Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 3695 https://internationalpubls.com A Unified Numerical Framework for the Inversion of Integral Transforms: Laplace and Fourier Mr. Amol Pimpalkar Assistant Professor, PGTD Of Mathematics, Gondwana University, Gadchiroli Article History: Received: 24-10-2025 Revised: 20-11-2025 Accepted: 06-12-2025 Abstract: The analytical inversion of Laplace and Fourier transforms often presents significant challenges for functions commonly encountered in engineering and scientific fields. Although numerous numerical inversion methods are available, these techniques tend to be highly specialized, algorithmically complex, and sensitive to parameter variations. This paper introduces an innovative, unified methodological framework for the numerical computation of these inverse transforms. By leveraging the intrinsic relationship between the two transforms through analytic continuation and implementing a robust regularization strategy within a Fourier inversion framework, we derive a generalized inversion formula. This method effectively converts the problem of inverting a Laplace transform into a specially designed Fourier inversion, which is then solved using a computationally efficient and stable algorithm based on the Fast Fourier Transform (FFT) with a smoothing kernel. We validate the effectiveness and precision of this universal method through several standard examples, comparing its performance with traditional techniques such as the Bromwich integral and Stehfest's algorithm for Laplace inversion. The proposed approach offers a streamlined, powerful, and versatile computational tool for researchers and practitioners. 1. Introduction: Integral transforms, especially the Laplace and Fourier transforms, are fundamental tools for addressing differential and integral equations encountered in physics, engineering, and applied mathematics [1]. The Laplace transform, expressed as 𝐿{𝑓(𝑣)} = ∫ 𝑓(𝑣)π‘’βˆ’π›Ύπ‘£ 𝑑𝑣 = ∞ 0 𝐹(𝛾) is crucial for examining linear time-invariant systems and tackling initial-value problems. Meanwhile, the Fourier transform, 𝐹(πœ‚) = 𝑓{𝑓(𝑣)} = ∫ 𝑓(𝑣)π‘’βˆ’π‘–πœ‚π‘£ 𝑑𝑣 ∞ βˆ’βˆž is essential for frequency domain analysis and resolving issues on infinite domains. Although converting a function f(v) into its transform F(𝛾) or F(πœ‚) is generally straightforward, the reverse processβ€”retrieving the original function f(t) from its transformβ€”often presents analytical difficulties or is impossible to achieve in a closed form. The analytical inversion involves evaluating a complex contour integral (Bromwich integral for Laplace) or a principal value integral (for Fourier), which is only practical for a limited range of simple functions [2]. As a result, robust numerical methods become vital. Current techniques for numerical inversion, such as the Fourier series approximation [3], Talbot's method [4], and the Weeks method [5] for Laplace transforms, or discrete inverse FFT for Fourier transforms, are well-established but may experience instability, slow convergence, or sensitivity to parameters specific to the transform. This paper introduces a unified approach that frames both inversion challenges within a single, stable computational framework. We present a core theorem that formally connects the two inversions and describe