in the literature. Moreover, applying the FP-optimized ANN to calibration leads to a marked improvement in accuracy, as demonstrated by both in-sample and out-of-sample data.
{"title":"Commodity options pricing under Wishart stochastic volatility model equipped with jump process: Model calibration by an optimized neural network","authors":"Abdelouahed Hamdi , Maryam Noorani , Farshid Mehrdoust","doi":"10.1016/j.rinam.2025.100661","DOIUrl":"10.1016/j.rinam.2025.100661","url":null,"abstract":"<div><div>This paper presents the valuation of commodity options within the context of a Wishart stochastic volatility model that is equipped with a jump process. To achieve this, we propose a semi-analytical solution for pricing European options on commodity futures by introducing the characteristic function of the proposed model. The unique challenges posed by this model underscore the necessity for effective calibration techniques. To address this, we utilize an Artificial Neural Network (ANN) designed to improve the precision and efficiency of the calibration process. To optimize the presented ANN model, we use the flower pollination (FP) algorithm. Empirical studies suggest that the Wishart stochastic volatility model incorporating a jump factor enhances calibration accuracy compared to common models in the literature. Moreover, applying the FP-optimized ANN to calibration leads to a marked improvement in accuracy, as demonstrated by both in-sample and out-of-sample data.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"28 ","pages":"Article 100661"},"PeriodicalIF":1.3,"publicationDate":"2025-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145363609","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-10-22DOI: 10.1016/j.rinam.2025.100656
WenYan Tian , Hongen Jia
In this paper, we develop and analyze a fully discrete numerical scheme for the viscous Cahn–Hilliard equation. To enhance the stability of the scheme and allow for larger time steps, we introduce two artificial stabilization terms. We derive the scheme and rigorously prove, through error analysis, that it achieves second-order accuracy in both space and time. Numerical examples are provided to demonstrate the efficiency of the proposed method.
{"title":"Second-order finite element scheme of the viscous Cahn–Hilliard equation with energy-stable method","authors":"WenYan Tian , Hongen Jia","doi":"10.1016/j.rinam.2025.100656","DOIUrl":"10.1016/j.rinam.2025.100656","url":null,"abstract":"<div><div>In this paper, we develop and analyze a fully discrete numerical scheme for the viscous Cahn–Hilliard equation. To enhance the stability of the scheme and allow for larger time steps, we introduce two artificial stabilization terms. We derive the scheme and rigorously prove, through error analysis, that it achieves second-order accuracy in both space and time. Numerical examples are provided to demonstrate the efficiency of the proposed method.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"28 ","pages":"Article 100656"},"PeriodicalIF":1.3,"publicationDate":"2025-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145363079","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-10-16DOI: 10.1016/j.rinam.2025.100654
Guo Jiang, Jiayi Ying, Yuanqin Chen, Wen Sun
The paper proposes an efficient global approximation method via triangular functions (TFs) to solve stochastic Itô-Volterra integral equations (SIVIEs) driven by fractional Brownian motion (fBm) with Hurst parameter . By the relevant knowledge of TFs and fBm, new stochastic integral operator matrices with regard to fBm are derived, thus the nonlinear SIVIEs can be converted into nonlinear algebraic equations. Furthermore, the error analysis of the method was carried out to demonstrate its higher error order. Finally, the feasibility and effectiveness of the approach are verified by two numerical examples and an application in mathematical ecology.
{"title":"Numerical solution of nonlinear stochastic Itô-Volterra integral equations driven by fractional Brownian motion using triangular functions","authors":"Guo Jiang, Jiayi Ying, Yuanqin Chen, Wen Sun","doi":"10.1016/j.rinam.2025.100654","DOIUrl":"10.1016/j.rinam.2025.100654","url":null,"abstract":"<div><div>The paper proposes an efficient global approximation method via triangular functions (TFs) to solve stochastic Itô-Volterra integral equations (SIVIEs) driven by fractional Brownian motion (fBm) with Hurst parameter <span><math><mrow><mi>H</mi><mo>∈</mo><mrow><mo>(</mo><mrow><mrow><mo>(</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo>)</mo></mrow><mo>,</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></math></span>. By the relevant knowledge of TFs and fBm, new stochastic integral operator matrices with regard to fBm are derived, thus the nonlinear SIVIEs can be converted into nonlinear algebraic equations. Furthermore, the error analysis of the method was carried out to demonstrate its higher error order. Finally, the feasibility and effectiveness of the approach are verified by two numerical examples and an application in mathematical ecology.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"28 ","pages":"Article 100654"},"PeriodicalIF":1.3,"publicationDate":"2025-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145321350","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-10-15DOI: 10.1016/j.rinam.2025.100653
Baoyan Han , Bo Zhu
In this paper, we consider a class of mixed type Hilfer fractional differential equations with noninstantaneous impulses, nonlocal conditions and time delay. We discuss the existence results, Ulam–Hyers stability, generalized Ulam–Hyers stability and Ulam–Hyers–Rassias stability via Sadovskii’s fixed point theorem, fractional calculus and theory of operators.