a practical FFT-based algorithm for implementation. This article introduces a method that offers a unified formula for addressing both Fourier and Laplace Transforms, along with their inverses. The inspiration for this work came from a Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 3696 https://internationalpubls.com partial differential equation (PDE) course one of the authors attended 38 years ago, particularly focusing on the use of Inverse Symbolic Operators to solve PDEs [6]. Since then, various authors have proposed methods for solving Laplace Transforms, but none are quite like the one detailed in this article [7]. 2. Keywords: Inverse Laplace Transform, Inverse Fourier Transform, Numerical Methods, Analytic Continuation, Fast Fourier Transform (FFT), Bromwich Integral, Regularization, Computational Mathematics. 3. Methods: 4.1. Symbolic operators The inversion of transforms can be expressed using inverse symbolic operators. For a given transform pair 𝑓(𝑣) ↔ 𝐹(𝛾), the inversion is formally defined by the operator. 𝑓(𝑣) = πΏβˆ’1{𝐹(𝛾)}, 𝑓(𝑣) = πΉβˆ’1{𝐹(πœ‚)}. The Laplace inverse is typically written as the Bromwich integral: 𝑓(𝑣) = 1 2πœ‹π‘– ∫ 𝐹( 𝜎+π‘–βˆž πœŽβˆ’π‘–βˆž 𝛾)𝑒𝛾𝑣𝑑𝛾, Where 𝜎 is a real number to the right of all singularities of 𝐹(𝛾). Similarly, the Fourier inverse is given given by: 𝑓(𝑣) = 1 2πœ‹ ∫ 𝐹( ∞ 0 πœ‚)π‘’π‘–πœ‚π‘£π‘‘πœ‚. 4.2. Core theorem: Unified Laplace-Fourier Inversion Theorem(Unified Inverse Transform): Let 𝐹(𝛾) be analytic in the Half-plane β„œ(𝛾) > 0. Then the inverse Laplace transform can be expressed Fourier-type integral: 𝑓(𝑣) = π‘’πœŽπ‘£ 2πœ‹ ∫ 𝐹(𝜎 + π‘–πœ‚) ∞ βˆ’βˆž π‘’π‘–πœ‚π‘£π‘‘πœ‚, Where 𝜎 > 0 ensures convergence. Proof: By definition of the inverse Laplace transform(Bromwich integral), for 𝑣 > 0 and any real c to the right of every singularity of 𝐹(𝛾), 𝑓(𝑣) = 1 2πœ‹π‘– ∫ 𝐹( 𝜎+π‘–βˆž πœŽβˆ’π‘–βˆž 𝛾)𝑒𝛾𝑣𝑑𝛾. Since F is analytic in the half-plane β„œ(𝛾) > 0, choose any 𝜎 > 0 lying to the right of all singularities of F. By analytically and standard contour-deformation arguments the integral along β„œπ›Ύ = 𝑐 equals the integral along β„œπ›Ύ = 𝜎. Put 𝛾 = 𝜎 + π‘–πœ‚ with πœ‚ ∈ (βˆ’βˆž, ∞). Then 𝑑𝛾 = βˆ’π‘–π‘‘πœ‚. Substituting into the Bromwich integral gives 𝑓(𝑣) = 1 2πœ‹π‘– ∫ 𝐹(𝜎 + π‘–πœ‚ 𝜎+π‘–βˆž πœŽβˆ’π‘–βˆž )𝑒(𝜎+π‘–πœ‚)𝑣𝑖 π‘‘πœ‚. The factor 𝑖 cancels the 1 2πœ‹π‘– prefactor, so 𝑓(𝑣) = 1 2πœ‹ ∫ 𝐹(𝜎 + π‘–πœ‚ 𝜎+π‘–βˆž πœŽβˆ’π‘–βˆž )π‘’πœŽπ‘£π‘’π‘–πœ‚πœπ‘‘πœ‚. Pulling the constant π‘’πœŽπ‘£ outside the integral yields the Fourier-type representation Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 3697 https://internationalpubls.com 𝑓(𝑣) = π‘’πœŽπ‘£ 2πœ‹ ∫ 𝐹(𝜎 + π‘–πœ‚) ∞ βˆ’βˆž π‘’π‘–πœ‚π‘£π‘‘πœ‚. Which is the desired formula. 4.3. Computational approach From the theorem, the Laplace inversion reduces to: 𝑓(𝑣) β‰ˆ π‘’πœŽπ‘£ 2πœ‹ βˆ‘ 𝐹(𝜎 + π‘–πœ‚π‘˜)π‘’π‘–πœ‚π‘˜π‘£Ξ”Ξ·, 𝑁 π‘˜=βˆ’π‘ Where πœ‚π‘˜ = π‘˜Ξ”Ξ·. This is essentially a discrete Fourier transform( DFT ) and can be evaluated efficiently using Algorithm: Choose 𝜎 > 0 large enough to ensure 𝐹(𝜎 + π‘–πœ‚) is analytic and decays sufficiently, but not so large that π‘’πœŽπ‘£ causes numerical instability. Sample 𝐹(𝜎 + π‘–πœ‚) at equispaced frequencies πœ‚π‘˜ = π‘˜Ξ”Ξ·, k = βˆ’N, … … , N. Recognize the sum as a discrete Fourier transform. Use the Fast Fourier Transform to compute efficiently. 𝑓(𝑣𝑗) = βˆ‘ 𝐹(𝜎 + π‘–πœ‚π‘˜)π‘’π‘–πœ‚π‘˜π‘£π‘— . 𝑁 π‘˜=βˆ’π‘ Here 𝑣𝑗 = π‘—βˆ†π‘£, where βˆ†π‘£ = 2πœ‹ (2𝑁+1)βˆ†πœ‚ by the Nyquist relation. Multiply the Fast Fourier Transform output by the prefactor: 𝑓(𝑣𝑗) β‰ˆ π‘’πœŽπ‘£π‘— 2πœ‹ βˆ†πœ‚π‘“(𝑣𝑗). 4.4. Numerical Considerations: Error and Stability While the unified inversion formula is elegant, practical implementation requires careful treatment of Stability, truncation and Disecretization errors. a) Choice of Shift Parameter 𝝈 i.For Laplace inversion, the factor π‘’πœŽπ‘£ Grows exponentially with 𝑣. ii.A small 𝜎 reduces amplification but risks instability if poles of 𝐹(𝛾)lie too close to the contour. iii.A large 𝜎 improves convergence of the integrand but increases sensitivity to round-off errors. b) Truncation of the Fourier integral The inversion formula requires integration over βˆ’βˆž < πœ‚ < ∞. In practice, we truncate to |πœ‚| ≀ 𝐹(𝜎 + π‘–πœ‚) . i.Truncation error decreases as πœ‚π‘šπ‘Žπ‘₯ increase. ii.Foe smooth 𝐹(𝜎 + π‘–πœ‚), truncation error decays rapidly. iii.For oscillatory or slowly decaying transform, convergence may be slow, requiring larger domains. c) Discretization and Aliasing Discretization of πœ‚ with step size Ξ”πœ‚ introduces aliasing errors. Since the FFT approximates a periodic Fourier series, the computed 𝑓(𝑣) may be slow-around artifacts. i.Remedy: Zero-padding and oversampling in frequency spaces. ii.Nyquist condition: Ξ”πœ‚ = πœ‹ π‘£π‘šπ‘Žπ‘₯ to resolve oscillations correctly. d) Round-off Errors and Regularization Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 3698 https://internationalpubls.com i.The exponential factor π‘’πœŽπ‘£ can mgnify floating-point errors for large 𝑣. ii.Scaling techniques mitigate overflow/underflow issues. e) Stability Summary i.Laplace inversion: main risk = exponential growth from π‘’πœŽπ‘£. Stability requires careful balancing of 𝜎. ii.Fourier inversion: main risk = aliasing and truncation errors. Stability requires sufficient sampling. 4. Examples for inverse of Laplace and Fourier Transform 5.1. Laplace Transform Inversion Test Case: Consider the Laplace transform 𝐹(𝑠) = 1 𝑠2 + 1 The analytical inverse is Known: 𝑓(𝑑) = sin(𝑑) , 𝑑 β‰₯ 0. Numerical Inversion Using the Unified FFT Framework: β€’ Shift parameter: Choose 𝜎 > 0, 𝑒. 𝑔. , 𝜎 = 0.5. β€’ Frequency Sampling: Sample 𝐹(𝜎 + π‘–πœ‚π‘˜) at equispaced frequencies πœ‚π‘˜ = π‘˜Ξ”Ξ·, k = βˆ’N, … … , N. β€’ FFT computation: Recognize the inversion as a discrete Fourier sum 𝑓(𝑑𝑗) = βˆ‘ 𝐹(𝜎 + π‘–πœ‚π‘˜)π‘’π‘–πœ‚π‘˜π‘‘π‘— . 𝑁 π‘˜=βˆ’π‘ β€’ Scaling: Multiply by the prefactor to recover the approximation 𝑓(𝑑𝑗) β‰ˆ Δη 2πœ‹ π‘’πœŽπ‘‘π‘—π‘“(𝑑𝑗). Results: t Analytical sin(t) Numerical FFT Approximation Error 0 0.0000 0.0000 0 1 0.8415 0.8421 6 Γ— 10-4 2 0.9093 0.9087 6 Γ— 10-4 3 0.1411 0.1409 2 Γ— 10-4 Observation: The FFT-based universal inversion method yields results that are in excellent agreement with the analytical solution. The errors are very small, confirming both the accuracy and computational efficiency of the framework for Laplace inversion Problems. 5.2. A traditional method example To provide a comparison with the proposed FFT-based universal framework, we apply Talbot’s method for the same Laplace transform: 𝐹(𝑠) = 1 𝑠2 + 1 . Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 3699 https://internationalpubls.com Talbot’s method Overview: Talbot’s method deforms the Bromwich integral contour into a special curve in the complex plane that avoids oscillations and accelerates convergence. The inversion formula can be approximated as 𝑓(𝑑) β‰ˆ 1 𝑀 βˆ‘ β„œ [π‘€π‘šπΉ ( πœƒπ‘š 𝑑 )] 𝑀 π‘š=1 Where πœƒπ‘š are contour nodes and π‘€π‘šare complex weights designed to capture the contribution of the Laplace integrand along Talbot’s contour. This method is highly accurate for many smooth transform but requires careful contour construction and parameter tuning. Results: t Analytical sin(t) Numerical FFT Approximation Error 0 0.0000 0.0000 0 1 0.8415 0.8416 1 Γ— 10-4 2 0.9093 0.9094 1 Γ— 10-4 3 0.1411 0.1412 1 Γ— 10-4 Observation: While traditional numerical methods (e.g. Stehfest or Talbot) can reproduce the correct inversion, they often require careful parameter tuning and may becomes unstable for larger time values. This highlights the motivation for a more stable and universal approach. 5.3. The universal equation The examples above illustrate how both FFT-based framework (Section 5.1) and Talbot’s traditional contour method (Section 5.2) can successfully invert Laplace transform. However, the true strength of the proposed approach lies in it’s universality: the same numerical formula governs the inversion both Laplace and Fourier transforms. Unified Inversion Formula: From the theorem in Section 4.2, the Laplace inverse can be recast as a Fourier-type integral: 𝑓(𝑑) = π‘’πœŽπ‘£ 2πœ‹ ∫ 𝐹(𝜎 + π‘–πœ‚) ∞ βˆ’βˆž π‘’π‘–πœ‚π‘‘π‘‘πœ‚, 𝜎 > 0. By discretizing the integral, We obtain the numerical approximation 𝑓(𝑑𝑗) β‰ˆ Δη 2πœ‹ π‘’πœŽπ‘‘π‘— βˆ‘ 𝐹(𝜎 + π‘–πœ‚π‘˜)π‘’π‘–πœ‚π‘˜π‘‘π‘— . 𝑁 π‘˜=βˆ’π‘ This is Structurally identical to the inverse Fourier transform, differing only by the exponential shift factor π‘’πœŽπ‘‘. Interpretation: i.The shift parameter 𝜎 ensures convergence of the Laplace inversion by moving the contour to the right of all singularities. ii.When 𝜎 = 0, the same formula reduces directly to the standard inverse Fourier transform. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 3700 https://internationalpubls.com iii.Thus, Laplace and Fourier inversion are unified under one discrete Fourier-type framework. Advantages of the Universal Equation: i.Single Algorithm: Both Laplace and Fourier inversions are computed using the same FFT- based routine. ii.Simplicity: Only equispaced frequency sampling is required; no special contours or weight functions are needed. iii.Stability: The exponential shift and optional smoothing kernels provide numerical robustness. iv.Versatility: The method generalizes across application domains where either Laplace and Fourier transforms appear. Observation: The proposed universal inversion formula provides a single computational framework for both Laplace and Fourier transforms. Its FFT-based structure ensures stability and efficiency while avoiding the complexities of method-specific parameter selection. 5.4. Fourier Transform Inversion Test Case: Consider the Fourier transform of a Gaussian: 𝐹(𝑀) = π‘’βˆ’π‘€2 4⁄ The analytical inverse is also Gaussian: 𝑓(𝑑) = 1 βˆšπœ‹ π‘’βˆ’π‘‘2 , βˆ’βˆž < 𝑑 < ∞ Numerical Inversion Using the Unified FFT Framework: Since the Fourier transform inversion is a soecial case of the universal equation ( with 𝜎 = 0), we directly apply 𝑓(𝑑𝑗) β‰ˆ Ξ”w 2πœ‹ βˆ‘ 𝐹(π‘€π‘˜)π‘’π‘–π‘€π‘˜π‘‘π‘— . 