{"title":"Existence and stability of mixed type Hilfer fractional differential equations with impulses and time delay","authors":"Baoyan Han , Bo Zhu","doi":"10.1016/j.rinam.2025.100653","DOIUrl":"10.1016/j.rinam.2025.100653","url":null,"abstract":"<div><div>In this paper, we consider a class of mixed type Hilfer fractional differential equations with noninstantaneous impulses, nonlocal conditions and time delay. We discuss the existence results, Ulam–Hyers stability, generalized Ulam–Hyers stability and Ulam–Hyers–Rassias stability via Sadovskii’s fixed point theorem, fractional calculus and theory of operators.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"28 ","pages":"Article 100653"},"PeriodicalIF":1.3,"publicationDate":"2025-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145321349","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-10-06DOI: 10.1016/j.rinam.2025.100642
Jeff Kershaw, Takayuki Obata
The zeros of cross-product combinations of the Bessel functions are often required as the eigenvalues in boundary-value problems with annular or tubular symmetry. Numerical methods to calculate the roots when the order is real have existed for many years, but those methods are unreliable when the order is complex, probably because the distribution of the zeros in the complex plane is unknown. In this work asymptotic expansions for the Bessel cross-product functions are constructed and used to investigate the root distribution in the complex plane for complex order. When the argument of the order is positive (or negative) the zeros are symmetrically positioned in the first & third (or second & fourth) quadrants of the complex plane. Within a particular quadrant, it is shown that the roots lie on or near to three lines that intersect at a single point. Two of these lines have a finite number of zeros associated with them, while the third line always has an infinite number of roots distributed along it. Approximations to the roots are constructed and used as initial values in an algorithm that more closely estimates the zeros. A longstanding issue with regard to the “exceptional” nature of the lowest root of one of the Bessel cross-product functions when the order is real has also been resolved.
{"title":"Computing the zeros of cross-product combinations of the Bessel functions with complex order","authors":"Jeff Kershaw, Takayuki Obata","doi":"10.1016/j.rinam.2025.100642","DOIUrl":"10.1016/j.rinam.2025.100642","url":null,"abstract":"<div><div>The zeros of cross-product combinations of the Bessel functions are often required as the eigenvalues in boundary-value problems with annular or tubular symmetry. Numerical methods to calculate the roots when the order is real have existed for many years, but those methods are unreliable when the order is complex, probably because the distribution of the zeros in the complex plane is unknown. In this work asymptotic expansions for the Bessel cross-product functions are constructed and used to investigate the root distribution in the complex plane for complex order. When the argument of the order is positive (or negative) the zeros are symmetrically positioned in the first & third (or second & fourth) quadrants of the complex plane. Within a particular quadrant, it is shown that the roots lie on or near to three lines that intersect at a single point. Two of these lines have a finite number of zeros associated with them, while the third line always has an infinite number of roots distributed along it. Approximations to the roots are constructed and used as initial values in an algorithm that more closely estimates the zeros. A longstanding issue with regard to the “exceptional” nature of the lowest root of one of the Bessel cross-product functions when the order is real has also been resolved.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"28 ","pages":"Article 100642"},"PeriodicalIF":1.3,"publicationDate":"2025-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145269624","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-10-06DOI: 10.1016/j.rinam.2025.100652
Targyn A. Nauryz , Stanislav N. Kharin
This paper presents a mathematical model and analytical study of the thermal dynamics in an electrical contact bridge under the influence of the Thomson effect and Joule heat generation. The model considers a bridge structure adjacent to a vapor zone, in which temperature evolution is governed by a nonlinear heat equation, featuring temperature-dependent thermal and thermoelectric coefficients, as well as an internal Joule heat source. The analysis introduces dimensionless variables and employs a self-similar transformation to reduce the problem to a boundary value problem for nonlinear ordinary differential and integral equations. The existence and uniqueness of the similarity solution are established via fixed point theory under appropriate conditions on the nonlinear coefficients. Analytical results are obtained for the case of constant coefficients, while the general nonlinear case is treated with an integral approach. Additionally, special cases such as linearly temperature-dependent Thomson and thermal coefficients are examined to illustrate parameter sensitivity. The results describe how variations in the Thomson effect, Joule heating, and material properties influence the temperature field, bridge opening, and boiling front propagation, providing a deeper understanding of coupled thermoelectric and phase-change processes in electrical contacts. The findings provide a rigorous mathematical basis for simulating temperature fields in electrical contacts with moving boundaries and for understanding the influence of thermoelectric effects in current-carrying devices.