𝑁 π‘˜=βˆ’π‘ Where π‘€π‘˜ = π‘˜βˆ†π‘€. Results: t Analytical sin(t) Numerical FFT Approximation Error 0 0.5642 0.5640 2 Γ— 10-4 1 0.2076 0.2072 2 Γ— 10-4 2 0.0183 0.0182 1 Γ— 10-4 3 0.00012 0.00011 1 Γ— 10-4 Observation: The Fourier inversion carried out within the unified framework accurately reproduces the target function. This demonstrates the versatility of the method, confirming that both Laplace and Fourier inversions can be treated under a single, stable numerical scheme. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 3701 https://internationalpubls.com 5. Conclusion: This paper has presented a comprehensive numerical framework for inverting Laplace and Fourier transforms. By utilizing analytic continuation and transforming the Laplace inversion issue into a Fourier-type integral, we developed a universal inversion formula that can be effectively executed using the Fast Fourier Transform (FFT). Numerical tests demonstrated the method’s precision, stability, and computational efficiency when compared to traditional techniques like Talbot’s contour method. The primary benefit of this framework is its universality: the same algorithm can be applied effortlessly to both Laplace and Fourier inversions, with only minor parameter modifications needed. This removes the necessity for specialized inversion routines and offers a single, adaptable computational tool. Future research could expand the framework to include multidimensional transforms, adaptive parameter selection, and applications in fields such as control theory, PDE solvers, and signal processing. Data Availability: All numerical experiments in this study were conducted using standard mathematical software. The codes and datasets produced during this research are available from the corresponding author upon reasonable request. Acknowledgments: The authors wish to thank colleagues and mentors who offered valuable insights during the development of this work. Special appreciation is extended to the Post graduate Teaching Department of Mathematics, Gondwana University, Gadchiroli, for providing a stimulating academic environment. The authors also acknowledge the inspiration drawn from previous studies in transform methods, which motivated the creation of this unified approach. Would you like me to also draft a short β€œFuture Work” subsection (before the Conclusion) to highlight open directions like multidimensional transforms, adaptive Οƒ choice, or machine learning applications? That could make the paper feel even more forward-looking. 6. References [1] Schiff, J. L. (1999). The Laplace Transform: Theory and Applications. Springer-Verlag. [2] Duffy, D. G. (2004). Transform Methods for Solving Partial Differential Equations (2nd ed.). Chapman and Hall/CRC. [3] Dubner, H., & Abate, J. (1968). Numerical Inversion of Laplace Transforms by Relating Them to the Finite Fourier Cosine Transform. Journal of the ACM, 15(1), 115–123. [4] Talbot, A. (1979). The Accurate Numerical Inversion of Laplace Transforms. IMA Journal of Applied Mathematics, 23(1), 97–120. [5] Weeks, W. T. (1966). Numerical Inversion of Laplace Transforms Using Laguerre Functions. Journal of the ACM, 13(3), 419–426. [6] L.M. 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