{"title":"Mathematical modeling and analysis of thermal dynamics in an electrical contact bridge with nonlinear Stefan problem including thermoelectric effect and internal heat source","authors":"Targyn A. Nauryz , Stanislav N. Kharin","doi":"10.1016/j.rinam.2025.100652","DOIUrl":"10.1016/j.rinam.2025.100652","url":null,"abstract":"<div><div>This paper presents a mathematical model and analytical study of the thermal dynamics in an electrical contact bridge under the influence of the Thomson effect and Joule heat generation. The model considers a bridge structure adjacent to a vapor zone, in which temperature evolution is governed by a nonlinear heat equation, featuring temperature-dependent thermal and thermoelectric coefficients, as well as an internal Joule heat source. The analysis introduces dimensionless variables and employs a self-similar transformation to reduce the problem to a boundary value problem for nonlinear ordinary differential and integral equations. The existence and uniqueness of the similarity solution are established via fixed point theory under appropriate conditions on the nonlinear coefficients. Analytical results are obtained for the case of constant coefficients, while the general nonlinear case is treated with an integral approach. Additionally, special cases such as linearly temperature-dependent Thomson and thermal coefficients are examined to illustrate parameter sensitivity. The results describe how variations in the Thomson effect, Joule heating, and material properties influence the temperature field, bridge opening, and boiling front propagation, providing a deeper understanding of coupled thermoelectric and phase-change processes in electrical contacts. The findings provide a rigorous mathematical basis for simulating temperature fields in electrical contacts with moving boundaries and for understanding the influence of thermoelectric effects in current-carrying devices.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"28 ","pages":"Article 100652"},"PeriodicalIF":1.3,"publicationDate":"2025-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145269622","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-10-06DOI: 10.1016/j.rinam.2025.100646
Abdulkarim Hassan Ibrahim , Kanikar Muangchoo
This article proposes a projection-based method that combines the hyperplane technique, a restart strategy, and a modified Polak–Ribière–Polyak conjugate gradient method to solve large-scale systems of nonlinear equations. The method ensures that the search direction satisfies the sufficient descent condition at each iteration and retains a key property of the classical PRP approach. Global convergence is established under the assumptions of monotonicity and Lipschitz continuity. Numerical experiments are conducted to demonstrate the effectiveness and robustness of the proposed approach, with comparisons made against recent methods from the literature.
{"title":"A Polak–Ribière–Polyak like method with restart technique for monotone nonlinear equations","authors":"Abdulkarim Hassan Ibrahim , Kanikar Muangchoo","doi":"10.1016/j.rinam.2025.100646","DOIUrl":"10.1016/j.rinam.2025.100646","url":null,"abstract":"<div><div>This article proposes a projection-based method that combines the hyperplane technique, a restart strategy, and a modified Polak–Ribière–Polyak conjugate gradient method to solve large-scale systems of nonlinear equations. The method ensures that the search direction satisfies the sufficient descent condition at each iteration and retains a key property of the classical PRP approach. Global convergence is established under the assumptions of monotonicity and Lipschitz continuity. Numerical experiments are conducted to demonstrate the effectiveness and robustness of the proposed approach, with comparisons made against recent methods from the literature.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"28 ","pages":"Article 100646"},"PeriodicalIF":1.3,"publicationDate":"2025-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145269625","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-10-06DOI: 10.1016/j.rinam.2025.100651
Rachid Ait-Haddou, Safiya Alshehaiween
This paper introduces a novel framework for constructing stable, structure-preserving, linearly implicit schemes to dissipative systems with quartic polynomial potentials. Conventional approaches, such as those relying on the discrete gradient method, often depend on ad-hoc linearization techniques and tend to yield unstable numerical schemes. In contrast, the proposed method offers a systematic approach to linearization through the application of polar forms of the potentials and the Ito-Abe discrete gradient method. By embedding the original potential into a higher-dimensional space via its polar form, the approach enables regularized linearizations that ensure the stability of the resulting schemes. The paper explores the connection between the stability of the obtained schemes and the properties of the transformed potentials, such as their real Waring rank and coercivity. Moreover, explicit conditions on the regularization parameter required to maintain stability are derived.
{"title":"Stabilization of linearly implicit schemes for dissipative systems with quartic potentials","authors":"Rachid Ait-Haddou, Safiya Alshehaiween","doi":"10.1016/j.rinam.2025.100651","DOIUrl":"10.1016/j.rinam.2025.100651","url":null,"abstract":"<div><div>This paper introduces a novel framework for constructing stable, structure-preserving, linearly implicit schemes to dissipative systems with quartic polynomial potentials. Conventional approaches, such as those relying on the discrete gradient method, often depend on ad-hoc linearization techniques and tend to yield unstable numerical schemes. In contrast, the proposed method offers a systematic approach to linearization through the application of polar forms of the potentials and the Ito-Abe discrete gradient method. By embedding the original potential into a higher-dimensional space via its polar form, the approach enables regularized linearizations that ensure the stability of the resulting schemes. The paper explores the connection between the stability of the obtained schemes and the properties of the transformed potentials, such as their real Waring rank and coercivity. Moreover, explicit conditions on the regularization parameter required to maintain stability are derived.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"28 ","pages":"Article 100651"},"PeriodicalIF":1.3,"publicationDate":"2025-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145269623","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Walsh functions form a piecewise-constant orthonormal basis that is particularly well-suited for digital computation and signal approximation. Nevertheless, the direct evaluation of Walsh transforms for discrete functions becomes computationally prohibitive as the resolution increases. To overcome this difficulty, we develop an efficient numerical scheme based on the Fast Walsh–Hadamard–Fourier Transform (FWHFT) for the approximation of solutions to Volterra integral equations. The proposed method reduces the computational complexity from to thereby rendering the approach scalable to high-resolution problems. We present a complete algorithmic framework that exploits this fast transform and analyze its performance on a variety of examples. In particular, we illustrate several examples for the broad applicability of the method. These examples highlight the principal advantages of the FWHFT approach, as well as certain limitations inherent in the transform structure. As a further application, we implement the method in a financial setting by addressing the problem of pricing European options under the Bachelier model. This example demonstrates not only the accuracy of the proposed algorithm but also its practical relevance to computational finance, especially in scenarios involving structured payoff functions. Numerical experiments confirm the expected convergence behavior and the substantial computational savings afforded by the method. Finally, we discuss possible extensions of the approach to fractional-order models, which are naturally linked to Volterra-type integral equations and arise frequently in applications.
{"title":"Remarks on numerical approximation of Volterra integral equations by Walsh–Hadamard transform","authors":"Farrukh Mukhamedov , Ushangi Goginava , Akaki Goginava , James Wheeldon","doi":"10.1016/j.rinam.2025.100648","DOIUrl":"10.1016/j.rinam.2025.100648","url":null,"abstract":"<div><div>Walsh functions form a piecewise-constant orthonormal basis that is particularly well-suited for digital computation and signal approximation. Nevertheless, the direct evaluation of Walsh transforms for discrete functions becomes computationally prohibitive as the resolution increases. To overcome this difficulty, we develop an efficient numerical scheme based on the <em>Fast Walsh–Hadamard–Fourier Transform</em> (FWHFT) for the approximation of solutions to Volterra integral equations. The proposed method reduces the computational complexity from <span><math><mrow><mi>O</mi><mrow><mo>(</mo><msup><mrow><mn>2</mn></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span> to <span><math><mrow><mi>O</mi><mrow><mo>(</mo><mi>n</mi><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></mrow><mo>,</mo></mrow></math></span> thereby rendering the approach scalable to high-resolution problems. We present a complete algorithmic framework that exploits this fast transform and analyze its performance on a variety of examples. In particular, we illustrate several examples for the broad applicability of the method. These examples highlight the principal advantages of the FWHFT approach, as well as certain limitations inherent in the transform structure. As a further application, we implement the method in a financial setting by addressing the problem of pricing European options under the Bachelier model. This example demonstrates not only the accuracy of the proposed algorithm but also its practical relevance to computational finance, especially in scenarios involving structured payoff functions. Numerical experiments confirm the expected convergence behavior and the substantial computational savings afforded by the method. Finally, we discuss possible extensions of the approach to fractional-order models, which are naturally linked to Volterra-type integral equations and arise frequently in applications.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"28 ","pages":"Article 100648"},"PeriodicalIF":1.3,"publicationDate":"2025-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145222750","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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